From 4f4f05c0cf8b9eb1c907d2a7a7c63030d302b0d0 Mon Sep 17 00:00:00 2001 From: Guyot Date: Sat, 14 Dec 2019 20:36:40 +0100 Subject: [PATCH] Commit principal. --- Images/koch1.eps | 122 + Images/koch2.eps | 317 + Images/koch3.eps | 953 + Images/kochIsland1.eps | 437 + Images/kochIsland2.eps | 869 + Images/kochIslandquadratic2.eps | 1097 + Images/kochrecursif1.eps | 965 + Images/plante1.eps | 1100 + Images/plante2.eps | 863 + Images/plante3.eps | 2429 +++ Images/plante4.eps | 1175 ++ Images/plante5.eps | 1160 ++ Images/plante6.eps | 1757 ++ Images/production.eps | 688 + Images/production.svg | 384 + Lsystem.aux | 126 + Lsystem.bbl | 44 + Lsystem.blg | 45 + Lsystem.brf | 0 Lsystem.dvi | Bin 0 -> 302960 bytes Lsystem.ent | 3 + Lsystem.idx | 0 Lsystem.log | 1397 ++ Lsystem.out | 11 + Lsystem.pdf | Bin 0 -> 365453 bytes Lsystem.ps | 32858 ++++++++++++++++++++++++++++++ Lsystem.tex | 1285 ++ Lsystem.tex.backup | 1285 ++ Lsystem.tex~ | 1285 ++ Lsystem.toc | 34 + Preambule.tex | 91 + Preambule.tex.backup | 90 + Preambule.tex~ | 91 + Python/Arbre1.py | 65 + Python/Arbre2.py | 65 + Python/ArbreFonctionnel.py | 93 + Python/ArbreFonctionnel2.py | 132 + Python/ArbreFonctionnel3.py | 192 + Python/ArbreRecursif1.py | 49 + Python/ArbreRecursif2.py | 45 + Python/ArbreRecursif3.py | 45 + Solutions.tex | 98 + algorithm2e.sty | 2636 +++ algorithme.sty | 661 + algorithmique.sty | 248 + tmp.mf | 31 + tmp.mp | 50 + tmp0.mp | 48 + 48 files changed, 57419 insertions(+) create mode 100644 Images/koch1.eps create mode 100644 Images/koch2.eps create mode 100644 Images/koch3.eps create mode 100644 Images/kochIsland1.eps create mode 100644 Images/kochIsland2.eps create mode 100644 Images/kochIslandquadratic2.eps create mode 100644 Images/kochrecursif1.eps create mode 100644 Images/plante1.eps create mode 100644 Images/plante2.eps create mode 100644 Images/plante3.eps create mode 100644 Images/plante4.eps create mode 100644 Images/plante5.eps create mode 100644 Images/plante6.eps create mode 100644 Images/production.eps create mode 100644 Images/production.svg create mode 100644 Lsystem.aux create mode 100644 Lsystem.bbl create mode 100644 Lsystem.blg create mode 100644 Lsystem.brf create mode 100644 Lsystem.dvi create mode 100644 Lsystem.ent create mode 100644 Lsystem.idx create mode 100644 Lsystem.log create mode 100644 Lsystem.out create mode 100644 Lsystem.pdf create mode 100644 Lsystem.ps create mode 100644 Lsystem.tex create mode 100644 Lsystem.tex.backup create mode 100644 Lsystem.tex~ create mode 100644 Lsystem.toc create mode 100644 Preambule.tex create mode 100644 Preambule.tex.backup create mode 100644 Preambule.tex~ create mode 100644 Python/Arbre1.py create mode 100644 Python/Arbre2.py create mode 100644 Python/ArbreFonctionnel.py create mode 100644 Python/ArbreFonctionnel2.py create mode 100644 Python/ArbreFonctionnel3.py create mode 100644 Python/ArbreRecursif1.py create mode 100644 Python/ArbreRecursif2.py create mode 100644 Python/ArbreRecursif3.py 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+1,44 @@ +\begin{thebibliography}{} +\expandafter\ifx\csname fonteauteurs\endcsname\relax +\def\fonteauteurs{\scshape}\fi +\expandafter\ifx\csname url\endcsname\relax + \def\url#1{{\tt #1}}% + \message{You should include the url package}\fi + +\bibitem[Buchs, 2011]{BD11} +\bgroup\fonteauteurs\bgroup Buchs\egroup\egroup{}, D. (2011). +\newblock Langages formels. + +\bibitem[Desgraupes, 2008]{DB08} +\bgroup\fonteauteurs\bgroup Desgraupes\egroup\egroup{}, B. (2008). +\newblock {\em Expressions r{\'e}guli{\`e}res.} +\newblock Pearson. +\newblock Simple, clair et efficace. Mais peu théorique. + +\bibitem[Gastin, 2009]{GP09} +\bgroup\fonteauteurs\bgroup Gastin\egroup\egroup{}, P. (2009). +\newblock Langages formels. + +\bibitem[Hammel, 1994]{PP94} +\bgroup\fonteauteurs\bgroup Hammel\egroup\egroup{}, P. P.~M. (1994). +\newblock {\em Language-Restricted Iterated Function Systems, Koch + Constructions, and L-systems}. +\newblock University of Calgary SIGGRAPH ’94 Course Notes. ACM Press. + +\bibitem[Prusinkiewicz, 2004]{LA04} +\bgroup\fonteauteurs\bgroup Prusinkiewicz\egroup\egroup{}, A. L.~P. (2004). +\newblock {\em The Algorithmic Beauty of Plants.} +\newblock Springer-Verlag. +\newblock Ouvrage référence publié sur + \url{http://algorithmicbotany.org/papers/}. + +\bibitem[Renault, 2006]{CJ06} +\bgroup\fonteauteurs\bgroup Renault\egroup\egroup{}, J. C.~F. (2006). +\newblock {\em Interpr{\'e}tation graphique d’une grammaire L-System}. +\newblock Ecole Polytechnique de l’Universite de Tours. + +\bibitem[Togni, 2007]{TO07} +\bgroup\fonteauteurs\bgroup Togni\egroup\egroup{}, O. 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0 0 596 842 +%%DocumentFonts: CMR10 CMMI8 CMR8 CMSY10 TeX-feymr10 MSAM10 CMMI10 CMSY8 +%%+ MSBM10 +%%DocumentPaperSizes: a4 +%%EndComments +%DVIPSWebPage: (www.radicaleye.com) +%DVIPSCommandLine: /usr/bin/dvips -o Lsystem.ps Lsystem.dvi +%DVIPSParameters: dpi=600 +%DVIPSSource: TeX output 2019.12.14:2033 +%%BeginProcSet: tex.pro 0 0 +%! +/TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S +N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 +mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 +0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ +landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize +mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ +matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round +exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ +statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] +N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin +/FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array +/BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 +array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N +df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A +definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get +}B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} +B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr +1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S +/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy +setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask +restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn +/BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put +}if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ +bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A +mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ +SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ +userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X +1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 +index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N +/dir 0 def/dyy{/dir 0 def}B/dyt{/dir 1 def}B/dty{/dir 2 def}B/dtt{/dir 3 +def}B/p{dir 2 eq{-90 rotate show 90 rotate}{dir 3 eq{-90 rotate show 90 +rotate}{show}ifelse}ifelse}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 +N/Ry 0 N/V{}B/RV/v{/Ry X/Rx X V}B statusdict begin/product where{pop +false[(Display)(NeXT)(LaserWriter 16/600)]{A length product length le{A +length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse} +forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{ +BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat +{BDot}imagemask grestore}}ifelse B/QV{gsave newpath transform round exch +round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 +rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B +/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M} +B/g{0 M}B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p +-3 w}B/n{p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{ +0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end + +%%EndProcSet +%%BeginProcSet: psfrag.pro 0 0 +%% +%% This is file `psfrag.pro', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% psfrag.dtx (with options: `filepro') +%% +%% Copyright (c) 1996 Craig Barratt, Michael C. Grant, and David Carlisle. +%% All rights reserved. +%% +%% This file is part of the PSfrag package. +%% +userdict begin +/PSfragLib 90 dict def +/PSfragDict 6 dict def +/PSfrag { PSfragLib begin load exec end } bind def +end +PSfragLib begin +/RO /readonly load def +/CP /currentpoint load def +/CM /currentmatrix load def +/B { bind RO def } bind def +/X { exch def } B +/MD { { X } forall } B +/OE { end exec PSfragLib begin } B +/S false def +/tstr 8 string def +/islev2 { languagelevel } stopped { false } { 2 ge } ifelse def +[ /sM /tM /srcM /dstM /dM /idM /srcFM /dstFM ] { matrix def } forall +sM currentmatrix RO pop +dM defaultmatrix RO idM invertmatrix RO pop +srcFM identmatrix pop +/Hide { gsave { CP } stopped not newpath clip { moveto } if } B +/Unhide { { CP } stopped not grestore { moveto } if } B +/setrepl islev2 {{ /glob currentglobal def true setglobal array astore + globaldict exch /PSfrags exch put glob setglobal }} + {{ array astore /PSfrags X }} ifelse B +/getrepl islev2 {{ globaldict /PSfrags get aload length }} + {{ PSfrags aload length }} ifelse B +/convert { + /src X src length string + /c 0 def src length { + dup c src c get dup 32 lt { pop 32 } if put /c c 1 add def + } repeat +} B +/Begin { + /saver save def + srcFM exch 3 exch put + 0 ne /debugMode X 0 setrepl + dup /S exch dict def { S 3 1 roll exch convert exch put } repeat + srcM CM dup invertmatrix pop + mark { currentdict { end } stopped { pop exit } if } loop + PSfragDict counttomark { begin } repeat pop +} B +/End { + mark { currentdict end dup PSfragDict eq { pop exit } if } loop + counttomark { begin } repeat pop + getrepl saver restore + 7 idiv dup /S exch dict def { + 6 array astore /mtrx X tstr cvs /K X + S K [ S K known { S K get aload pop } if mtrx ] put + } repeat +} B +/Place { + tstr cvs /K X + S K known { + bind /proc X tM CM pop + CP /cY X /cX X + 0 0 transform idtransform neg /aY X neg /aX X + S K get dup length /maxiter X + /iter 1 def { + iter maxiter ne { /saver save def } if + tM setmatrix aX aY translate + [ exch aload pop idtransform ] concat + cX neg cY neg translate cX cY moveto + /proc load OE + iter maxiter ne { saver restore /iter iter 1 add def } if + } forall + /noXY { CP /cY X /cX X } stopped def + tM setmatrix noXY { newpath } { cX cY moveto } ifelse + } { + Hide OE Unhide + } ifelse +} B +/normalize { + 2 index dup mul 2 index dup mul add sqrt div + dup 4 -1 roll exch mul 3 1 roll mul +} B +/replace { + aload pop MD + CP /bY X /lX X gsave sM setmatrix + str stringwidth abs exch abs add dup 0 eq + { pop } { 360 exch div dup scale } ifelse + lX neg bY neg translate newpath lX bY moveto + str { /ch X ( ) dup 0 ch put false charpath ch Kproc } forall + flattenpath pathbbox [ /uY /uX /lY /lX ] MD + CP grestore moveto + currentfont /FontMatrix get dstFM copy dup + 0 get 0 lt { uX lX /uX X /lX X } if + 3 get 0 lt { uY lY /uY X /lY X } if + /cX uX lX add 0.5 mul def + /cY uY lY add 0.5 mul def + debugMode { gsave 0 setgray 1 setlinewidth + lX lY moveto lX uY lineto uX uY lineto uX lY lineto closepath + lX bY moveto uX bY lineto lX cY moveto uX cY lineto + cX lY moveto cX uY lineto stroke + grestore } if + dstFM dup invertmatrix dstM CM srcM + 2 { dstM concatmatrix } repeat pop + getrepl /temp X + S str convert get { + aload pop [ /rot /scl /loc /K ] MD + /aX cX def /aY cY def + loc { + dup 66 eq { /aY bY def } { % B + dup 98 eq { /aY lY def } { % b + dup 108 eq { /aX lX def } { % l + dup 114 eq { /aX uX def } { % r + dup 116 eq { /aY uY def } % t + if } ifelse } ifelse } ifelse } ifelse pop + } forall + K srcFM rot tM rotate dstM + 2 { tM concatmatrix } repeat aload pop pop pop + 2 { scl normalize 4 2 roll } repeat + aX aY transform + /temp temp 7 add def + } forall + temp setrepl +} B +/Rif { + S 3 index convert known { pop replace } { exch pop OE } ifelse +} B +/XA { bind [ /Kproc /str } B /XC { ] 2 array astore def } B +/xs { pop } XA XC +/xks { /kern load OE } XA /kern XC +/xas { pop ax ay rmoveto } XA /ay /ax XC +/xws { c eq { cx cy rmoveto } if } XA /c /cy /cx XC +/xaws { ax ay rmoveto c eq { cx cy rmoveto } if } + XA /ay /ax /c /cy /cx XC +/raws { xaws { awidthshow } Rif } B +/rws { xws { widthshow } Rif } B +/rks { xks { kshow } Rif } B +/ras { xas { ashow } Rif } B +/rs { xs { show } Rif } B +/rrs { getrepl dup 2 add -1 roll //restore exec setrepl } B +PSfragDict begin +islev2 not { /restore { /rrs PSfrag } B } if +/show { /rs PSfrag } B +/kshow { /rks PSfrag } B +/ashow { /ras PSfrag } B +/widthshow { /rws PSfrag } B +/awidthshow { /raws PSfrag } B +end PSfragDict RO pop +end + +%%EndProcSet +%%BeginProcSet: pstricks.pro 0 0 +% $Id: pstricks.pro 872 2018-12-21 20:39:31Z herbert $ +% +%% PostScript prologue for pstricks.tex. +%% Version 1.30, 2018/12/17 +%% +%% This program can be redistributed and/or modified under the terms +%% of the LaTeX Project Public License Distributed from CTAN archives +%% in directory macros/latex/base/lppl.txt. +% +% +% Define the follwing gs-functions if not known, eg when using distiller +% +systemdict /.setopacityalpha known not {/.setopacityalpha { pop } def } if +systemdict /.setblendmode known not {/.setblendmode { pop } def } if +systemdict /.setshapealpha known not {/.setshapealpha { pop } def } if +% +/tx@Dict 200 dict def % the main PSTricks dictionary +tx@Dict begin +/ADict 25 dict def % The arrow dictionary +/CM { matrix currentmatrix } bind def +/SLW /setlinewidth load def +/CLW /currentlinewidth load def +/CP /currentpoint load def +/ED { exch def } bind def +/L /lineto load def +/T /translate load def +/TMatrix { } def +/RAngle { 0 } def +/Sqrt { dup 0 lt { pop 0 } { sqrt } ifelse } def % return 0 for negative arguments +/Atan { /atan load stopped { pop pop 0 } if } def % return 0 if atan not known +/ATAN1 {neg -1 atan 180 sub } def % atan(x) (only one parameter) +/Div { dup 0 eq { pop } { div } ifelse } def % control the division +/Log { dup 1e-20 lt { pop -1e30 }{ log } ifelse } def % control the log +/tan { dup cos abs 1.e-10 lt + { pop 1.e10 } % return 1.e10 as infinit + { dup sin exch cos div } ifelse % default sin/cos +} def +/Tan { dup sin exch cos Div } def % sin(x)/cos(x) x in degrees +/Acos {dup dup mul neg 1 add dup 0 lt { % arc cos, returns 0 when negative root + pop pop 0 }{ sqrt exch atan} ifelse } def +/Acos2 { 2 dict begin + /x ED /y ED + y abs 1.0e-20 lt { 1.0e30 } if + x y div + dup dup mul neg 1 add dup 0 lt { % arc cos needs two values x,y + pop pop 0 }{ sqrt exch atan} ifelse + y 0 lt { 180 add } if +} def +/Power { % a^b latest ghostscript don't allow -4^-3.1 + 2 dict begin % hold all local + /b ED + /a ED + a 0 lt % prevent something like (-4)^(-3.1)=> 1/(-4)^3 + { b 0 lt + { a b cvi exp } + { a b exp } ifelse + } + { a 0 eq { 0 }{ a b exp } ifelse + } ifelse + end +} def +% +/NET { neg exch neg exch T } def % change coordinate system to the negative one +/Pyth { dup mul exch dup mul add sqrt } def % Pythagoras, expects 2 parameter +/Pyth2 { % Pythagoras, xA yA xB yB + 3 -1 roll % xA xB yB yA + sub % xA xB yB-yA + 3 1 roll % yB-yA xA xB + sub % yB-yA xA-xB + Pyth } def +/PtoC { 2 copy cos mul 3 1 roll sin mul } def % Polar to Cartesian (origimal) +/PtoCrel { pst@angleunit PtoC } def % Polar to Cartesian with \degrees[??] +/PtoCab { dup cos 4 -1 roll mul 3 1 roll sin mul } def % Polar to Cartesian (Ellipse) a b phi-> x y +/AnytoDeg { pst@angleunit } def +/DegtoAny { 1 pst@angleunit div} def +/AnytoRad { AnytoDeg DegtoRad } def +/RadtoAny { RadtoDeg DegtoAny } def +% +%/Rand { rand 4294967295 div } def % a real random number +/Rand { rand 2147483447 div } def % a real random number between 0 and 1 +%----------------- hv added 20050516 --------------- +/PiDiv2 1.57079632680 def +/Pi 3.14159265359 def +/TwoPi 6.28318530718 def +/Euler 2.71828182846 def +%/e Euler bind def +% +/RadtoDeg { 180 mul Pi div } bind def % convert from radian to degrees +/DegtoRad { Pi mul 180 div } bind def % viceversa +% +/startGlobal { true setglobal globaldict begin } bind def +/endGlobal { end false setglobal } bind def +/pssetRGBcolor /setrgbcolor load def +/pssetCMYKcolor /setcmykcolor load def +/pssetGraycolor /setgray load def +% +%----------------- hv end--------------------------- +/PathLength@ { /z z y y1 sub x x1 sub Pyth add def /y1 y def /x1 x def } def +% +/PathLength { + flattenpath /z 0 def + { /y1 ED /x1 ED /y2 y1 def /x2 x1 def } + { /y ED /x ED PathLength@ } + {} + { /y y2 def /x x2 def PathLength@ } + /pathforall load stopped { pop pop pop pop } if + z +} def +% +/STP { .996264 dup scale } def % BP/PT scaling +/STV { SDict begin normalscale end STP } def % +% +/DashLine { + dup 0 gt + { /a .5 def PathLength exch div } + { pop /a 1 def PathLength } ifelse + /b ED % pattern should fit evenly in b + dup /X ED % pattern array + 0 get /y ED % length of first black segment + /z 0 X {add} forall def % length of the full pattern + %% Computation of the scaling factor as described by van Zandt: + b a .5 sub 2 mul y mul sub z Div round + z mul a .5 sub 2 mul y mul add b exch Div + %%%% scaling factor on stack. + /z ED %% now, z is the scaling factor + false % for the length test below + X { z mul } forall X astore %% modification TN 04-08-07 + %%% Checking whether at least one dash in X has positive length: + {0 gt or} forall + { X 1 a sub y mul } + { [ 1 0 ] 0 } + ifelse + setdash stroke +} def +% +/DotLine { + /b PathLength def + /a ED /z ED /y CLW def + /z y z add def + a 0 gt { + /b b a div def + }{ + a 0 eq { + /b b y sub def + }{ a -3 eq { + /b b y add def } if + } ifelse + } ifelse + [ 0 b b z Div round Div dup 0 le { pop 1 } if ] + a 0 gt { 0 }{ y 2 div a -2 gt { neg }if } ifelse + setdash 1 setlinecap stroke +} def +% +/SymbolLine { % on stack [ x y x y ... + counttomark % number of elements + 2 div cvi /n ED % n pairs + /YA ED /XA ED % the start point + n 1 sub { + /YB ED /XB ED + /XLength XB XA sub def + /YLength YB YA sub def + /PAngle YLength XLength Atan def + /XYLength XLength YLength Pyth def + %% for negative SymStep we calculate the distance + SymStep 0 lt + { %XYLength SymStep div abs cvi + /nSym SymStep abs cvi def } + { /nSym XYLength SymStep div cvi def } + ifelse + 0.5 setflat + /Shift Symbol stringwidth pop 2 div def + /deltaX XLength nSym div def + /deltaY YLength nSym div def + curveticks + { XA YA moveto } + { XA Shift sub YA Shift sub moveto } + ifelse + nSym { + gsave + curveticks + { PAngle 180 sub CorrAngle sub tickAngle add /rotAngle ED + currentpoint translate rotAngle rotate + 0 SymbolWidth 2 div moveto 0 SymbolWidth 2 div neg lineto + SymbolLinewidth setlinewidth stroke + } + { + rotateSymbol { PAngle 180 sub CorrAngle sub rotate } if + Symbol show + } + ifelse + grestore + deltaX deltaY rmoveto + } repeat + /YA YB def /XA XB def + } repeat + curveticks + { XA YA moveto } + { XA Shift sub YA Shift sub moveto } + ifelse + gsave + curveticks + { PAngle 180 sub CorrAngle sub tickAngle add /rotAngle ED + XA YA translate rotAngle rotate + 0 SymbolWidth 2 div moveto 0 SymbolWidth 2 div neg lineto + SymbolLinewidth setlinewidth stroke + } + { + rotateSymbol { PAngle 180 sub CorrAngle sub rotate } if + Symbol show + } + ifelse + grestore + pop % delete the mark symbol +} def +% +/LineFill { % hv ------------ patch 7 ------------- + gsave + abs /hatchWidthInc ED + abs /hatchSepInc ED + abs CLW add /a ED + a 0 dtransform round exch round exch + 2 copy idtransform + exch Atan rotate + idtransform pop /a ED + .25 .25 itransform pathbbox + /y2 ED + a Div ceiling cvi /x2 ED /y1 ED + a Div cvi /x1 ED /y2 y2 y1 sub def + clipType % must be defined in pstricks.tex: clip -- eoclip + newpath + 2 setlinecap + systemdict + /setstrokeadjust known { true setstrokeadjust } if + x2 x1 sub 1 add { + x1 a mul y1 moveto 0 y2 rlineto stroke + /x1 x1 1 add + hatchWidthInc 0 gt { CLW add } if + def + hatchSepInc 0 gt hatchWidthInc 0 gt or { + /a a hatchSepInc add def + CLW hatchWidthInc add SLW + } if + } repeat + grestore + pop pop } def +% +/DotFill {% on stack: dot radius + /dotRadius ED + abs CLW add /a ED + a 0 dtransform round exch round exch + 2 copy idtransform + exch Atan rotate + idtransform pop /a ED + .25 .25 itransform + pathbbox % llx lly urx ury of smallest bounding box + /y2 ED /x2 ED /y1 ED /x1 ED + y2 y1 sub a div 2 add cvi /Ny ED + x2 x1 sub a div 2 add cvi /Nx ED + clipType % must be defined in pstricks.tex: clip -- eoclip + newpath + /yA y1 dotRadius add CLW add def + /xA0 x1 dotRadius add CLW add def + Ny { + /xA xA0 def + Nx { + newpath + xA yA dotRadius 0 360 arc + SolidDot { gsave fill grestore } if + stroke + xA a add /xA ED + } repeat + yA a add /yA ED + } repeat + grestore +} def +% +/PenroseFill {% on stack: scaling factor + /Scale ED +% 1 exch div round /penroseFactor ED +% a 0 dtransform round exch round exch +% 2 copy idtransform +% exch Atan rotate +% idtransform pop /a ED +% .25 .25 itransform pathbbox +% /y2 ED +% a Div ceiling cvi /x2 ED /y1 ED +% a Div cvi /x1 ED /y2 y2 y1 sub def + clip + newpath +gsave + 220 150 translate + Scale dup scale + systemdict /setstrokeadjust known { true setstrokeadjust } if + /I/S/L/W/G/+/Z/F/E/D[/def/exch/for{E D}/add{s E get mul} + { Z -36.2001 1 33 }{25 E S rlineto}{/q Z dup q G E q 1 + G}{Z 2 2}]{cvx def}forall + [0 72 1008 {dup sin E cos }F ]1 setlinejoin/s W{/a W{/b I 10{/i I 4{/m I moveto + i m +/j I 10{/l Z b m l + G a l G sub s m get div .2 + floor .3 + 25 + mul j l + S rmoveto}F i L j L stroke }F}F}F}F + grestore +% pop pop +} def +% +/PenroseFillA {% on stack: scaling factor, border color, kite color, dart color + /Scale ED + Scale dup scale + /border_colour ED + /kite_colour ED + /dart_colour ED + clip + newpath + gsave + 100 100 translate + 6 + Scale 1 lt { 1 Scale dup add div mul cvi } if %%%% Number of iterations + 10 %%%% Long side length in millimeters + /border_width { L 0.06 mul }def %%%% Choose the scalefactor for the borders + /L exch 25.4 div 72 mul def %%%% Conversion: mm -> inches -> points + /f{-1 5 sqrt add 2 div}bind def %%%% The reciprocal of the golden ratio + /l{L f mul}bind def %%%% Short side length l = L*f + /Ll{L l add}bind def %%%% Ll = L + l + /c{36 cos L mul}bind def %%%% c = L*cos(36) + /s{36 sin L mul}bind def %%%% s = L*sin(36) + /draw_tile { 0 0 moveto c s lineto 0 lineto gsave closepath gsave fill grestore + 0 setlinewidth stroke grestore border_colour stroke } bind def + /half_kite { dup dup 0 gt{ 1 sub gsave f f neg scale -36 rotate half_dart + Ll 0 translate 144 rotate kite grestore } + { kite_colour L draw_tile }ifelse + pop } bind def + /half_dart { dup dup 0 gt{ 1 sub gsave f f scale half_kite + -144 rotate Ll neg 0 translate half_dart grestore } + { dart_colour l draw_tile }ifelse + pop } bind def + /kite{ gsave half_kite 1 -1 scale half_kite grestore }bind def + border_width setlinewidth 1 setlinejoin 1 setlinecap +% 450 0 translate + dup f exch neg exp dup scale + 5 {kite 72 rotate } repeat stroke + grestore +} def +% +% +/TruchetFill { % on stack: scaling factor + 10 dict begin + dup dup scale + 1 exch div round /penroseFactor ED + a 0 dtransform round exch round exch + 2 copy idtransform + exch Atan rotate + idtransform pop /a ED + .25 .25 itransform pathbbox + /y2 ED + a Div ceiling cvi /x2 ED /y1 ED + a Div cvi /x1 ED /y2 y2 y1 sub def + clip + newpath + systemdict + /setstrokeadjust known { true setstrokeadjust } if + /ma a neg def + /ha a 2 div def + /mha ha neg def + /tile { + rand dup 2 idiv 2 mul eq { 90 rotate } if + mha mha moveto ha mha lineto + ha ha lineto mha ha lineto +% closepath .1 setlinewidth stroke + contents + } def + /contents{ + 0 ha moveto ha 0 lineto + 0 mha moveto mha 0 lineto +% 1 setlinewidth stroke + } def + /dotiling { + f ma mul a f a mul { + /i exch def + f ma mul a f a mul { + /j exch def + gsave i j translate + tile stroke grestore + } for + } for + } def +% + /f 3 def + 5 srand dotiling + end % local user dict +} def +% +/BeginArrow { + ADict begin % hold it local, for end see EndArrow + /@mtrx CM def + gsave + 2 copy T + 2 index sub neg exch + 3 index sub exch Atan + rotate newpath +} def +% +/EndArrow { @mtrx setmatrix CP grestore end } def % end the ADict +% +/Arrow { + CLW mul add dup + 2 div /w ED + mul dup /h ED + mul /a ED + { 0 h T 1 -1 scale } if + w neg h moveto + 0 0 L w h L w neg a neg rlineto + gsave fill grestore +} def +% +/ArrowD { % the sides are drawn as curves (hv 20071211) + CLW mul add dup + 2 div /w ED + mul dup /h ED + mul /Inset ED + { 0 h T 1 -1 scale } if % changes the direction +% we use y=w/h^2 * x^2 as equation for the control points +% for the coordinates the arrow is seen from top to bottom +% the bottom (tip) is (0;0) + w neg h moveto % lower left of > + w 9 div 4 mul neg h 3 div 2 mul + w 9 div neg h 3 div + 0 0 curveto % tip of > + w 9 div h 3 div + w 9 div 4 mul h 3 div 2 mul + w h curveto % upper left of > + w neg Inset neg rlineto % move to x=0 and inset + gsave fill grestore +} def +% +/Tbar { + CLW mul add /z ED + z -2 div CLW 2 div moveto + z 0 rlineto stroke + 0 CLW moveto +} def +% +/Bracket { + CLW mul add dup CLW sub 2 div + /x ED mul CLW add /y ED /z CLW 2 div def + x neg y moveto + x neg CLW 2 div L x CLW 2 div L x y L stroke + 0 CLW moveto +} def +% +/RoundBracket { + CLW mul add dup 2 div + /x ED mul /y ED /mtrx CM def + 0 CLW 2 div T x y mul 0 ne { x y scale } if + 1 1 moveto + .85 .5 .35 0 0 0 curveto + -.35 0 -.85 .5 -1 1 curveto + mtrx setmatrix stroke 0 CLW moveto +} def +% +/SD { 0 360 arc fill } def +% +/EndDot { % DS is the dot size + { /z DS def } { /z 0 def } ifelse % outer or inner dimen + /b ED % the color definition + 0 z DS SD + b { 0 z DS CLW sub SD } if + 0 DS z add CLW 4 div sub + moveto +} def +% +/Shadow { [ { /moveto load } { /lineto load } { /curveto load } { + /closepath load } /pathforall load stopped { pop pop pop pop CP /moveto + load } if ] cvx newpath 3 1 roll T exec } def +% +/NArray { % holds the coordinates and on top of stack the showpoints boolean + /showpoints ED + counttomark 2 div dup cvi /n ED % n 2 div on stack + n eq not { exch pop } if % even numbers of points? delete one + ] aload /Points ED + showpoints not { Points aload pop } if +% { ] aload /Points ED } +% { n 2 mul 1 add -1 roll pop } ifelse % delete the mark symbol +} def +% +/Line { + NArray n 0 eq not + { n 1 eq { 0 0 /n 2 def } if ArrowA /n n 2 sub def + n { Lineto } repeat + CP 4 2 roll ArrowB L pop pop + } if +} def +% +/LineToYAxis { + /Ox ED % Save the x origin value + NArray % all x-y pairs on stack + n { 2 copy moveto % go to current point + Ox exch Lineto % line to y-axis + pop % delete old x-value + } repeat +} def +% +/LineToXAxis{ + /Oy ED % Save the y origin value + NArray % all x-y pairs on stack + n 0 eq not + { n 1 eq { 0 0 /n 2 def } if + ArrowA + /n n 2 sub def + CP 2 copy moveto pop Oy Lineto + n { 2 copy moveto pop Oy Lineto } repeat + CP + 4 2 roll + ArrowB + 2 copy moveto pop Oy + L + pop pop } if +} def +% +/Arcto { + /a [ 6 -2 roll ] cvx def + a r + /arcto load stopped { 5 } { 4 } ifelse { pop } repeat + a +} def +% +/CheckClosed { + dup n 2 mul 1 sub index eq 2 index n 2 mul 1 add index eq + and { pop pop /n n 1 sub def } if +} def +% +/Polygon { + NArray n 2 eq { 0 0 /n 3 def } if + n 3 lt + { n { pop pop } repeat } + { n 3 gt { CheckClosed } if + n 2 mul -2 roll + /y0 ED /x0 ED /y1 ED /x1 ED + x1 y1 + /x1 x0 x1 add 2 div def + /y1 y0 y1 add 2 div def + x1 y1 moveto + /n n 2 sub def + n { Lineto } repeat + x1 y1 x0 y0 6 4 roll Lineto + Lineto pop pop closepath } ifelse +} def +% +/SymbolPolygon { % on stack [ x y x y ... + counttomark % number of elements + 2 add /m ED + 2 copy m 2 roll % copy last two + m 2 div cvi /n ED % n pairs + /YA ED /XA ED % the start point + n 1 sub { + /YB ED /XB ED + /XLength XB XA sub def + /YLength YB YA sub def + /PAngle YLength XLength Atan def + /XYLength XLength YLength Pyth def + /nSym XYLength SymStep Div cvi def + /Shift Symbol stringwidth pop 2 Div def + /deltaX XLength nSym Div def + /deltaY YLength nSym Div def + XA Shift sub YA Shift sub moveto + nSym { + gsave rotateSymbol { PAngle 180 sub CorrAngle sub rotate } if + Symbol show + grestore + deltaX deltaY rmoveto + } repeat +% XB Shift sub YB Shift sub moveto Symbol show + /YA YB def /XA XB def + } repeat + pop % delete the mark symbol +} def +% +/Diamond { + /mtrx CM def + T rotate + /h ED + /w ED + dup 0 eq { pop } { CLW mul neg + /d ED + /a w h Atan def + /h d a sin Div h add def + /w d a cos Div w add def } ifelse + mark w 2 div h 2 div w 0 0 h neg w neg 0 0 h w 2 div h 2 div + /ArrowA { moveto } def + /ArrowB { } def + false Line + closepath mtrx setmatrix } def +% +/Triangle { + /mtrx CM def + translate + rotate /h ED 2 div /w ED + dup CLW mul /d ED + /h h d w h Atan sin Div sub def + /w w d h w Atan 2 div dup cos exch sin Div mul sub def + mark + 0 d w neg d 0 h w d 0 d + /ArrowA { moveto } def + /ArrowB { } def + false + Line + closepath + mtrx +% DG/SR modification begin - Jun. 1, 1998 - Patch 3 (from Michael Vulis) +% setmatrix } def + setmatrix pop +} def +% DG/SR modification end +% +/CCA { + /y ED /x ED + 2 copy y sub /dy1 ED + x sub /dx1 ED + /l1 dx1 dy1 Pyth def +} def +% +/CC { + /l0 l1 def + /x1 x dx sub def + /y1 y dy sub def + /dx0 dx1 def + /dy0 dy1 def + CCA + /dx dx0 l1 c exp mul dx1 l0 c exp mul add def + /dy dy0 l1 c exp mul dy1 l0 c exp mul add def + /m dx0 dy0 Atan dx1 dy1 Atan sub 2 div cos abs b exp a mul dx dy Pyth Div 2 div def + /x2 x l0 dx mul m mul sub def + /y2 y l0 dy mul m mul sub def + /dx l1 dx mul m mul neg def + /dy l1 dy mul m mul neg def +} def +% +/IC { + /c c 1 add def + c 0 lt { /c 0 def } { c 3 gt { /c 3 def } if } ifelse + /a a 2 mul 3 div 45 cos b exp div def + CCA + /dx 0 def + /dy 0 def +} def +% +/BOC { IC CC x2 y2 x1 y1 ArrowA CP 4 2 roll x y curveto } def +/NC { CC x1 y1 x2 y2 x y curveto } def +/EOC { x dx sub y dy sub 4 2 roll ArrowB 2 copy curveto } def +/BAC { IC CC x y moveto CC x1 y1 CP ArrowA } def +/NAC { x2 y2 x y curveto CC x1 y1 } def +/EAC { x2 y2 x y ArrowB curveto pop pop } def +% +/OpenCurve { + NArray n 3 lt + { n { pop pop } repeat } + { BOC /n n 3 sub def n { NC } repeat EOC } ifelse +} def +% +/CurvePath { + %% for negative SymStep we calculate the distance + SymStep 0 lt { gsave PathLength SymStep div abs /SymStep ED grestore } if + 0.5 setflat + flattenpath /z 0 def /z0 0 def + { /y1 ED /x1 ED /y2 y1 def /x2 x1 def + x1 Shift sub y1 Shift sub moveto + gsave + curveticks + { x1 y1 translate startAngle rotate + 0 SymbolWidth 2 div moveto 0 SymbolWidth 2 div neg lineto + SymbolLinewidth setlinewidth stroke + } + { startAngle rotate Symbol show } + ifelse + grestore /z0 z def } + { /y ED /x ED PathLength@ z z0 sub SymStep ge { + x Shift sub y Shift sub moveto + gsave + curveticks + { y yOld sub x xOld sub Atan 180 sub CorrAngle sub /rotAngle ED + x y translate rotAngle rotate + 0 SymbolWidth 2 div moveto 0 SymbolWidth 2 div neg lineto + SymbolLinewidth setlinewidth stroke + } + { + rotateSymbol { y yOld sub x xOld sub Atan 180 sub CorrAngle sub rotate } if + Symbol show + } + ifelse + grestore /z0 z def } if + /yOld y def /xOld x def } + {} %% the lineto part + { /y y2 def /x x2 def PathLength@ + x Shift sub y Shift sub moveto + gsave + curveticks + { y yOld sub x xOld sub Atan 180 sub /rotAngle ED + x y translate rotAngle rotate + 0 SymbolWidth 2 div moveto 0 SymbolWidth 2 div neg lineto + SymbolLinewidth setlinewidth stroke + } + { + x Shift sub y Shift sub moveto + rotateSymbol { y yOld sub x xOld sub Atan 180 sub CorrAngle sub rotate } if + Symbol show + } + ifelse + grestore + } + pathforall +% curveticks +% { gsave +% x y translate rotAngle rotate +% 0 SymbolWidth 2 div moveto 0 SymbolWidth 2 div neg lineto +% SymbolLinewidth setlinewidth stroke grestore +% } if + z +} def +% +/OpenSymbolCurve { + OpenCurve + 0.1 setflat + /Shift Symbol stringwidth pop 2 div def + CurvePath +} def +% +/AltCurve { + { false NArray n 2 mul 2 roll + [ n 2 mul 3 sub 1 roll ] aload + /Points ED + n 2 mul -2 roll } + { false NArray } ifelse + n 4 lt { n { pop pop } repeat } { BAC /n n 4 sub def n { NAC } repeat EAC } ifelse +} def +% +/AltOpenSymbolCurve { + AltCurve + 0.1 setflat + /Shift Symbol stringwidth pop 2 div def + CurvePath +} def +% +/ClosedCurve { + NArray n 3 lt + { n { pop pop } repeat } + { n 3 gt { CheckClosed } if + 6 copy n 2 mul 6 add 6 roll + IC CC x y moveto n { NC } repeat + closepath pop pop + } ifelse +} def +% +/ClosedSymbolCurve { + ClosedCurve + 0.1 setflat + /Shift Symbol stringwidth pop 2 div def + CurvePath +} def +% +/CalcBezierSpline {% Christoph Bersch + 10 dict begin + /getX { Points exch 2 mul get } def + /getY { Points exch 2 mul 1 add get } def + /n Points length 1 sub 2 idiv def + /GetFirstControlPoints { + /x n array def + /tmp n array def + /b 2 def + x 0 rhs 0 get b div put + 1 1 n 1 sub { + /i exch def + tmp i 1 b div dup 4 1 roll put + i n 1 sub lt { 4 }{ 3.5 } ifelse exch sub /b exch def + x i rhs i get x i 1 sub get sub b div put + } for + 1 1 n 1 sub { + n exch sub + dup dup x exch 1 sub 2 copy 6 2 roll + get 3 1 roll tmp exch get + exch x exch get mul sub + put + } for + x + } def + % + n 1 eq { + 0 getX 2 mul 1 getX add 3 div + 0 getY 2 mul 1 getY add 3 div + exch dup 3 1 roll 2 mul 0 getX sub + exch dup 3 1 roll 2 mul 0 getY sub + [ 0 getX 0 getY 7 3 roll 1 getX 1 getY ] /outPoints exch def + } { + /outPoints 6 n mul 2 add array def + 0 1 n { + dup dup 6 mul dup 1 add + outPoints exch 5 -1 roll getY put + outPoints exch 3 -1 roll getX put + } for + /rhs n array def + 1 1 n 2 sub { + rhs exch dup dup getX 4 mul exch 1 add getX 2 mul add put + } for + rhs 0 0 getX 1 getX 2 mul add put + rhs n 1 sub dup getX 8 mul n getX add 2 div put + GetFirstControlPoints + 1 1 n 2 sub { + rhs exch dup dup getY 4 mul exch 1 add getY 2 mul add put + } for + rhs 0 0 getY 1 getY 2 mul add put + rhs n 1 sub dup getY 8 mul n getY add 2 div put + GetFirstControlPoints + 0 1 n 1 sub { + /i exch def + 2 copy + i get outPoints 6 i mul 3 add 3 -1 roll put + i get outPoints 6 i mul 2 add 3 -1 roll put + 2 copy + i n 1 sub lt { + i 1 add get i 1 add getY 2 mul exch sub outPoints 6 i mul 5 add 3 -1 roll put + i 1 add get i 1 add getX 2 mul exch sub outPoints 6 i mul 4 add 3 -1 roll put + }{ + n 1 sub get n getY add 2 div outPoints 6 n 1 sub mul 5 add 3 -1 roll put + n 1 sub get n getX add 2 div outPoints 6 n 1 sub mul 4 add 3 -1 roll put + } ifelse + } for + pop pop + } ifelse + outPoints + end +} def +/Spline { + /showpoints ED + counttomark 2 div dup cvi /n ED + n eq not { exch pop } if + ] /Points ED + n 1 gt { + CalcBezierSpline + mark exch aload pop + ArrowA + n 2 sub { + 6 2 roll 4 2 roll curveto + } repeat + 6 2 roll 4 2 roll ArrowB curveto + } if +} def +/OpenSymbolSpline { + Spline + 0.1 setflat + /Shift Symbol stringwidth pop 2 div def + CurvePath +} def +% +/SQ { /r ED r r moveto r r neg L r neg r neg L r neg r L fill } def +/ST { /y ED /x ED x y moveto x neg y L 0 x L fill } def +/SP { /r ED gsave 0 r moveto 4 { 72 rotate 0 r L } repeat fill grestore } def +% +/FontDot { + DS 2 mul dup + matrix scale matrix concatmatrix exch matrix + rotate matrix concatmatrix exch + findfont exch makefont setfont +} def +% +/Rect { + x1 y1 y2 add 2 div moveto +% x1 y2 lineto +% x2 y2 lineto +% x2 y1 lineto +% x1 y1 lineto + x1 y1 lineto % counter clockwise path + x2 y1 lineto + x2 y2 lineto + x1 y2 lineto + closepath +} def +% +/OvalFrame { + x1 x2 eq y1 y2 eq or + { pop pop x1 y1 moveto x2 y2 L } + { y1 y2 sub abs x1 x2 sub abs 2 copy gt + { exch pop } { pop } ifelse + 2 div exch { dup 3 1 roll mul exch } if + 2 copy lt { pop } { exch pop } ifelse + /b ED + x1 y1 y2 add 2 div moveto + x1 y2 x2 y2 b arcto + x2 y2 x2 y1 b arcto + x2 y1 x1 y1 b arcto + x1 y1 x1 y2 b arcto + 16 { pop } repeat + closepath + } ifelse +} def +% +/Frame { + CLW mul /a ED + 3 -1 roll + 2 copy gt { exch } if + a sub /y2 ED + a add /y1 ED + 2 copy gt { exch } if + a sub /x2 ED + a add /x1 ED + 1 index 0 eq { pop pop Rect } { OvalFrame } ifelse +} def +% +/BezierNArray { + /f ED + counttomark 2 div dup cvi /n ED + n eq not { exch pop } if + n 1 sub neg 3 mod 3 add 3 mod { 0 0 /n n 1 add def } repeat + f { ] aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse +} def +% +/OpenBezier { + BezierNArray + n 1 eq + { pop pop } + { ArrowA n 4 sub 3 idiv + { 6 2 roll 4 2 roll curveto } repeat + 6 2 roll 4 2 roll ArrowB curveto } ifelse +} def +% +/OpenSymbolBezier { + OpenBezier + 0.1 setflat + /Shift Symbol stringwidth pop 2 div def + CurvePath +} def +% +/ClosedBezier { + BezierNArray + n 1 eq + { pop pop } + { moveto n 1 sub 3 idiv + { 6 2 roll 4 2 roll curveto } repeat + closepath } ifelse +} def +% +/ClosedSymbolBezier { + /f ED % save showpoints value + 2 copy /yEnd ED /xEnd ED + counttomark -2 roll 2 copy /yStart ED /xStart ED + counttomark 2 roll + f + ClosedBezier + 0.1 setflat + /Shift Symbol stringwidth pop 2 div def + CurvePath + [ xEnd yEnd xStart yStart SymbolLine +} def +% +/BezierShowPoints { + gsave + Points aload length 2 div cvi /n ED + moveto + n 1 sub { lineto } repeat + CLW 2 div SLW [ 4 4 ] 0 setdash stroke + grestore +} def +% +/Parab { + /y0 ED /x0 ED /y1 ED /x1 ED + /dx x0 x1 sub 3 div def + /dy y0 y1 sub 3 div def + x0 dx sub y0 dy add x1 y1 ArrowA + x0 dx add y0 dy add x0 2 mul x1 sub y1 ArrowB + curveto + /Points [ x1 y1 x0 y0 x0 2 mul x1 sub y1 ] def +} def +% +/Parab1 { % 1 end | 0 SP + /ySP ED /xSP ED /y1 ED /x1 ED + /dx xSP x1 sub 3 div def + /dy ySP y1 sub 3 div def + newpath x1 y1 moveto xSP y1 lineto xSP ySP lineto + x1 ySP lineto closepath clip + currentpoint + newpath moveto + xSP dx sub ySP dy add x1 y1 ArrowA + xSP dx add ySP dy add xSP 2 mul x1 sub y1 ArrowB + curveto + /Points [ x1 y1 xSP ySP xSP 2 mul x1 sub y1 ] def +} def +% +/Grid { + newpath + /a 4 string def + /b ED % psk@gridlabels in pt + /c ED % { \pst@usecolor\psgridlabelcolor } + /n ED % psk@griddots + cvi dup 1 lt { pop 1 } if + /s ED % \psk@subgriddiv + s div dup 0 eq { pop 1 } if + /dy ED s div dup 0 eq { pop 1 } if % \pst@number\psyunit abs + /dx ED dy div round dy mul % \pst@number\psxunit abs + /y0 ED dx div round dx mul + /x0 ED dy div round cvi + /y2 ED dx div round cvi + /x2 ED dy div round cvi + /y1 ED dx div round cvi + /x1 ED + /h y2 y1 sub 0 gt { 1 } { -1 } ifelse def + /w x2 x1 sub 0 gt { 1 } { -1 } ifelse def + b 0 gt { + /z1 b 4 div CLW 2 div add def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Helvetica findfont b scalefont setfont +% is set in pstricks.tex +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% /NimbusSanL-Regu findfont b scalefont setfont + /b b .95 mul CLW 2 div add def } if + systemdict /setstrokeadjust known + { true setstrokeadjust /t { } def } + { /t { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add + exch itransform } bind def } ifelse + gsave n 0 gt { 1 setlinecap [ 0 dy n div ] dy n div 2 div setdash } { 2 setlinecap } ifelse + /i x1 def + /f y1 dy mul n 0 gt { dy n div 2 div h mul sub } if def + /g y2 dy mul n 0 gt { dy n div 2 div h mul add } if def + x2 x1 sub w mul 1 add dup 1000 gt { pop 1000 } if + { i dx mul dup xGridOffset add y0 moveto + b 0 gt + { gsave c i a cvs dup stringwidth pop + /z2 ED w 0 gt {z1} {z1 z2 add neg} ifelse + h 0 gt {b neg}{z1} ifelse + rmoveto show grestore } if + dup t f moveto + g t L stroke + /i i w add def + } repeat + grestore + gsave + n 0 gt + % DG/SR modification begin - Nov. 7, 1997 - Patch 1 + %{ 1 setlinecap [ 0 dx n div ] dy n div 2 div setdash } + { 1 setlinecap [ 0 dx n div ] dx n div 2 div setdash } + % DG/SR modification end + { 2 setlinecap } ifelse + /i y1 def + /f x1 dx mul n 0 gt { dx n div 2 div w mul sub } if def + /g x2 dx mul n 0 gt { dx n div 2 div w mul add } if def + y2 y1 sub h mul 1 add dup 1000 gt { pop 1000 } if + { newpath i dy mul dup yGridOffset add x0 exch moveto + b 0 gt { gsave c i a cvs dup stringwidth pop + /z2 ED + w 0 gt {z1 z2 add neg} {z1} ifelse + h 0 gt {z1} {b neg} ifelse + rmoveto show grestore } if + dup f exch t moveto + g exch t L stroke + /i i h add def + } repeat + grestore +} def +% +/ArcArrow { + /d ED /b ED /a ED + gsave + newpath 0 -1000 moveto clip + newpath + 0 1 0 0 b + grestore + c mul + /e ED + pop pop pop r a e d PtoC y add exch x add + exch r a PtoC y add exch x add exch b pop pop pop pop a e d CLW 8 div c + mul neg d +} def +% +% +/isbool { type (booleantype) cvn eq } def +% +/Ellipse { + dup isbool { /MoveToStart ED }{ /MoveToStart false def }ifelse % false or true + /rotAngle ED + /mtrx CM def + T + rotAngle rotate + scale + MoveToStart { 0 0 moveto 1 0 rmoveto } if % move to the start position + 0 0 1 5 3 roll arc + mtrx setmatrix +} def +% +/ArcAdjust { %%%% Vincent Guirardel +% given a target length (targetLength) and an initial angle (angle0) [in the stack], +% let M(angle0)=(rx*cos(angle0),ry*sin(angle0))=(x0,y0). +% This computes an angle t such that (x0,y0) is at distance +% targetLength from the point M(t)=(rx*cos(t),ry*sin(t)). +% NOTE: this an absolute angle, it does not have to be added or substracted to angle0 +% contrary to TvZ's code. +% To achieve, this, one iterates the following process: start with some angle t, +% compute the point M' at distance targetLength of (x0,y0) on the semi-line [(x0,y0) M(t)]. +% Now take t' (= new angle) so that (0,0) M(t') and M' are aligned. +% +% Another difference with TvZ's code is that we need d (=add/sub) to be defined. +% the value of d = add/sub is used to know on which side we have to move. +% It is only used in the initialisation of the angle before the iteration. +% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Input stack: 1: target length 2: initial angle +% variables used : rx, ry, d (=add/sub) +% + /targetLength ED /angle0 ED + /x0 rx angle0 cos mul def + /y0 ry angle0 sin mul def +% we are looking for an angle t such that (x0,y0) is at distance targetLength +% from the point M(t)=(rx*cos(t),ry*sin(t))) +%initialisation of angle (using 1st order approx = TvZ's code) + targetLength 57.2958 mul + angle0 sin rx mul dup mul + angle0 cos ry mul dup mul + add sqrt div +% if initialisation angle is two large (more than 90 degrees) set it to 90 degrees +% (if the ellipse is very curved at the point where we draw the arrow, % +% the value can be much more than 360 degrees !) +% this should avoid going on the wrong side (more than 180 degrees) or go near +% a bad attractive point (at 180 degrees) + dup 90 ge { pop 90 } if + angle0 exch d % add or sub +% maximum number of times to iterate the iterative procedure: +% iterative procedure: takes an angle t on top of stack, computes a +% better angle (and put it on top of stack) + 30 { dup +% compute distance D between (x0,y0) and M(t) + dup cos rx mul x0 sub dup mul exch sin ry mul y0 sub dup mul add sqrt +% if D almost equals targetLength, we stop + dup targetLength sub abs 1e-5 le { pop exit } if +% stack now contains D t +% compute the point M(t') at distance targetLength of (x0,y0) on the semi-line [(x0,y0) M(t)]: +% M(t')= ( (x(t)-x0)*targetLength/d+x0 , (y(t)-y0)*targetLength/d+y0 ) + exch dup cos rx mul x0 sub exch sin ry mul y0 sub +% stack contains: y(t)-y0, x(t)-x0, d + 2 index Div targetLength mul y0 add ry Div exch + 2 index Div targetLength mul x0 add rx Div +% stack contains x(t')/rx , y(t')/ry , d +% now compute t', and remove D from stack + atan exch pop + } repeat +% we don't look at what happened... in particular, if targetLength is greater +% than the diameter of the ellipse... +% the final angle will be around /angle0 + 180. maybe we should treat this pathological case... +% after iteration, stack contains an angle t such that M(t) is the tail of the arrow +% to give back the result as a an angle relative to angle0 we could add the following line: +% angle0 sub 0 exch d +% +% begin bug fix 2006-01-11 +% we want to adjust the new angle t' by a multiple of 360 so that | t'-angle0 | <= 180 +%(we don't want to make the ellipse turn more or less than it should)... +dup angle0 sub dup abs 180 gt { 180 add 360 div floor 360 mul sub } { pop } ifelse +% end bug fix +} def +% +/EllipticArcArrow { + /d ED % is add or sub + /b ED % arrow procedure + /a1 ED % angle + gsave + newpath + 0 -1000 moveto + clip % Set clippath far from arrow. + newpath + 0 1 0 0 b % Draw arrow to determine length. + grestore +% Length of arrow is on top of stack. Next 3 numbers are junk. +% + a1 exch ArcAdjust % Angular position of base of arrow. + /a2 ED + pop pop pop + a2 cos rx mul xOrig add % hv 2007-08-29 x->xOrig + a2 sin ry mul yOrig add % hv 2007-08-29 y->yOrig + a1 cos rx mul xOrig add % + a1 sin ry mul yOrig add % +% Now arrow tip coor and base coor are on stack. + b pop pop pop pop % Draw arrow, and discard coordinates. + a2 CLW 8 div +% change value of d (test it by looking if `` 1 1 d '' gives 2 or not ) + 1 1 d 2 eq { /d { sub } def } { /d { add } def } ifelse + ArcAdjust +% resets original value of d + 1 1 d 2 eq { /d { sub } def } { /d { add } def } ifelse % Adjust angle to give overlap. +} def +%%------------------ tvz/DG/hv (2004-05-10) end -------------------%% +% +/Rot { CP CP translate 3 -1 roll neg rotate NET } def +% +/RotBegin { + tx@Dict /TMatrix known not { /TMatrix { } def /RAngle { 0 } def } if + /TMatrix [ TMatrix CM ] cvx def + /a ED + a Rot /RAngle [ RAngle dup a add ] cvx def +} def +% +/RotEnd { + /TMatrix [ TMatrix setmatrix ] cvx def + /RAngle [ RAngle pop ] cvx def +} def +% +/PutCoor { gsave CP T CM STV exch exec moveto setmatrix CP grestore } def +/PutBegin { /TMatrix [ TMatrix CM ] cvx def CP 4 2 roll T moveto } def +/PutEnd { CP /TMatrix [ TMatrix setmatrix ] cvx def moveto } def +% +/Uput { + /a ED + add 2 div /h ED 2 + div /w ED + /s a sin def + /c a cos def + /b s abs c abs 2 copy gt dup + /q ED + { pop } { exch pop } ifelse def + /w1 c b div w mul def + /h1 s b div h mul def + q { w1 abs w sub dup c mul abs }{ h1 abs h sub dup s mul abs } ifelse +} def +% +/UUput { + 5 dict begin + /z ED + abs /y ED + /x ED + q { x s div c mul abs y gt }{ x c div s mul abs y gt } ifelse + { x x mul y y mul sub z z mul add sqrt z add } + { q { x s div } { x c div } ifelse abs + } ifelse + a PtoC + h1 add exch + w1 add exch + end +} def +% +end +%-----------------------------------------------------------------------------% +% +% END pstricks.pro + +%%EndProcSet +%%BeginProcSet: pst-algparser.pro 0 0 +% $Id: pst-algparser.pro 446 2017-04-19 11:40:55Z herbert $ +%% +%% PostScript prologue for PSTricks algorithm parser +%% Version 0.07, 2014/08/01 +%% +%% This program can be redistributed and/or modified under the terms +%% of the LaTeX Project Public License Distributed from CTAN archives +%% in directory macros/latex/base/lppl.txt. +%% +%%-----------------------------------------------------------------------------% +% +%currentdict /Pi known not { /Pi 3.14159265359 def } if +% +/AlgParser { tx@AlgToPs begin AlgToPs end } def % Dominique Rodriguez +/I2P { AlgParser cvx exec } def % Infix to Postfix +% +/tx@CoreAnalyzerDict 100 dict def tx@CoreAnalyzerDict begin +% +% PS ANALYZER FOR ALGEBRAIC EXPRESSION V1.13 +% +% 09/2011 DR factorial with ! added +% +% E->T|E+T +% T->FS|T*FS +% FS -> F | +FS | -FS +% F->P|F^SF|P! +% P->(E)|literal +% literal->number|var|var[E]|func(params) +% params->E|E,param +% number->TOBEFINISHED +% +%% E expression, T term, SF signed factor, F factor, P power +% +%% parser +% +%% str +% +%% C->EE +%% STR index -> STR index+lenExpr +/AnalyzeCond { AnalyzeExpr ReadCondOp AnalyzeExpr EvalCondOp } def +% +%% analyze Expression List (separator , or | ) +%% STR index -> STR index+lenExpr +%% /AnalyzeListOfE { +%% { NextNonBlankChar pop AnalyzeExpr%%dup Strlen eq { exit } if NextNonBlankChar +%% NextNonBlankChar dup 0 eq { pop exit } if +%% dup 44 ne 1 index 124 ne and { dup 41 ne { PROBLEMCONTACTBILLOU } { pop exit } ifelse } if +%% pop NextNonBlankChar dup 0 eq { exit } if 124 ne { PROBLEMCONTACTBILLOU } if 1 add NextNonBlankChar 0 eq {toto} if } loop +%% AnalyzeListOfEPostHook +%% } def +/AnalyzeListOfE { + /NotFirst false def + { NextNonBlankChar pop AnalyzeExpr + NotFirst { EvalListOfExpr } { /NotFirst true def } ifelse + dup Strlen eq { exit } if NextNonBlankChar + dup 44 ne 1 index 124 ne and + { dup 41 ne { PROBLEMCONTACTBILLOU } { pop exit } ifelse } + if pop 1 add } loop + AnalyzeListOfEPostHook +} def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% E->T|E+T +%% STR index -> STR index+lenExpr +/AnalyzeExpr { + AnalyzePreHook AnalyzeTerm IsEndingExpr + { dup 0 ne { 32 eq { NextNonBlankChar } if } { pop } ifelse } + { { RollOp 1 add NextNonBlankChar pop AnalyzeTerm PreEvalHook EvalAddSub IsEndingExpr { pop exit } if } loop } + ifelse + AnalyzePostHook +} def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% T->FS|T*FS +%% STR index +/AnalyzeTerm { + AnalyzePreHook AnalyzeSignedFactor IsEndingTerm + { dup 0 ne { 32 eq { NextNonBlankChar } if } { pop } ifelse } + { { RollOp 1 add NextNonBlankChar pop AnalyzeSignedFactor PreEvalHook EvalMulDiv IsEndingTerm { pop exit } if} loop } + ifelse + AnalyzePostHook +} def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% FS -> F | +FS | -FS +%% STR index +/AnalyzeSignedFactor { + AnalyzePreHook 2 copy get dup IsUnaryOp + { RollOp 1 add NextNonBlankChar pop AnalyzeSignedFactor EvalUnaryOp } + { pop AnalyzeFactor } + ifelse AnalyzePostHook +} def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% F->P|F^P|P! +%% STR index +/AnalyzeFactor { + AnalyzePreHook AnalyzePower IsEndingFactor + { dup 0 ne { 32 eq { NextNonBlankChar } if } { pop } ifelse } + { { dup 33 eq%% is there a ! DR 09/2011 + { pop 1 add NextNonBlankChar pop EvalFactorial } + { RollOp 1 add NextNonBlankChar pop AnalyzePower PreEvalHook EvalPower } + ifelse + IsEndingFactor { pop exit } if } loop } + ifelse AnalyzePostHook +} def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% P->(E)|literal +%% STR index +/AnalyzePower { + %% depending of first char either a number, or a literal + 2 copy get dup 40 eq%%an open par + { pop 1 add NextNonBlankChar pop AnalyzeExpr 1 add NextNonBlankChar pop } + { AnalyzeLiteral } + ifelse +} def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% STR index STR[index] -> STR index +%/AnalyzeLiteral { IsNumber { EvalNumber } { EvalLiteral } ifelse } def +/AnalyzeLiteral { dup IsUnaryOp exch IsNumber or { EvalNumber } { EvalLiteral } ifelse } def%%dr 09102006 +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% recognize + or - +%% chr -> T/F +/IsUnaryOp { dup 43 eq exch 45 eq or } bind def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% a number can contain only : 0123456789. +%% chr -> T/F +/IsNumber { dup 48 ge exch dup 57 le 3 -1 roll and exch 46 eq or } bind def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% STR index -> STR index number +%% a number can be of the form [0-9]*.[0-9]*\([eE][+-]?[0-9]+\)? +%% STR index -> STR index' number +/ReadNumber { + exch dup 3 -1 roll dup 3 1 roll + %%read mantissa + { 1 add 2 copy dup Strlen eq { pop pop 0 exit } if get dup IsNumber not { exit } if pop } loop + dup 101 eq exch 69 eq or + %%% there is a "e" or "E" -> read exponant + { 1 add 2 copy get dup IsUnaryOp + { pop 1 add 2 copy get } if + { IsNumber not { exit } if 1 add 2 copy get } loop } + if + dup 4 1 roll + 3 -1 roll exch 1 index sub getinterval +} def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% a number can contain only : 0123456789. +%% chr -> T/F +/IsCondOp { dup 30 eq exch dup 60 ge exch 62 le and or } bind def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% STR index -> STR index number +%% a number can be of the form [0-9]*.[0-9]*\([eE][+-]?[0-9]+\)? +%% STR index -> STR index' number +/ReadCondOp { + NextNonBlankChar 1 index 4 1 roll + { IsCondOp not { exit } if 1 add 2 copy get } loop + 2 copy 5 -1 roll + exch 1 index sub getinterval 3 1 roll +} def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% a literal can contain only : 0123456789. +%% chr -> T/F +/IsLiteral {% + dup 48 ge exch dup 57 le 3 -1 roll and exch + dup 65 ge exch dup 90 le 3 -1 roll and 3 -1 roll or exch + dup 97 ge exch 122 le and or } bind def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% a literal can be of the form [a-zA-Z][a-zA-Z0-9]*\(\((Expression)\)|\(\[Expression\]\)\)? +%% STR index -> literal STR index' nextchr +/ReadLiteral { + exch dup 3 -1 roll dup 3 1 roll + %%read literal core + { 2 copy dup Strlen eq { pop pop 0 exit } if get dup IsLiteral not { exit } if pop 1 add } loop + 4 1 roll dup 5 1 roll 3 -1 roll exch 1 index sub getinterval 4 1 roll +} def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% expr is ended by end of str or a clpar +%% STR index -> STR index STR[index] T/F +/IsEndingExpr {% + 2 copy dup Strlen eq + %% if end of str is reached -> end ! + { pop pop 0 true } + %% ending chr -> clpar, comma, |, <, >, =, !, + {get dup dup 41 eq + exch dup 124 eq + exch dup 93 eq + exch dup 44 eq + exch dup 30 eq + exch dup 60 ge exch 62 le and or or or or or} + ifelse } def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% expr is ended by end of str or a +- +%% STR index -> STR index STR[index] T/F +/IsEndingTerm { IsEndingExpr { true } { dup dup 43 eq exch 45 eq or } ifelse } def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% expr is ended by end of str or */ +%% STR index -> STR index STR[index] T/F +/IsEndingFactor { IsEndingTerm { true } { dup dup 42 eq exch 47 eq or } ifelse } def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% expr is ended by end of str or ^ +%% STR index -> STR index STR[index] T/F +/IsEndingPower { IsEndingFactor { true } { dup 94 eq } ifelse } def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% STR index -> STR index STR[index] +/NextNonBlankChar { { dup Strlen eq { 0 exit } if 2 copy get dup neBlkChar { exit } if pop 1 add } loop } bind def +/neBlkChar { dup 32 ne exch dup 10 ne exch 9 ne and and } bind def +%%%%%%%%%%%%%%%%%%%%%%%% +%% DEBUG +/BRK {false} def +/BRKtrue {/BRK true def} def +/BRKStop {BRK {BRKtoto} if } def +/BRKEvalStop {BRK exch if } def +/BRKBRK2true {BRK {BRK2true} if } def +/BRK2 {false} def +/BRK2true {/BRK2 true def} def +/BRK2Stop {BRK2 {BRK2toto} if } def/BRK {false} def +end +% +%-------------------------------------------------------------------------------% +% +/tx@AlgToPs 12 dict def tx@AlgToPs begin +% +%% algExpr -> PSVector +/AlgToPs { tx@CoreAnalyzerDict begin InitParser AnalyzeListOfE pop pop EndingSequence end } def +/EndingSequence { ExpressionVector aload length /end cvx exch 1 add array astore } def +/InitParser { /ExpressionVector [ /tx@AddMathFunc cvx /begin cvx ] def dup length /Strlen exch def 0 } def +/Strlen 0 def +/EvalListOfExpr {} def% +/EvalNumber {% + ReadNumber cvr /ExpressionVector ExpressionVector aload length dup 3 add -1 roll cvx + exch 1 add array astore def NextNonBlankChar pop } def +/EvalAddSub {% + /ExpressionVector ExpressionVector aload length dup 5 add -1 roll + 43 eq { /add } { /sub } ifelse cvx exch 1 add array astore def +} def +/EvalMulDiv {% + /ExpressionVector ExpressionVector aload length dup 5 add -1 roll + 42 eq { /mul } { /div } ifelse cvx exch 1 add array astore def +} def +/EvalPower {% + /ExpressionVector ExpressionVector aload length dup 5 add -1 roll + pop /exp cvx exch 1 add array astore def +} def +/EvalFactorial {% DR 09/2011 + /ExpressionVector ExpressionVector aload length + /fact cvx exch 1 add array astore def +} def +/EvalLiteral {% + ReadLiteral + dup 40 eq%%% there is an open par -> function call + { pop 2 index + dup (Sum) eq { EvalSum } + { dup (IfTE) eq { EvalCond } + { dup (Derive) eq { pop EvalDerive } + { pop 1 add NextNonBlankChar pop AnalyzeListOfE 2 index TrigoFunc + /ExpressionVector ExpressionVector aload length dup 5 add -1 roll cvn cvx + exch 1 add array astore def 1 add NextNonBlankChar pop } ifelse } ifelse} ifelse } + { /ExpressionVector ExpressionVector aload length dup 6 add -1 roll cvn cvx exch 1 add array astore def + dup 91 eq%%% there is an open bracket -> vector element + { pop 1 add NextNonBlankChar pop AnalyzeExpr + /ExpressionVector ExpressionVector aload length /cvi cvx exch /get cvx exch 2 add array astore def 1 add } + { pop NextNonBlankChar pop } + ifelse} + ifelse +} def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% the derive function : Derive(n,f(x)) +%% firstparindex lastparindex -> +/EvalDerive { + %% manage the function descripiton + 1 add ReadNumber 3 1 roll NextNonBlankChar + 44 ne { ANALYZER_ERROR_missing_second_comma_in_Sum } if + 1 add NextNonBlankChar pop + 3 -1 roll cvi + dup 0 eq + { pop AnalyzeExpr 3 -1 roll pop 1 add } + { 1 sub 3 1 roll (x) exch tx@Derive begin DeriveIndexed end 4 -1 roll + { (x) tx@Derive begin Derive end } repeat + ExpressionVector exch /ExpressionVector [] def + AlgToPs aload length + /ExpressionVector 1 index 3 add -1 roll aload length dup 3 add -1 roll /l2 exch def /l1 exch def + l1 l2 add 1 add l2 neg roll l1 l2 add array astore def 3 -1 roll pop 1 add + 1 index length /Strlen exch def } ifelse +} def +/EvalSum {% + pop 1 add NextNonBlankChar pop + %% read the variable name + ReadLiteral pop NextNonBlankChar + 44 ne { ANALYZER_ERROR_missing_first_comma_in_Sum } if + %% read the initial value + 1 add NextNonBlankChar pop ReadNumber cvi 3 1 roll + 2 copy get 44 ne { ANALYZER_ERROR_missing_second_comma_in_Sum } if + %% read the increment value + 1 add NextNonBlankChar pop ReadNumber cvi 3 1 roll + 2 copy get 44 ne { ANALYZER_ERROR_missing_second_comma_in_Sum } if + %% read the limit value + 1 add NextNonBlankChar pop ReadNumber cvi 3 1 roll + 2 copy get 44 ne { ANALYZER_ERROR_missing_second_comma_in_Sum } if + /ExpressionVector ExpressionVector aload length dup 7 add -3 roll 0 4 1 roll + 5 -1 roll 4 add array astore def + %% keep ExpressionVector for later and create a new one for internal Sum computation + ExpressionVector 3 1 roll /ExpressionVector [ 6 -1 roll cvn /exch cvx /def cvx ] def + 1 add NextNonBlankChar pop AnalyzeExpr + %% add each term + /ExpressionVector ExpressionVector aload length 1 add /add cvx exch array astore def + /ExpressionVector 4 -1 roll aload length ExpressionVector cvx /for cvx 3 -1 roll 2 add + array astore def 3 -1 roll pop 1 add +} def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Convert to radians if trigo function call +%% (name) -> +/TrigoFunc { + dup (cos) eq 1 index (sin) eq or exch (tan) eq or + { /ExpressionVector ExpressionVector aload length 3.1415926 /div cvx 180 /mul cvx 5 -1 roll 4 add + array astore def + } if +} def +/EvalCond {% + pop 1 add AnalyzeCond NextNonBlankChar + 44 ne { ANALYZER_ERROR_missing_first_comma_in_IfTE } if + ExpressionVector 3 1 roll /ExpressionVector [] def + 1 add AnalyzeExpr ExpressionVector 3 1 roll /ExpressionVector [] def + NextNonBlankChar 44 ne { ANALYZER_ERROR_missing_second_comma_in_IfTE } if + 1 add AnalyzeExpr + NextNonBlankChar 41 ne { ANALYZER_ERROR_missing_ending parenthesis_in_IfTE } if + ExpressionVector + /ExpressionVector 6 -1 roll aload length dup + 6 add -1 roll cvx exch dup 4 add -1 roll cvx /ifelse cvx 3 -1 roll 3 add array astore def + 1 add 3 -1 roll pop +} def +%% CondOp STR index +/EvalCondOp {% + 3 -1 roll + dup (=) eq { /eq } {% + dup (<) eq { /lt } {% + dup (>) eq { /gt } {% + dup (>=) eq { /ge } {% + dup (<=) eq { /ge } {% + dup (!=) eq { /ne } { ERROR_non_valid_conditional_operator } + ifelse } ifelse } ifelse } ifelse } ifelse } ifelse + cvx exch pop + /ExpressionVector ExpressionVector aload length dup 3 add -1 roll exch 1 add array astore def } def +/EvalUnaryOp { + 3 -1 roll 45 eq { /ExpressionVector ExpressionVector aload length /neg cvx exch 1 add array astore def } if +} def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% H O O K S +/AnalyzePreHook {} bind def +/PreEvalHook {} bind def +/AnalyzeListOfEPostHook {} bind def +/AnalyzePostHook {} def +/RollOp { 3 1 roll } bind def +end %tx@CoreAnalyzerDict +% +%--------------------------------------------------------------------% +% +/tx@Derive 41 dict def tx@Derive begin +%%increase ^^ for each function added +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% algExpr variable -> PSVector +/Derive { + 10240 string 3 1 roll 0 3 1 roll + /Variable exch def + tx@CoreAnalyzerDict begin InitParser AnalyzeListOfE end +} def +/Strlen 0 def +/InitParser { dup length /Strlen exch def 0 } def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% algExpr variable index -> PSVector +/DeriveIndexed {% + 3 1 roll 10240 string 3 1 roll 0 3 1 roll + /Variable exch def + tx@CoreAnalyzerDict begin InitParser pop 4 -1 roll AnalyzeExpr 4 -2 roll pop pop 4 -2 roll exch pop end +} def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% (u,v)'=-(u',v') +/EvalListOfExpr {% + 4 2 roll 2 copy 9 -1 roll dup length 4 1 roll putinterval add AddPipe + 2 copy 7 -1 roll dup length 4 1 roll putinterval add + 6 -2 roll pop pop + 2 copy pop 0 6 2 roll GetIntervalNewStr 5 1 roll 2 copy 0 exch getinterval 6 1 roll } def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% (-u)'=-(u') +/EvalUnaryOp { + 4 -2 roll 4 index (0) eq + { (0) StrConcat 7 -1 roll pop } + { 7 -1 roll 45 eq + { AddSub AddOpPar true } { false } ifelse + 3 1 roll 5 index StrConcat 3 -1 roll { AddClPar } if } ifelse + 2 copy pop 0 6 2 roll GetIntervalNewStr + 7 -2 roll pop pop 2 index 6 index dup 4 index exch sub getinterval exch 6 2 roll +} def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% (number)'=0 +/EvalNumber { ReadNumber (0) 6 2 roll } def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% (u+v)'=u'+v' +/EvalAddSub {% + 7 index dup (0) eq + { pop true }%% du=0 nothing added + { dup length exch 5 index 5 index 3 -1 roll putinterval 4 -1 roll add 3 1 roll false } + ifelse + 5 index dup (0) eq + { pop { (0) } { 4 -2 roll 2 copy pop 0 6 2 roll GetIntervalNewStr } ifelse }%%dv=0 + { exch + { 5 -2 roll 7 index 45 eq { AddSub } if false } %%nothing yet added + { 5 -2 roll 7 index 43 eq%%something yet added + { AddAdd false } { AddSub AddOpPar true } ifelse } + ifelse 11 1 roll + 3 -1 roll StrConcat 10 -1 roll { AddClPar } if + 2 copy pop 0 6 2 roll GetIntervalNewStr } + ifelse + mark 11 -5 roll cleartomark 2 index 6 index dup 4 index exch sub getinterval exch 6 2 roll +} def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% (u*v)' or (u/v)' +/EvalMulDiv { 6 index 42 eq {EvalMul} {EvalDiv} ifelse } def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% (u*v)'=u'*v+u*v' +/EvalMul {% + 4 -2 roll 7 index dup (0) eq + { pop false }%%du=0 + { (1) eq%%du=1 + { false } + { AddOpPar 7 index StrConcat AddClPar AddMul AddOpPar true } ifelse + 3 1 roll 6 index StrConcat 3 -1 roll { AddClPar } if + true }%%du!=0 + ifelse + 5 1 roll 5 index (0) eq + { 5 -1 roll not { (0) StrConcat } if }%%dv=0 + { 5 -1 roll { AddAdd } if + 4 index (1) eq + { 8 index StrConcat } + { AddOpPar 8 index StrConcat AddClPar AddMul AddOpPar 4 index StrConcat AddClPar } + ifelse + }%%dv!=0 + ifelse + 2 copy pop 0 6 2 roll GetIntervalNewStr + mark 11 -5 roll cleartomark 2 index 6 index dup 4 index exch sub getinterval exch 6 2 roll +} def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% (u/v)'=(u'*v-u*v')/v^2 +/EvalDiv {% + 4 -2 roll + 4 index (0) eq%%dv=0 -> u'/v + { 7 index (0) eq { (0) StrConcat } { AddOpPar 7 index StrConcat AddClPar AddDiv 5 index StrConcat } ifelse } + { 7 index dup (0) eq + { pop }%%du=0 + { (1) eq%%du=1 + { false } + { AddOpPar 7 index StrConcat AddClPar AddMul AddOpPar true } ifelse + 3 1 roll 6 index StrConcat 3 -1 roll { AddClPar } if}%%du!=0 + ifelse + AddSub + 4 index (1) eq + { 8 index StrConcat } + { AddOpPar 8 index StrConcat AddClPar AddMul AddOpPar 4 index StrConcat AddClPar } + ifelse + %}%%dv!=0 + 2 copy GetIntervalNewStr 3 1 roll pop 0 AddOpPar 3 -1 roll StrConcat AddClPar + AddDiv AddOpPar 5 index StrConcat AddClPar 2 copy (^2) putinterval 2 add } + ifelse + 2 copy pop 0 6 2 roll GetIntervalNewStr + mark 11 -5 roll cleartomark 2 index 6 index dup 4 index exch sub getinterval exch 6 2 roll +} def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% str1 index str2 -> str1 index +/StrConcat { dup length 4 2 roll 2 copy 6 -1 roll putinterval 3 -1 roll add } bind def +/GetIntervalNewStr { 0 exch getinterval dup length string copy } bind def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% (u^v)'=(u^v)'=u'vu^(v-1)+v'u^(v)ln(u) +/EvalPower {% + 4 -2 roll 7 index (0) eq + {%%if du=0 then (u^v)'=v'ln(u)u^v + 4 index (0) eq + { (0) StrConcat }%%if dv=0 then (u^v)'=0 + { 4 index (1) ne { AddOpPar 4 index StrConcat (\)*) StrConcat } if + 8 index (e) ne { (ln\() StrConcat 8 index StrConcat (\)*) StrConcat } if + AddOpPar 8 index StrConcat (\)^\() StrConcat 5 index StrConcat AddClPar } ifelse + } + {%%du!=0 + 4 index (0) eq + {%%if dv=0 then (u^v)'=vu'u^(v-1) + 5 index dup IsStrNumber + { dup (0) eq + { StrConcat } + { dup dup (1) eq exch (1.0) eq or + { StrConcat } + { StrConcat + 7 index dup (1) ne exch (1.0) ne and%%%dr 09102006 insert du if <> 1 + { (*\() StrConcat 7 index StrConcat (\)) StrConcat } if%%%dr 09102006 + (*\() StrConcat 8 index StrConcat (\)) StrConcat + 5 index dup dup (2) eq exch (2.0) eq or + { pop } { cvr 1 sub 20 string cvs 3 1 roll (^) StrConcat 3 -1 roll StrConcat } ifelse } ifelse } ifelse } + { pop AddOpPar 5 index StrConcat (\)*\() StrConcat 8 index StrConcat (\)^\() StrConcat + 5 index StrConcat (-1\)) StrConcat } ifelse + } + {%%if dv!=0 and du!=0 then (u^v)'=u'vu^(v-1)+v'u^(v)ln(u) + 7 index (1) ne { AddOpPar 7 index StrConcat (\)*) StrConcat } if + AddOpPar 5 index StrConcat (\)*\() StrConcat + 8 index StrConcat (\)^\() StrConcat + 5 index StrConcat (-1\)+\() StrConcat + 4 index (1) ne { 4 index StrConcat (\)*\() StrConcat } if + 8 index StrConcat (\)^\() StrConcat + 5 index StrConcat (\)*ln\() StrConcat + 8 index StrConcat AddClPar + } ifelse + } ifelse + 2 copy pop 0 6 2 roll GetIntervalNewStr + mark 11 -5 roll cleartomark 2 index 6 index dup 4 index exch sub getinterval exch 6 2 roll +} def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% No derivative for factorial ! only cst => null derivative +/EvalFactorial {% DR 09/2011 + 4 index (0) eq + { (0) mark 8 -2 roll cleartomark 2 index 7 index dup 4 index exch sub getinterval exch 6 2 roll } + { DERIVATIVE_ENGINE_ERROR_no_variable_in_factorial } ifelse +} def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% str -> true/false +/IsStrNumber {% + true exch + { dup 48 lt exch dup 57 gt 3 -1 roll or + exch dup 46 ne%%. + exch dup 43 ne%%+ + exch 45 ne%%- + and and and { pop false } if } forall +} def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% literal switch -> func call, vector, variables +/EvalLiteral {% + ReadLiteral dup 40 eq%%% there is an open par -> function call + { pop (EvalFunc_ ) 9 4 index StrConcat 0 exch getinterval cvn cvx exec } + { dup 91 eq%%% there is an open bracket -> vector element + { DERIVATIVE_ENGINE_ERROR_vector_not_yet_implemented } + { pop EvalVariable } + ifelse } + ifelse +} def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% first last parpos Expr[first:parpos-1] -> +/EvalVariable { 2 index Variable eq { (1) } { (0) } ifelse 4 -1 roll exch 6 2 roll } def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% (f(u))'=u'f'(u) +/EvalFunc { + 4 2 roll 4 index (1) ne + { AddOpPar 4 index StrConcat (\)*) StrConcat } if + (Eval ) 4 8 index StrConcat 0 exch getinterval cvn cvx exec + 2 copy pop 0 6 2 roll GetIntervalNewStr + mark 9 -3 roll cleartomark 2 index 6 index dup 4 index exch sub getinterval exch 6 2 roll +} def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Func derivative -> Eval +/EvalFunc_sin {% + PreCommonFunc + { (cos\() StrConcat 5 index StrConcat AddClPar } if + PostCommonFunc } def +/EvalFunc_cos {% + PreCommonFunc + { (\(-sin\() StrConcat 5 index StrConcat (\)\)) StrConcat } if + PostCommonFunc } def +/EvalFunc_tan {% + PreCommonFunc + { dup 0 eq { (1) StrConcat } { 1 sub } ifelse (/cos\() StrConcat 5 index StrConcat (\)^2) StrConcat } if + PostCommonFunc } def +/EvalFunc_asin {% + PreCommonFunc + { (1/sqrt\(1-\() StrConcat 5 index StrConcat (\)^2\)\)) StrConcat } if + PostCommonFunc } def +/EvalFunc_acos {% + PreCommonFunc + { (-1/sqrt\(1-\() StrConcat 5 index StrConcat (\)^2\)\)) StrConcat } if + PostCommonFunc } def +/EvalFunc_atg {% + PreCommonFunc + { (1/\(1+\() StrConcat 5 index StrConcat (\)^2\)\)) StrConcat } if + PostCommonFunc } def +/EvalFunc_floor {% + PreCommonFunc + { dup 0 eq { (1) StrConcat } { 1 sub } ifelse (/\() StrConcat 5 index StrConcat AddClPar } if + PostCommonFunc } def +/EvalFunc_ln {% + PreCommonFunc + { dup 0 eq { (1) StrConcat } { 1 sub } ifelse (/\() StrConcat 5 index StrConcat AddClPar } if + PostCommonFunc } def +/EvalFunc_exp {% + PreCommonFunc + { (exp\() StrConcat 5 index StrConcat AddClPar } if + PostCommonFunc } def +/EvalFunc_EXP {% + PreCommonFunc + { (EXP\() StrConcat 5 index StrConcat AddClPar } if + PostCommonFunc } def +/EvalFunc_sqrt {% + PreCommonFunc + { dup 0 eq { (1) StrConcat } { 1 sub } ifelse (/\(2*sqrt\() StrConcat 5 index StrConcat (\)\)) StrConcat } if + PostCommonFunc } def +/EvalFunc_Fact {% + PreCommonFunc { DERIVATIVE_ENGINE_ERROR_no_variable_expression_in_Fact } if + PostCommonFunc } def +/EvalFunc_sh {% + PreCommonFunc + { (ch\() StrConcat 5 index StrConcat AddClPar } if + PostCommonFunc } def +/EvalFunc_ch {% + PreCommonFunc + { (sh\() StrConcat 5 index StrConcat AddClPar } if + PostCommonFunc } def +/EvalFunc_th {% + PreCommonFunc + { dup 0 eq { (1) StrConcat } { 1 sub } ifelse (/ch\() StrConcat 5 index StrConcat (\)^2) StrConcat } if + PostCommonFunc } def +/EvalFunc_Argsh {% + PreCommonFunc + { (1/sqrt\(1+\() StrConcat 5 index StrConcat (\)^2\)\)) StrConcat } if + PostCommonFunc } def +/EvalFunc_Argch {% + PreCommonFunc + { (1/sqrt\(\() StrConcat 5 index StrConcat (\)^2-1\)\)) StrConcat } if + PostCommonFunc } def +/EvalFunc_Argth {% + PreCommonFunc + { (1/\(1-\() StrConcat 5 index StrConcat (\)^2\)\)) StrConcat } if + PostCommonFunc } def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +/PreCommonFunc { + 1 add NextNonBlankChar pop 3 -1 roll 5 1 roll AnalyzeExpr 1 add NextNonBlankChar pop + 4 2 roll 4 index (0) eq + { (0) StrConcat false } + { 4 index (1) ne { AddOpPar 4 index StrConcat (\)*) StrConcat } if true } ifelse +} def +/PostCommonFunc { + 2 copy pop 0 6 2 roll GetIntervalNewStr + mark 9 -3 roll cleartomark 2 index 6 index dup 4 index exch sub getinterval exch 6 2 roll +} def +/EvalFunc_Derive {% + 1 add ReadNumber cvi 1 add dup cvr log 1 add cvi string cvs + 4 -1 roll pop 5 1 roll 1 add NextNonBlankChar pop AnalyzeExpr 1 add + 4 -2 roll (Derive\() StrConcat 7 -1 roll StrConcat (,) StrConcat 6 -1 roll StrConcat AddClPar + 2 copy pop 0 6 2 roll GetIntervalNewStr 6 -1 roll pop 2 index 6 index dup 4 index exch sub getinterval + exch 6 2 roll } def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% literal switch -> func call, vector, variables +/EvalFunc_Sum {% + 1 add NextNonBlankChar pop + %% read the variable name + ReadLiteral pop 3 -1 roll pop NextNonBlankChar + 44 ne { ANALYZER_ERROR_missing_first_comma_in_Sum } if + %% read the initial value + 1 add NextNonBlankChar pop ReadNumber pop + 2 copy get 44 ne { ANALYZER_ERROR_missing_second_comma_in_Sum } if + %% read the increment value + 1 add NextNonBlankChar pop ReadNumber pop + 2 copy get 44 ne { ANALYZER_ERROR_missing_third_comma_in_Sum } if + %% read the limit value + 1 add NextNonBlankChar pop ReadNumber pop + 2 copy get 44 ne { ANALYZER_ERROR_missing_fourth_comma_in_Sum } if + 1 add NextNonBlankChar pop dup 6 1 roll 3 -1 roll pop AnalyzeExpr 1 add NextNonBlankChar pop + 4 -2 roll 3 index 8 index dup 9 index exch sub getinterval StrConcat + 4 index StrConcat AddClPar + 2 copy pop 0 6 2 roll GetIntervalNewStr + mark 9 -3 roll cleartomark 2 index 6 index dup 4 index exch sub getinterval exch 6 2 roll +} def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% literal switch -> func call, vector, variables +/EvalFunc_IfTE {% + 3 -1 roll pop 1 add NextNonBlankChar pop SkipCond + NextNonBlankChar + 44 ne { ANALYZER_ERROR_missing_first_comma_in_IfTE } if + 1 add NextNonBlankChar pop dup 5 1 roll + AnalyzeExpr NextNonBlankChar + 44 ne { ANALYZER_ERROR_missing_second_comma_in_IfTE } if + 1 add NextNonBlankChar pop + AnalyzeExpr 1 add NextNonBlankChar pop + 4 -2 roll 3 index 10 index dup 11 index exch sub getinterval StrConcat + 6 index StrConcat (,) StrConcat 4 index StrConcat AddClPar + 2 copy pop 0 6 2 roll GetIntervalNewStr + mark 11 -5 roll cleartomark 2 index 6 index dup 4 index exch sub getinterval exch 6 2 roll +} def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% advance in str until a comma is found (no error detection!) +%% str index -> str index' +/SkipCond { { 1 add 2 copy get 44 eq {exit } if } loop } bind def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Convert to radians if trigo function call +%% (name) -> +/TrigoFunc { + dup (cos) eq 1 index (sin) eq or exch (tan) eq or + { /ExpressionVector ExpressionVector aload length Pi /div cvx 180 /mul cvx 5 -1 roll 4 add + array astore def + } if +} def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% No derivative for condition.... +/EvalCondOp { 3 -1 roll pop } bind def +/PutIntervalOneAdd {putinterval 1 add} bind def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Add open parenthesis in string at the given index +%% str index -> str index+1 +/AddOpPar {2 copy (\() PutIntervalOneAdd} bind def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Add close parenthesis in string at the given index +%% str index -> str index+1 +/AddClPar {2 copy (\)) PutIntervalOneAdd} bind def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Add 0 in string at the given index +%% str index -> str index+1 +/AddZero {2 copy (0) PutIntervalOneAdd} bind def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Add open parenthesis in string at the given index +%% str index -> str index+1 +/AddMul {2 copy (*) PutIntervalOneAdd} bind def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Add open parenthesis in string at the given index +%% str index -> str index+1 +/AddDiv {2 copy (/) PutIntervalOneAdd} bind def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Add a plus sign in string at the given index +%% str index -> str index+1 +/AddAdd {2 copy (+) PutIntervalOneAdd} bind def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Add a minus sign in string at the given index +%% str index -> str index+1 +/AddSub {2 copy (-) PutIntervalOneAdd} bind def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Add a pipe sign in string at the given index +%% str index -> str index+1 +/AddPipe {2 copy (|) PutIntervalOneAdd} bind def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% H O O K S +/AnalyzePreHook { dup 5 1 roll } bind def +/PreEvalHook {} def +/AnalyzePostHook { 7 -1 roll pop } bind def +/AnalyzeListOfEPostHook { 6 -1 roll mark 6 1 roll cleartomark } bind def +/RollOp { 5 1 roll } bind def +end%%%tx@CoreAnalyzerDict +/tx@AddMathFunc 12 dict def tx@AddMathFunc begin +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% NEW FUNC +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% function arcsine in radians asin(x)=atan(x/sqrt(1-x^2)) +%% x -> theta +/asin {% + dup abs 1 gt { EQDFasinrangeerror } if + dup dup dup mul 1 exch sub sqrt atan exch 0 lt { 360 sub } if DegtoRad +} def +%% function arccosine in radians acos(x)=atan(sqrt(1-x^2)/x) +%% x -> theta +/acos {% + dup abs 1 gt { EQDFacosrangeerror } if + dup dup mul 1 exch sub sqrt exch atan DegtoRad +} def +%% function arctangent in radians +%% x -> theta +/atg { 1 atan dup 90 gt { 360 sub } if DegtoRad } bind def +%% HYPERBOLIC FUNCTIONS +/sh { dup Ex exch neg Ex sub 2 div } def +/ch { dup Ex exch neg Ex add 2 div } def +/th { dup sh exch ch div } def +/Argsh { dup dup mul 1 add sqrt add ln } def +/Argch { dup dup mul 1 sub sqrt add ln } def +/Argth { dup 1 add exch 1 exch sub div ln 2 div } def +%% modified exponential funtion for 0 +%% x n -> x^n +/Exp { dup 0 eq { pop pop 1 } { exp } ifelse } bind def +%% modified exponential funtion for 0 +%% x -> e^x +/Ex { Euler exch exp } bind def +%% +%% factorial function +%% n -> n! +/Fact { 1 exch 2 exch 1 exch { mul } for } bind def +/fact { Fact } bind def +/! { Fact } bind def +end +% +% END pst-algparser.pro + +%%EndProcSet +%%BeginProcSet: pst-tools.pro 0 0 +% $Id: pst-tools.pro 842 2013-10-26 10:25:41Z herbert $ +% +%% PostScript tools prologue for pstricks.tex. +%% Version 0.06, 2017/12/03 +%% +%% This program can be redistributed and/or modified under the terms +%% of the LaTeX Project Public License Distributed from CTAN archives +%% in directory macros/latex/base/lppl.txt. +% +% +/Pi2 1.57079632679489661925640 def +/factorial { % n on stack, returns n! + dup 0 eq { 1 }{ + dup 1 gt { dup 1 sub factorial mul } if } + ifelse } def +% +/MoverN { % m n on stack, returns the binomial coefficient m over n + 2 dict begin + /n exch def /m exch def + n 0 eq { 1 }{ + m n eq { 1 }{ + m factorial n factorial m n sub factorial mul div } ifelse } ifelse + end +} def +% +/ps@ReverseOrderOfPoints { % on stack [P1 P2 P3 ...Pn]=>[Pn,Pn-1,...,P2,P1] + 5 dict begin % all local + aload length /n ED % number of coors + n 2 div cvi /m ED % number of Points + /n1 n def + m { n1 2 roll /n1 n1 2 sub def } repeat + n array astore + end +} def +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% subroutines for complex numbers, given as an array [a b] +% which is a+bi = Real+i Imag +% +/cxadd { % [a1 b1] [a2 b2] = [a1+a2 b1+b2] + dup 0 get % [a1 b1] [a2 b2] a2 + 3 -1 roll % [a2 b2] a2 [a1 b1] + dup 0 get % [a2 b2] a2 [a1 b1] a1 + 3 -1 roll % [a2 b2] [a1 b1] a1 a2 + add % [a2 b2] [a1 b1] a1+a2 + 3 1 roll % a1+a2 [a2 b2] [a1 b1] + 1 get % a1+a2 [a2 b2] b1 + exch 1 get % a1+a2 b1 b2 + add 2 array astore +} def +% +/cxneg { % [a b] + dup 1 get % [a b] b + exch 0 get % b a + neg exch neg % -a -b + 2 array astore +} def +% +/cxsub { cxneg cxadd } def % same as negative addition +% +% [a1 b1][a2 b2] = [a1a2-b1b2 a1b2+b1a2] = [a3 b3] +/cxmul { % [a1 b1] [a2 b2] + dup 0 get % [a1 b1] [a2 b2] a2 + exch 1 get % [a1 b1] a2 b2 + 3 -1 roll % a2 b2 [a1 b1] + dup 0 get % a2 b2 [a1 b1] a1 + exch 1 get % a2 b2 a1 b1 + dup % a2 b2 a1 b1 b1 + 5 -1 roll dup % b2 a1 b1 b1 a2 a2 + 3 1 roll mul % b2 a1 b1 a2 b1a2 + 5 -2 roll dup % b1 a2 b1a2 b2 a1 a1 + 3 -1 roll dup % b1 a2 b1a2 a1 a1 b2 b2 + 3 1 roll mul % b1 a2 b1a2 a1 b2 a1b2 + 4 -1 roll add % b1 a2 a1 b2 b3 + 4 2 roll mul % b1 b2 b3 a1a2 + 4 2 roll mul sub % b3 a3 + exch 2 array astore +} def +% +% [a b]^2 = [a^2-b^2 2ab] = [a2 b2] +/cxsqr { % [a b] square root + dup 0 get exch 1 get % a b + dup dup mul % a b b^2 + 3 -1 roll % b b^2 a + dup dup mul % b b^2 a a^2 + 3 -1 roll sub % b a a2 + 3 1 roll mul 2 mul % a2 b2 + 2 array astore +} def +% +/cxsqrt { % [a b] +% dup cxnorm sqrt /r exch def +% cxarg 2 div RadtoDeg dup cos r mul exch sin r mul cxmake2 + cxlog % log[a b] + 2 cxrdiv % log[a b]/2 + aload pop exch % b a + 2.781 exch exp % b exp(a) + exch cxconv exch % [Re +iIm] exp(a) + cxrmul % +} def +% +/cxarg { % [a b] + aload pop % a b + exch atan % arctan b/a + DegtoRad % arg(z)=atan(b/a) +} def +% +% log[a b] = [a^2-b^2 2ab] = [a2 b2] +/cxlog { % [a b] + dup % [a b][a b] + cxnorm % [a b] |z| + log % [a b] log|z| + exch % log|z|[a b] + cxarg % log|z| Theta + cxmake2 % [log|z| Theta] +} def +% +% square of magnitude of complex number +/cxnorm2 { % [a b] + dup 0 get exch 1 get % a b + dup mul % a b^2 + exch dup mul add % a^2+b^2 +} def +% +/cxnorm { % [a b] + cxnorm2 sqrt +} def +% +/cxconj { % conjugent complex + dup 0 get exch 1 get % a b + neg 2 array astore % [a -b] +} def +% +/cxre { 0 get } def % real value +/cxim { 1 get } def % imag value +% +% 1/[a b] = ([a -b]/(a^2+b^2) +/cxrecip { % [a b] + dup cxnorm2 exch % n2 [a b] + dup 0 get exch 1 get % n2 a b + 3 -1 roll % a b n2 + dup % a b n2 n2 + 4 -1 roll exch div % b n2 a/n2 + 3 1 roll div % a/n2 b/n2 + neg 2 array astore +} def +% +/cxmake1 { 0 2 array astore } def % make a complex number, real given +/cxmake2 { 2 array astore } def % dito, both given +% +/cxdiv { cxrecip cxmul } def +% +% multiplikation by a real number +/cxrmul { % [a b] r + exch aload pop % r a b + 3 -1 roll dup % a b r r + 3 1 roll mul % a r b*r + 3 1 roll mul % b*r a*r + exch 2 array astore % [a*r b*r] +} def +% +% division by a real number +/cxrdiv { % [a b] r + 1 exch div % [a b] 1/r + cxrmul +} def +% +% exp(i theta) = cos(theta)+i sin(theta) polar<->cartesian +/cxconv { % theta + RadtoDeg dup sin exch cos cxmake2 +} def + +%%%%% ### bubblesort ### +%% syntax : array bubblesort --> array2 trie par ordre croissant +%% code de Bill Casselman +%% http://www.math.ubc.ca/people/faculty/cass/graphics/text/www/ +/bubblesort { % on stack must be an array [ ... ] +4 dict begin + /a exch def + /n a length 1 sub def + n 0 gt { + % at this point only the n+1 items in the bottom of a remain to + % the sorted largest item in that blocks is to be moved up into + % position n + n { + 0 1 n 1 sub { + /i exch def + a i get a i 1 add get gt { + % if a[i] > a[i+1] swap a[i] and a[i+1] + a i 1 add + a i get + a i a i 1 add get + % set new a[i] = old a[i+1] + put + % set new a[i+1] = old a[i] + put + } if + } for + /n n 1 sub def + } repeat + } if + a % return the sorted array +end +} def +% +/concatstringarray{ % [(a) (b) ... (z)] --> (ab...z) 20100422 + 0 1 index { length add } forall + string + 0 3 2 roll + { 3 copy putinterval length add }forall + pop +} bind def +% +/concatstrings{ % (a) (b) -> (ab) + 1 dict begin + exch dup length + 2 index length add string + dup dup 4 2 roll copy length + 4 -1 roll putinterval + end +} def +% +/reversestring { % (aBC) -> (CBa) + 5 dict begin + /str exch def + /L str length def + /strTemp L string def + /i 0 def + L { + /I L 1 sub i sub def + strTemp i str I 1 getinterval putinterval + /i i 1 add def + } repeat + strTemp + end +} def +% +/concatarray{ % [a c] [b d] -> [a c b d] + 2 dict begin + /a2 exch def + /a1 exch def + [ a1 aload pop a2 aload pop ] + end +} def +% +/dot2comma {% on stack a string (...) + 2 dict begin + /Output exch def + 0 1 Output length 1 sub { + /Index exch def + Output Index get 46 eq { Output Index 44 put } if + } for + Output + end +} def +% +/rightTrim { % on stack the string and the character number to be stripped + /charNo exch def + dup + length 1 sub -1 0 { + /i exch def dup i get charNo ne { exit } if + } for + 0 i 1 add getinterval + dup length string copy +} bind def % leaves the stripped string on the stack + +/psStringwidth /stringwidth load def +/psShow /show load def + +%/stringwidth{ 32 rightTrim psStringwidth } bind def + +%/show { 32 rightTrim psShow } bind def +%-----------------------------------------------------------------------------% + +/pgffunctions { + /pgfsc{}bind def% stroke color is empty by default + /pgffc{}bind def% fill color is empty by default + /pgfstr{stroke}bind def% + /pgffill{fill}bind def% + /pgfeofill{eofill}bind def% + /pgfe{a dup 0 rlineto exch 0 exch rlineto neg 0 rlineto closepath}bind def% rectangle + /pgfw{setlinewidth}bind def% setlinewidth + /pgfs{save pgfpd 72 Resolution div 72 VResolution div neg scale + magscale{1 DVImag div dup scale}if + pgfx neg pgfy neg translate pgffoa .setopacityalpha}bind def% save + /pgfr{pgfsd restore}bind def %restore + userdict begin% + /pgfo{pgfsd /pgfx currentpoint /pgfy exch def def @beginspecial}bind def %open + /pgfc{newpath @endspecial pgfpd}bind def %close + /pgfsd{globaldict /pgfdelta /delta where {pop delta} {0} ifelse put}bind def% save delta + /pgfpd{/delta globaldict /pgfdelta get def}bind def % put delta + /.setopacityalpha where {pop} {/.setopacityalpha{pop}def} ifelse % install .setopacityalpha + /.pgfsetfillopacityalpha{/pgffoa exch def + /pgffill{gsave pgffoa .setopacityalpha fill 1 .setopacityalpha newpath fill grestore newpath}bind def + /pgfeofill{gsave pgffoa .setopacityalpha eofill 1 .setopacityalpha newpath eofill grestore newpath}bind def}bind def + /.pgfsetstrokeopacityalpha{/pgfsoa exch def /pgfstr{gsave pgfsoa .setopacityalpha stroke grestore newpath}bind def}bind def + /pgffoa 1 def + /pgfsoa 1 def + end +} def +%-----------------------------------------------------------------------------% +% END pst-tools.pro + +%%EndProcSet +%%BeginProcSet: pst-dots.pro 0 0 +% $Id: pst-dots.pro 446 2017-04-19 11:40:55Z herbert $ +% +%% PostScript prologue for pstricks.tex. +%% Version 2.02, 2009/06/16 +%% +%% For distribution, see pstricks.tex. +%% +%% Timothy Van Zandt +%% +%% This program can be redistributed and/or modified under the terms +%% of the LaTeX Project Public License Distributed from CTAN archives +%% in directory macros/latex/base/lppl.txt. +%% +%% Modified by Etienne Riga - Dec. 16, 1999 +%% Modified by Etienne Riga - 2005/01/01 (er) +%% to add /Diamond, /SolidDiamond and /BoldDiamond +%% Modified by Herbert Voss (hv) - 2008/04/17 +% +10 dict dup begin % hold local + /FontType 3 def + /FontMatrix [.001 0 0 .001 0 0] def +% /FontBBox [-571.5 -742.5 571.5 742.5] def % changed to next line 20060616 hv + /FontBBox [-1000 -1000 1000 1000] def % See end of file in /BuildGlyph + /Encoding 256 array def + 0 1 255 {Encoding exch /.notdef put} for % fill the array with /.notdef + Encoding % replace with given dot names + dup (b) 0 get /Bullet put % get the numerical position of b in ASCII +% % and save /Bullet at this place in Encoding + dup (c) 0 get /Circle put + dup (C) 0 get /BoldCircle put % 67 + dup (u) 0 get /SolidTriangle put + dup (t) 0 get /Triangle put + dup (T) 0 get /BoldTriangle put + dup (r) 0 get /SolidSquare put + dup (s) 0 get /Square put + dup (S) 0 get /BoldSquare put + dup (q) 0 get /SolidPentagon put + dup (p) 0 get /Pentagon put + dup (P) 0 get /BoldPentagon put +%%% + dup (k) 0 get /Asterisk put + dup (K) 0 get /BoldAsterisk put + dup (J) 0 get /SolidAsterisk put + dup (h) 0 get /Hexagon put + dup (H) 0 get /BoldHexagon put + dup (G) 0 get /SolidHexagon put + dup (f) 0 get /Octogon put % 2008-04-18 hv + dup (F) 0 get /BoldOctogon put % 2008-04-18 hv + dup (g) 0 get /SolidOctogon put % 2008-04-18 hv + dup (a) 0 get /Add put + dup (A) 0 get /BoldAdd put % 65 + dup (x) 0 get /Mul put + dup (X) 0 get /BoldMul put + dup (m) 0 get /Oplus put + dup (M) 0 get /BOplus put + dup (e) 0 get /SolidOplus put + dup (n) 0 get /Otimes put + dup (N) 0 get /BOtimes put + dup (E) 0 get /SolidOtimes put + dup (i) 0 get /Bar put + dup (I) 0 get /BoldBar put + dup (l) 0 get /SolidDiamond put + dup (d) 0 get /Diamond put + (D) 0 get /BoldDiamond put +%%% +/CharProcs 47 dict def +CharProcs begin + /CirclePath {0 0 500 0 360 arc closepath} def + /Bullet {CirclePath fill} def + /Circle {CirclePath .9 .9 scale CirclePath eofill} def + /BoldCircle {CirclePath .8 .8 scale CirclePath eofill} def + /TrianglePath {0 660 moveto -571.5 -330 lineto 571.5 -330 lineto closepath} def + /SolidTriangle {TrianglePath fill} def + /Triangle {TrianglePath .85 .85 scale TrianglePath eofill} def + /BoldTriangle {TrianglePath .7 .7 scale TrianglePath eofill} def + /SquarePath {-450 450 moveto 450 450 lineto 450 -450 lineto -450 -450 lineto closepath} def + /SolidSquare {SquarePath fill} def + /Square {SquarePath .89 .89 scale SquarePath eofill} def + /BoldSquare {SquarePath .78 .78 scale SquarePath eofill} def + /PentagonPath { + -337.8 -465 moveto 337.8 -465 lineto 546.6 177.6 lineto + 0 574.7 lineto -546.6 177.6 lineto closepath + } def + /SolidPentagon {PentagonPath fill} def + /Pentagon {PentagonPath .89 .89 scale PentagonPath eofill} def + /BoldPentagon {PentagonPath .78 .78 scale PentagonPath eofill} def +%-------------- hv begin 2004/07/25 from: er 2003/03/24 + /HexagonPath { + 0 550 moveto -476 275 lineto -476 -275 lineto + 0 -550 lineto 476 -275 lineto 476 275 lineto closepath + } def + /SolidHexagon {HexagonPath fill} def + /Hexagon {HexagonPath .89 .89 scale HexagonPath eofill} def + /BoldHexagon {HexagonPath .79 .79 scale HexagonPath eofill} def +% 2008-04-18 hv + /OctogonPath { + 550 dup 22.5 tan mul dup neg dup add /xMove exch def + exch moveto 7 { xMove 0 rlineto 45 rotate } repeat closepath } def + /SolidOctogon { OctogonPath fill } def + /Octogon { OctogonPath .89 .89 scale OctogonPath eofill } def + /BoldOctogon { OctogonPath .79 .79 scale OctogonPath eofill } def +% + /AsteriskPath { + 20 0 moveto 10 250 180 500 0 500 curveto + -180 500 -10 250 -20 0 curveto closepath + } def + /Asterisk { + AsteriskPath 60 rotate AsteriskPath 60 rotate AsteriskPath + 60 rotate AsteriskPath 60 rotate AsteriskPath 60 rotate AsteriskPath fill + } def +% + /Basterp {50 250 220 500 0 500 curveto -220 500 -50 250 -50 30 cos 100 mul curveto} def + /BoldAsteriskPath { + 50 30 cos 100 mul moveto Basterp + 60 rotate Basterp 60 rotate Basterp + 60 rotate Basterp 60 rotate Basterp + 60 rotate Basterp closepath + } def + /BoldAsterisk {BoldAsteriskPath fill} def + /SolidAsterisk {CirclePath .9 .9 scale BoldAsteriskPath eofill} def + /CrossPath { + 40 550 moveto -40 550 lineto -40 40 lineto -550 40 lineto + -550 -40 lineto -40 -40 lineto -40 -550 lineto 40 -550 lineto + 40 -40 lineto 550 -40 lineto 550 40 lineto 40 40 lineto closepath + } def + /BoldCrossPath {80 550 moveto -80 550 lineto -80 80 lineto -550 80 lineto + -550 -80 lineto -80 -80 lineto -80 -550 lineto 80 -550 lineto + 80 -80 lineto 550 -80 lineto 550 80 lineto 80 80 lineto closepath + } def + /Add {CrossPath fill} def + /Mul {45 rotate CrossPath fill} def + /BoldAdd {BoldCrossPath fill} def + /BoldMul {45 rotate BoldCrossPath fill} def + /Oplus {CirclePath .9 .9 scale CirclePath eofill .775 .775 scale CrossPath fill } def + /SolidOplus {CirclePath .775 .775 scale BoldCrossPath eofill} def + /BOplus {CirclePath .8 .8 scale CirclePath eofill .775 .775 scale BoldCrossPath fill} def + /Otimes {CirclePath .9 .9 scale CirclePath eofill 45 rotate .775 .775 scale CrossPath fill} def + /BOtimes {CirclePath .8 .8 scale CirclePath eofill 45 rotate .775 .775 scale BoldCrossPath fill } def + /SolidOtimes {CirclePath 45 rotate .775 .775 scale BoldCrossPath eofill} def + /BarPath {40 660 moveto -40 660 lineto -40 -660 lineto 40 -660 lineto closepath} def + /Bar {BarPath fill} def + /BoldBarPath {80 660 moveto -80 660 lineto -80 -660 lineto 80 -660 lineto closepath} def + /BoldBar {BoldBarPath fill} def + /DiamondPath {0 742.5 moveto -428.5 0 lineto 0 -742.5 lineto 428.5 0 lineto closepath} def + /SolidDiamond {DiamondPath fill} def + /Diamond {DiamondPath .865 .865 scale DiamondPath eofill} def + /BoldDiamond {DiamondPath .73 .73 scale DiamondPath eofill} def +%%% + /.notdef { } def +end +% +/BuildGlyph { + exch + begin +% Metrics 1 index get exec 0 + 0 0 +% BBoxes 3 index get exec + -1000 -1000 1000 1000 +% -571.5 -742.5 571.5 742.5 + setcachedevice + CharProcs begin load exec end + end +} def +% +/BuildChar { + 1 index /Encoding get exch get + 1 index /BuildGlyph get exec +} bind def +% +end +/PSTricksDotFont exch definefont pop +% +%% end + +%%EndProcSet +%%BeginProcSet: pst-node.pro 0 0 +% $Id: pst-node.pro 395 2017-03-21 09:24:38Z herbert $ +%% +%% PostScript prologue for pst-node.tex. +%% Version 1.15, 2014/01/27. +%% +%% This program can be redistributed and/or modified under the terms +%% of the LaTeX Project Public License Distributed from CTAN archives +%% in directory macros/latex/base/lppl.txt. +% +%%%% EMPTY lines are not alowed!!! Problem with pst-eps -> \par +% +% +tx@Dict begin % from main pstricks dict + /T /translate load def + /CP /currentpoint load def +% /startGlobal { true setglobal globaldict begin } bind def +% /endGlobal { end false setglobal } bind def +end +/tx@NodeDict 400 dict def tx@NodeDict begin +/NewNode { % on stack: { x y } boolean N@name type InitXnode + gsave + NodeScale % a bugfix for xelatex, it's empty for dvips + /next exch def % { x y } boolean N@name type + dict dup % { x y } boolean N@name dict dict + 3 1 roll def % { x y } boolean dict N@name dict def + exch { dup 3 1 roll def } if % { x y } dict boolean + begin % { x y } dict begin + tx@Dict begin + STV CP T exec % set scaling + end + /NodeMtrx CM def % save CM + next % InitXNode + end + grestore +} def +% +/InitPnode { + /Y ED /X ED + /NodePos { NodeSep Cos mul NodeSep Sin mul } def +} def +% +/InitCnode { + /r ED /Y ED /X ED + /NodePos { NodeSep r add dup Cos mul exch Sin mul } def +} def +% +/GetRnodePos { + Cos 0 gt { /dx r NodeSep add def } { /dx l NodeSep sub def } ifelse + Sin 0 gt { /dy u NodeSep add def } { /dy d NodeSep sub def } ifelse + dx Sin mul abs dy + Cos mul abs gt { dy Cos mul Sin div dy } { dx dup Sin mul Cos Div } ifelse +} def +% +/InitRnode { + /Y ED /X ED X sub /r ED /l X neg def Y add neg /d ED Y sub /u ED + /NodePos { GetRnodePos } def +} def +% +/DiaNodePos { + w h mul w Sin mul abs h Cos mul abs add Div NodeSep add dup + Cos mul exch Sin mul +} def +% +/TriNodePos { + Sin s lt + { d NodeSep sub dup Cos mul Sin Div exch } + { w h mul w Sin mul h Cos abs mul add Div + NodeSep add dup Cos mul exch Sin mul + } ifelse +} def +% +/InitTriNode { + sub 2 div exch + 2 div exch + 2 copy T + 2 copy 4 index index /d ED + pop pop pop pop + -90 mul rotate + /NodeMtrx CM def + /X 0 def /Y 0 def + d sub abs neg /d ED + d add /h ED + 2 div h mul h d sub Div /w ED + /s d w Atan sin def + /NodePos { TriNodePos } def +} def +% +/OvalNodePos { + /ww w NodeSep add def + /hh h NodeSep add def + Sin ww mul Cos hh mul Atan dup cos ww mul exch sin hh mul +} def +% +/GetCenter { begin X Y NodeMtrx transform CM itransform end } def +% +/XYPos { + dup sin exch cos Do + /Cos ED /Sin ED /Dist ED + Cos 0 gt + { Dist Dist Sin mul Cos div } + { Cos 0 lt + { Dist neg Dist Sin mul Cos div neg } + { 0 Dist Sin mul } ifelse + } ifelse + Do +} def +% +/GetEdge { + dup 0 eq + { pop begin 1 0 NodeMtrx dtransform + CM idtransform + exch atan sub + dup + sin /Sin ED + cos /Cos ED + /NodeSep ED + NodePos NodeMtrx dtransform CM idtransform end } + { 1 eq {{exch}} {{}} ifelse /Do ED pop XYPos } ifelse +} def +% +/AddOffset { + 1 index 0 eq + { pop pop } + { 2 copy 5 2 roll cos mul add 4 1 roll sin mul sub exch } ifelse +} def +% +/GetEdgeA { + NodeSepA AngleA NodeA NodeSepTypeA GetEdge + OffsetA AngleA AddOffset + yA add /yA1 ED + xA add /xA1 ED +} def +% +/GetEdgeB { + NodeSepB AngleB NodeB NodeSepTypeB GetEdge + OffsetB AngleB AddOffset + yB add /yB1 ED + xB add /xB1 ED +} def +% +/GetArmA { + ArmTypeA 0 eq + { /xA2 ArmA AngleA cos mul xA1 add def + /yA2 ArmA AngleA sin mul yA1 add def } + { ArmTypeA 1 eq {{exch}} {{}} ifelse + /Do ED + ArmA AngleA XYPos OffsetA AngleA AddOffset + yA add /yA2 ED + xA add /xA2 ED } ifelse +} def +% +/GetArmB { + ArmTypeB 0 eq + { /xB2 ArmB AngleB cos mul xB1 add def + /yB2 ArmB AngleB sin mul yB1 add def } + { ArmTypeB 1 eq {{exch}} {{}} ifelse + /Do ED + ArmB AngleB XYPos OffsetB AngleB AddOffset + yB add /yB2 ED + xB add /xB2 ED } ifelse +} def +% +/InitNC { + /b ED /a ED % second and first node + /NodeSepTypeB ED /NodeSepTypeA ED + /NodeSepB ED /NodeSepA ED + /OffsetB ED /OffsetA ED + tx@NodeDict a known tx@NodeDict b known and dup { + /NodeA a load def + /NodeB b load def + NodeA GetCenter /yA ED /xA ED + NodeB GetCenter /yB ED /xB ED } if +} def +% +/LPutLine { + 4 copy + 3 -1 roll sub neg 3 1 roll sub Atan /NAngle ED + 1 t sub mul + 3 1 roll 1 t sub mul + 4 1 roll t mul add /Y ED + t mul add /X ED +} def +% +/LPutLines { + mark LPutVar counttomark 2 div 1 sub /n ED +% t floor dup n gt + t floor dup n ge % to allow npos<= hv 2008-08-14 + { pop n 1 sub /t 1 def } { dup t sub neg /t ED } ifelse + cvi 2 mul { pop } repeat + LPutLine + cleartomark +} def +% +/BezierMidpoint { + /y3 ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /y0 ED /x0 ED /t ED + /cx x1 x0 sub 3 mul def + /cy y1 y0 sub 3 mul def + /bx x2 x1 sub 3 mul cx sub def + /by y2 y1 sub 3 mul cy sub def + /ax x3 x0 sub cx sub bx sub def + /ay y3 y0 sub cy sub by sub def + ax t 3 exp mul bx t t mul mul add + cx t mul add x0 add ay t 3 exp mul + by t t mul mul add cy t mul add + y0 add 3 ay t t mul mul mul 2 + by t mul mul add cy add 3 ax t t mul mul mul + 2 bx t mul mul add cx add atan /NAngle ED + /Y ED /X ED +} def +% +/HPosBegin { yB yA ge { /t 1 t sub def } if /Y yB yA sub t mul yA add def +} def +% +/HPosEnd { /X Y yyA sub yyB yyA sub Div xxB xxA sub mul xxA add def + /NAngle yyB yyA sub xxB xxA sub Atan def +} def +/HPutLine { HPosBegin /yyA ED /xxA ED /yyB ED /xxB ED HPosEnd } def +% +/HPutLines { HPosBegin yB yA ge + { /check { le } def } { /check { ge } def } ifelse + /xxA xA def + /yyA yA def + mark xB yB LPutVar + { dup Y check { exit } { /yyA ED /xxA ED } ifelse } + loop + /yyB ED /xxB ED cleartomark HPosEnd +} def +% +/VPosBegin { + xB xA lt { /t 1 t sub def } if /X xB xA sub t mul xA add def +} def +% +/VPosEnd { /Y X xxA sub xxB xxA sub Div yyB yyA sub mul yyA add def +/NAngle yyB yyA sub xxB xxA sub Atan def } def +/VPutLine { VPosBegin /yyA ED /xxA ED /yyB ED /xxB ED VPosEnd } def +/VPutLines { VPosBegin xB xA ge { /check { le } def } { /check { ge } def +} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { 1 index X check { +exit } { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark +VPosEnd } def +/HPutCurve { gsave newpath /SaveLPutVar /LPutVar load def LPutVar 8 -2 +roll moveto curveto flattenpath /LPutVar [ {} {} {} {} pathforall ] cvx +def grestore exec /LPutVar /SaveLPutVar load def +} def +% +/NCCoor { + /AngleA yB yA sub xB xA sub Atan def + /AngleB AngleA 180 add def + GetEdgeA GetEdgeB + /LPutVar [ xB1 yB1 xA1 yA1 ] cvx def + /LPutPos { LPutVar LPutLine } def + /HPutPos { LPutVar HPutLine } def + /VPutPos { LPutVar VPutLine } def + LPutVar +} def +% +/NCLine { + NCCoor + tx@Dict begin + ArrowA CP 4 2 roll + ArrowB + lineto pop pop + end +} def +% +/NCLines { + false NArray + n 0 eq + { NCLine } + { 2 copy yA sub exch xA sub Atan /AngleA ED + n 2 mul dup index exch index yB sub exch xB sub + Atan /AngleB ED + GetEdgeA GetEdgeB + /LPutVar [ xB1 yB1 n 2 mul 4 add 4 roll xA1 yA1 ] cvx def + mark LPutVar + tx@Dict begin false Line end + /LPutPos { LPutLines } def + /HPutPos { HPutLines } def + /VPutPos { VPutLines } def + } ifelse +} def +% +/NCCurve { + GetEdgeA + GetEdgeB + xA1 xB1 sub yA1 yB1 sub Pyth 2 div dup 3 -1 +roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def /ArmTypeB 0 def GetArmA +GetArmB xA2 yA2 xA1 yA1 tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict +begin ArrowB end curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 ] +cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines +} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } def +% +/NCAngles { + GetEdgeA GetEdgeB GetArmA GetArmB + /mtrx AngleA matrix rotate def + xA2 yA2 mtrx transform pop + xB2 yB2 mtrx transform exch pop + mtrx itransform + /y0 ED /x0 ED + mark ArmB 0 ne { xB1 yB1 } if + xB2 yB2 x0 y0 xA2 yA2 + ArmA 0 ne { xA1 yA1 } if + tx@Dict begin false Line end + /LPutVar [ xB1 yB1 xB2 yB2 x0 y0 xA2 yA2 xA1 yA1 ] cvx def + /LPutPos { LPutLines } def + /HPutPos { HPutLines } def + /VPutPos { VPutLines } def } def +% +/NCAngle { + GetEdgeA GetEdgeB GetArmB + /mtrx AngleA matrix rotate def + xB2 yB2 mtrx itransform pop xA1 yA1 mtrx itransform exch pop mtrx transform + /y0 ED /x0 ED + mark + ArmB 0 ne { xB1 yB1 } if + xB2 yB2 x0 y0 xA1 yA1 + tx@Dict begin false Line end + /LPutVar [ xB1 yB1 xB2 yB2 x0 y0 xA1 yA1 ] cvx def + /LPutPos { LPutLines } def + /HPutPos { HPutLines } def + /VPutPos { VPutLines } def +} def +% +/NCBar { + GetEdgeA GetEdgeB GetArmA GetArmB + /mtrx AngleA matrix rotate def + xA2 yA2 mtrx itransform pop + xB2 yB2 mtrx itransform pop + sub dup 0 mtrx transform + 3 -1 roll 0 gt + { /yB2 exch yB2 add def /xB2 exch xB2 add def } + { /yA2 exch neg yA2 add def /xA2 exch neg xA2 add def } ifelse + mark + ArmB 0 ne { xB1 yB1 } if + xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if + tx@Dict begin false Line end + /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def + /LPutPos { LPutLines } def + /HPutPos { HPutLines } def + /VPutPos { VPutLines } def +} def +% +/NCDiag { + /lineAngle ED + GetEdgeA GetEdgeB GetArmA GetArmB mark + lineAngle abs 0 gt { + /xTemp xA2 10 add def + /yTemp yA2 lineAngle dup sin exch cos div 10 mul add def + /dY1 yTemp yA2 sub def + /dX1 xTemp xA2 sub def + /dY2 yB2 yB1 sub def + /dX2 xB2 xB1 sub def + dX1 abs 0.01 lt { + /m2 dY2 dX2 div def + /xB2 xA2 def + /yB2 xA2 xB1 sub m2 mul yB1 add def + }{ + dX2 abs 0.01 lt { + /m1 dY1 dX1 div def + /xB2 xB1 def + /yB2 xB1 xA2 sub m1 mul yA2 add def + }{% + /m1 dY1 dX1 div def + /m2 dY2 dX2 div def + /xB2 m1 xA2 mul m2 xB1 mul sub yA2 sub yB1 add m1 m2 sub div def + /yB2 xB2 xA2 sub m1 mul yA2 add def + } ifelse + } ifelse + } if + ArmB 0 ne { xB1 yB1 } if + xB2 yB2 xA2 yA2 + ArmA 0 ne { xA1 yA1 } if + tx@Dict begin false Line end + /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def + /LPutPos { LPutLines } def + /HPutPos { HPutLines } def + /VPutPos { VPutLines } def +% +% GetEdgeA GetEdgeB GetArmA GetArmB mark +% ArmB 0 ne { xB1 yB1 } if +% xB2 yB2 xA2 yA2 +% ArmA 0 ne { xA1 yA1 } if +% tx@Dict begin false Line end +% /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def +% /LPutPos { LPutLines } def +% /HPutPos { HPutLines } def +% /VPutPos { VPutLines } def +} def +% +/NCDiagg { + /lineAngle ED + GetEdgeA GetArmA + lineAngle abs 0 gt + { lineAngle } + { yB yA2 sub xB xA2 sub Atan 180 add } ifelse + /AngleB ED + GetEdgeB mark + lineAngle abs 0 gt { + /dY2 yA2 yA1 sub def + /dX2 xA2 xA1 sub def + lineAngle abs 90 eq { + /m2 dY2 dX2 div def + /yA2 xB xA2 sub m2 mul yA2 add def + /xA2 xB def + }{ + /m1 lineAngle dup sin exch cos div def % tan alpha + dX2 abs 0.01 lt { + /yA2 xA1 xB sub m1 mul yB add def + /xA2 xA1 def + }{% + /m2 dY2 dX2 div def + /xA2 m1 xB mul m2 xA2 mul sub yA2 add yB sub m1 m2 sub div def + /yA2 xA2 xB sub m1 mul yB add def + } ifelse + } ifelse + } if + xB1 yB1 xA2 yA2 + ArmA 0 ne { xA1 yA1 } if + tx@Dict begin false Line end + /LPutVar [ xB1 yB1 xA2 yA2 xA1 yA1 ] cvx def + /LPutPos { LPutLines } def + /HPutPos { HPutLines } def + /VPutPos { VPutLines } def +% +% GetEdgeA GetArmA +% yB yA2 sub xB xA2 sub Atan 180 add /AngleB ED +% GetEdgeB +% mark +% xB1 yB1 xA2 yA2 +% ArmA 0 ne { xA1 yA1 } if +% tx@Dict begin false Line end +% /LPutVar [ xB1 yB1 xA2 yA2 xA1 yA1 ] cvx def +% /LPutPos { LPutLines } def +% /HPutPos { HPutLines } def +% /VPutPos { VPutLines } def +} def +% +/NCLoop { + GetEdgeA GetEdgeB GetArmA GetArmB + /mtrx AngleA matrix rotate def + xA2 yA2 mtrx transform loopsize add /yA3 ED /xA3 ED + /xB3 xB2 yB2 mtrx transform pop def + xB3 yA3 mtrx itransform /yB3 ED /xB3 ED + xA3 yA3 mtrx itransform /yA3 ED /xA3 ED + mark ArmB 0 ne { xB1 yB1 } if + xB2 yB2 xB3 yB3 xA3 yA3 xA2 yA2 ArmA 0 ne { xA1 yA1 } if + tx@Dict begin false Line end + /LPutVar [ xB1 yB1 xB2 yB2 xB3 yB3 xA3 yA3 xA2 yA2 xA1 yA1 ] cvx def + /LPutPos { LPutLines } def + /HPutPos { HPutLines } def + /VPutPos { VPutLines } def +} def +% +% DG/SR modification begin - May 9, 1997 - Patch 1 +%/NCCircle { 0 0 NodesepA nodeA \tx@GetEdge pop xA sub 2 div dup 2 exp r +%r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add +%exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360 +%mul add dup 5 1 roll 90 sub \tx@PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED +% +/NCCircle { + NodeSepA 0 NodeA 0 GetEdge pop + 2 div dup 2 exp r r mul sub abs sqrt + atan 2 mul /a ED + r AngleA 90 add PtoC yA add exch xA add + exch 2 copy + /LPutVar [ 4 2 roll r AngleA ] cvx def + /LPutPos { + LPutVar t 360 mul add dup 5 1 roll 90 sub PtoC + 3 -1 roll add + /Y ED add /X ED /NAngle ED +% DG/SR modification end + } def + /HPutPos { LPutPos } def + /VPutPos { LPutPos } def + r AngleA 90 sub a add AngleA 270 add a sub + tx@Dict begin + /angleB ED /angleA ED /r ED + /c 57.2957 r Div def + /y ED /x ED +} def +% +/NCBox { + /d ED /h ED + /AngleB yB yA sub xB xA sub Atan def + /AngleA AngleB 180 add def + GetEdgeA GetEdgeB + /dx d AngleB sin mul def + /dy d AngleB cos mul neg def + /hx h AngleB sin mul neg def + /hy h AngleB cos mul def + /LPutVar [ + xA1 hx add yA1 hy add xB1 hx add yB1 hy add + xB1 dx add yB1 dy add xA1 dx add yA1 dy add ] cvx def + /LPutPos { LPutLines } def + /HPutPos { xB yB xA yA LPutLine } def + /VPutPos { HPutPos } def + mark + LPutVar tx@Dict begin false Polygon end +} def +% +/NCArcBox { + /l ED neg /d ED /h ED /a ED + /AngleA yB yA sub xB xA sub Atan def + /AngleB AngleA 180 add def + /tA AngleA a sub 90 add def + /tB tA a 2 mul add def + /r xB xA sub tA cos tB cos sub Div dup 0 eq { pop 1 } if def + /x0 xA r tA cos mul add def + /y0 yA r tA sin mul add def + /c 57.2958 r div def + /AngleA AngleA a sub 180 add def + /AngleB AngleB a add 180 add def + GetEdgeA GetEdgeB + /AngleA tA 180 add yA yA1 sub xA xA1 sub Pyth c mul sub def + /AngleB tB 180 add yB yB1 sub xB xB1 sub Pyth c mul add def + l 0 eq { + x0 y0 r h add AngleA AngleB arc x0 y0 r d add AngleB AngleA arcn + }{ + x0 y0 translate + /tA AngleA l c mul add def + /tB AngleB l c mul sub def + 0 0 r h add tA tB arc r h add + AngleB PtoC r d add + AngleB PtoC 2 copy + 6 2 roll l arcto 4 { pop } repeat + r d add tB PtoC l arcto 4 { pop } repeat + 0 0 r d add tB tA arcn r d add + AngleA PtoC r h add + AngleA PtoC 2 copy 6 2 roll + l arcto 4 { pop } repeat + r h add tA PtoC l arcto 4 { pop } repeat + } ifelse + closepath + /LPutVar [ x0 y0 r AngleA AngleB h d ] cvx def + /LPutPos { + LPutVar /d ED /h ED + /AngleB ED /AngleA ED + /r ED /y0 ED /x0 ED + t 1 le { + r h add AngleA 1 t sub mul AngleB t mul add dup 90 add /NAngle ED PtoC + }{t 2 lt { + /NAngle AngleB 180 add def r 2 t sub + h mul t 1 sub d mul add add AngleB PtoC + }{ + t 3 lt { + r d add AngleB 3 t sub mul AngleA 2 t sub + mul add dup 90 sub /NAngle ED PtoC + }{ + /NAngle AngleA 180 add def + r 4 t sub d mul t 3 sub h mul add add AngleA PtoC + } ifelse + } ifelse + } ifelse + y0 add /Y ED x0 add /X ED + } def + /HPutPos { LPutPos } def + /VPutPos { LPutPos } def +} def +% +/Tfan { /AngleA yB yA sub xB xA sub Atan def GetEdgeA w xA1 xB sub yA1 yB +sub Pyth Pyth w Div CLW 2 div mul 2 div dup AngleA sin mul yA1 add /yA1 +ED AngleA cos mul xA1 add /xA1 ED /LPutVar [ xA1 yA1 m { xB w add yB xB +w sub yB } { xB yB w sub xB yB w add } ifelse xA1 yA1 ] cvx def /LPutPos +{ LPutLines } def /VPutPos@ { LPutVar flag { 8 4 roll pop pop pop pop } +{ pop pop pop pop 4 2 roll } ifelse } def /VPutPos { VPutPos@ VPutLine } +def /HPutPos { VPutPos@ HPutLine } def mark LPutVar tx@Dict begin +/ArrowA { moveto } def /ArrowB { } def false Line closepath end } def +% +/LPutCoor { + NAngle + tx@Dict begin /NAngle ED end + gsave + CM STV + CP Y sub neg exch X sub neg exch moveto + setmatrix CP + grestore +} def +% +/LPut { + tx@NodeDict /LPutPos known + { LPutPos } { CP /Y ED /X ED /NAngle 0 def } ifelse + LPutCoor +} def +% +/HPutAdjust { + Sin Cos mul 0 eq + { 0 } + { d Cos mul Sin div flag not { neg } if + h Cos mul Sin div flag { neg } if + 2 copy gt { pop } { exch pop } ifelse + } ifelse + s add flag { r add neg }{ l add } ifelse + X add /X ED +} def +% +/VPutAdjust { + Sin Cos mul + 0 eq + { 0 } + { l Sin mul Cos div flag { neg } if + r Sin mul Cos div flag not { neg } if + 2 copy gt { pop } { exch pop } ifelse + } ifelse + s add flag { d add } { h add neg } ifelse + Y add /Y ED +} def +% +% +end +% +% END pst-node.pro + +%%EndProcSet +%%BeginProcSet: pst-3d.pro 0 0 +%% $Id: pst-3d.pro 247 2010-01-04 22:45:42Z herbert $ +% PostScript prologue for pst-3d.tex. +% Version 0.01, 2010/01/01 +% +/tx@3Ddict 300 dict def +tx@3Ddict begin +% +/SetMatrixThreeD { + dup sin /e ED cos /f ED + /p3 ED /p2 ED /p1 ED + p1 0 eq + { /a 0 def /b p2 0 le { 1 } { -1 } ifelse def + p3 p2 abs + } + { p2 0 eq + { /a p1 0 lt { -1 } { 1 } ifelse def /b 0 def + p3 p1 abs + } + { p1 dup mul p2 dup mul add sqrt dup + p1 exch div /a ED + p2 exch div neg /b ED + p3 p1 a div + } + ifelse + } + ifelse + atan dup sin /c ED cos /d ED + /Matrix3D + [ + b f mul c a mul e mul sub + a f mul c b mul e mul add + d e mul + b e mul neg c a mul f mul sub + a e mul neg c b mul f mul add + d f mul + ] def +} def +% +/ProjThreeD { + /z ED /y ED /x ED + Matrix3D aload pop + z mul exch y mul add exch x mul add + 4 1 roll + z mul exch y mul add exch x mul add + exch +} def +% +/SetMatrixEmbed { + SetMatrixThreeD + Matrix3D aload pop + /z3 ED /z2 ED /z1 ED /x3 ED /x2 ED /x1 ED + SetMatrixThreeD + [ + Matrix3D aload pop + z3 mul exch z2 mul add exch z1 mul add 4 1 roll + z3 mul exch z2 mul add exch z1 mul add + Matrix3D aload pop + x3 mul exch x2 mul add exch x1 mul add 4 1 roll + x3 mul exch x2 mul add exch x1 mul add + 3 -1 roll 3 -1 roll 4 -1 roll 8 -3 roll 3 copy + x3 mul exch x2 mul add exch x1 mul add 4 1 roll + z3 mul exch z2 mul add exch z1 mul add + ] + concat +} def +% +/TMSave { + tx@Dict /TMatrix known not { /TMatrix { } def /RAngle { 0 } def } if + /TMatrix [ TMatrix CM ] cvx def +} def +% +/TMRestore { CP /TMatrix [ TMatrix setmatrix ] cvx def moveto } def +% +/TMChange { + TMSave + /cp [ currentpoint ] cvx def % ??? Check this later. + CM + CP T STV + CM matrix invertmatrix % Inv(M') + matrix concatmatrix % M Inv(M') + exch exec + concat cp moveto +} def +% +end % of tx@3Ddict +%% +%% End of file `pst-3d.pro'. + +%%EndProcSet +%%BeginProcSet: pst-math.pro 0 0 +% -*- Mode: Postscript -*- +% pst-math.pro --- PostScript header file pst-math.pro +% +% Author : Christophe JORSSEN +% Author : Herbert Voß +% Last Mod : $Date: 2018/12/16 $ +% Version : 0.65 $ +% +/PI 3.14159265359 def +/ENeperian 2.71828182846 def +% +/DegToRad {PI mul 180 div} bind def +/RadToDeg {180 mul PI div} bind def +% +/COS {RadToDeg cos} bind def +/SIN {RadToDeg sin} bind def +/TAN {dup SIN exch COS Div} bind def +/tan {dup sin exch cos Div} bind def +/ATAN {neg -1 atan 180 sub DegToRad} bind def +/ACOS {dup dup mul neg 1 add sqrt exch atan DegToRad} bind def +/acos {dup dup mul neg 1 add sqrt exch atan} bind def +/ASEC {1 exch Div ACOS} bind def +/ASIN {neg dup dup mul neg 1 add sqrt neg atan 180 sub DegToRad} bind def +/asin {neg dup dup mul neg 1 add sqrt neg atan 180 sub} bind def +/ACSC {1 exch Div ASIN} bind def +% +/EXP {ENeperian exch exp} bind def +% +/COSH {dup EXP exch neg EXP add 2 div} bind def +/SINH {dup EXP exch neg EXP sub 2 div} bind def +/TANH {dup SINH exch COSH div} bind def +/ACOSH {dup dup mul 1 sub sqrt add ln} bind def +/ASINH {dup dup mul 1 add sqrt add ln} bind def +/ATANH {dup 1 add exch neg 1 add Div ln 2 div} bind def +% +%/SINC {dup SIN exch Div} bind def +/SINC { dup 0 eq { pop 1 } { dup SIN exch div } ifelse } bind def + +/GAUSS {dup mul 2 mul dup 4 -2 roll sub dup mul exch div neg EXP exch PI mul sqrt div} bind def +% +/GAMMA { 2 dict begin % hv 2007-08-30 + /z exch def + 1.000000000190015 % p(0) + 0 1 5 { % on stack is 0 1 2 3 4 5 + dup % n-1 n-1 + [ 76.18009172947146 + -86.50532032941677 + 24.0140982483091 + -1.231739572450155 + 0.1208650973866179E-2 + -0.5395239384953E-5 ] exch get exch % p(n) n-1 + 1 add z add div % p(n)/(z+n) + add % build the sum + } for + Pi 2 mul sqrt z div mul + z 5.5 add z 0.5 add exp mul ENeperian z 5.5 add neg exp mul + end } bind def +% +/GAMMALN {dup dup dup 5.5 add dup ln 3 -1 roll .5 add mul sub neg 1.000000000190015 + 0 1 5 { + [76.18009172947146 -86.50532032941677 24.0140982483091 -1.231739572450155 + .1208650973866179E-2 -.5395239384953E-5 2.5066282746310005] exch get + 4 -1 roll 1 add dup 5 1 roll div add} for + 4 -1 roll div 2.5066282746310005 mul ln add exch pop} bind def +/BETA {2 copy add GAMMALN neg exch GAMMALN 3 -1 roll GAMMALN EXP} bind def +% +/HORNER {aload length + dup 2 add -1 roll + exch 1 sub { + dup 4 1 roll + mul add exch + } repeat + pop +} bind def +% +/BESSEL_J0 {dup abs 8 lt { + dup mul dup [57568490574 -13362590354 651619640.7 -11214424.18 77392.33017 -184.9052456] HORNER + exch [57568490411 1029532985 9494680.718 59272.64853 267.8532712 1] HORNER + Div} + {abs dup .636619772 exch div sqrt exch dup .785398164 sub exch 8 exch div dup dup mul dup + [1 -1.098628627E-2 .2734510407E-4 -.2073370639E-5 .2093887211E-6] HORNER + 3 index COS mul + exch [-.1562499995E-1 .1430488765E-3 -.6911147651E-5 .7621095161E-6 -.934945152E-7] HORNER + 4 -1 roll SIN mul 3 -1 roll mul neg add mul} + ifelse} bind def +% +/BESSEL_Y0 {dup 8 lt { + dup dup mul dup [-2957821389 7062834065 -512359803.6 10879881.29 -86327.92757 228.4622733] HORNER + exch [40076544269 745249964.8 7189466.438 47447.26470 226.1030244 1] HORNER + Div exch dup ln exch BESSEL_J0 .636619772 mul mul add} + {dup .636619772 exch div sqrt exch dup .785398164 sub exch 8 exch div dup dup mul dup + [1 -.1098628627E-2 .2734510407E-4 -.2073370639E-5 .2093887211E-6] HORNER + 3 index SIN mul + exch [-.1562499995E-1 .1430488765E-3 -.6911147651E-5 .7621095161E-6 -.934945152E-7] HORNER + 4 -1 roll COS mul 3 -1 roll mul add mul} + ifelse} bind def +% +/BESSEL_J1 {dup abs 8 lt { + dup dup mul dup 3 -2 roll [72362614232 -7895059235 242396853.1 -2972611.439 15704.48260 -30.16036606] HORNER mul + exch [144725228442 2300535178 18583304.74 99447.43394 376.9991397 1] HORNER + Div} + {dup abs dup .636619772 exch div sqrt exch dup 2.356194491 sub exch 8 exch div dup dup mul dup + [1 .183105E-2 -.3516396496E-4 .2457520174E-5 -.240337019E-6] HORNER + 3 index COS mul + exch [.04687499995 6.2002690873E-3 .8449199096E-5 -.88228987E-6 .105787412E-6] HORNER + 4 -1 roll SIN mul 3 -1 roll mul neg add mul exch dup abs Div mul} + ifelse} bind def +% +/BESSEL_Y1 {dup 8 lt { + dup dup dup mul dup [-.4900604943E13 .1275274390E13 -.5153428139E11 .7349264551E9 -.4237922726E7 .8511937935E4] HORNER + exch [.2499580570E14 .4244419664E12 .3733650367E10 .2245904002E8 .1020426050E6 .3549632885E3 1] HORNER + Div mul exch dup dup ln exch BESSEL_J1 mul exch 1 exch div sub .636619772 mul add} + {dup .636619772 exch div sqrt exch dup 2.356194491 sub exch 8 exch div dup dup mul dup + [1 .183105E-2 -.3516396496E-4 .2457520174E-5 -.240337019E-6] HORNER + 3 index SIN mul + exch [.04687499995 -.2002690873E-3 .8449199096E-5 6.88228987E-6 .105787412E-6] HORNER + 4 -1 roll COS mul 3 -1 roll mul add mul} + ifelse} bind def +% +% En cours... +/BESSEL_Yn {dup 0 eq {pop BESSEL_Y0}{dup 1 eq {pop BESSEL_Y1}{ + exch dup BESSEL_Y0 exch dup BESSEL_Y1 exch 2 exch Div { + mul 3 -1 roll mul 2 index sub pstack} for + } ifelse } ifelse } bind def +% +/SIMPSON { 1 dict begin %% on stack a b var f ierr Dominik Rodriguez + 3 index 5 index sub % compute h + 1 % a b var f ierr h n + 4 index 7 index def 3 index exec % a b var f ierr h n f(a) + 5 index 7 index def 4 index exec add % a b var f ierr h n f(a)+f(b) + 5 index 8 index 4 index 2 div add def 4 index exec % a b var f ierr h n f(a)+f(b) f(a+h/2) + exch 1 index 4 mul add 0 % a b var f ierr h n old=f(a+h/2) Estim=f(a)+f(b)+4f(a+h/2) NbLoop + { % a b var f ierr h n old Estim NbLoop + 5 -1 roll 2 div dup 6 1 roll % h<-h/2 + 5 -1 roll 2 mul 5 1 roll % n<-2n + % a b var f ierr h n old Estim NbLoop h + 2 div 10 index add 0 % a b var f ierr h n old Estim NbLoop a+h/2 Cumul + 5 index { + 1 index 10 index exch def 8 index exec add exch 6 index add exch + } repeat % a b var f ierr h n old Estim NbLoop a+nh/2 Cumul + exch pop % a b var f ierr h n old Estim NbLoop New + 2 index 1 index 4 mul 6 -1 roll 2 mul sub sub % a b var f ierr h n Estim NbLoop New Diff + 4 -1 roll 2 mul 1 index sub 4 1 roll % a b var f ierr h n Estim NbLoop New Diff + exch 4 1 roll % a b var f ierr h n old Estim NbLoop Diff + 5 index 6 div mul abs 6 index lt { exit } if + 1 add dup 9 eq { exit } if + } loop % a b var f ierr h n old Estim NbLoop + exch 5 -1 roll 6 div mul mark 10 2 roll cleartomark +end +} def +% ------------------------------------ math stuff ---------------------------------- +% +% Matrix A in arrays of rows A[[row1][row2]...] +% with [row1]=[a11 a12 ... b1] +% returns on stack solution vector X=[x1 x2 ... xn] +/SolveLinEqSystem { % on stack matrix M=[A,b] (A*x=b) + 10 dict begin % hold all ocal + /A exch def + /Rows A length def % Rows = number of rows + /Cols A 0 get length def % Cols = number of columns + /Index [ 0 1 Rows 1 sub { } for ] def % Index = [0 1 2 ... Rows-1] + /col 0 def + /row 0 def + /PR Rows array def % PR[c] = pivot row for row row + { % starts the loop, find pivot entry in row r + col Cols ge row Rows ge or { exit } if % col < Cols and row < Rows else exit + /pRow row def % pRow = pivot row + /max A row get col get abs def % get A[row[col]], first A[0,0] + row 1 add 1 Rows 1 sub { % starts for loop 1 1 Rows-1 + /j exch def % index counter + /x A j get col get abs def % get A[j[r]] + x max gt { % x>max, then save position + /pRow j def + /max x def + } if + } for % now we have the row with biggest A[0,1] + % with pRow = the pivot row + max 0 gt { % swap entries pRow and row in i + /tmp Index row get def + Index row Index pRow get put + Index pRow tmp put % and columns pRow and row in A + /tmp A row get def + A row A pRow get put + A pRow tmp put % pivot + /row0 A row get def % the pivoting row + /p0 row0 col get def % the pivot value + row 1 add 1 Rows 1 sub { % start for loop + /j exch def + /c1 A j get def + /p c1 col get p0 div def + c1 col p put % subtract (p1/p0)*row[i] from row[j] + col 1 add 1 Cols 1 sub { % start for loop + /i exch def + c1 dup i exch % c1 i c1 + i get row0 i get p mul sub put + } for + } for + PR row col put + /col col 1 add def + /row row 1 add def + }{ % all zero entries + /row row 1 add def % continue loop with same row + } ifelse + } loop + /X A def % solution vector + A Rows 1 sub get dup + Cols 1 sub get exch + Cols 2 sub get div + X Rows 1 sub 3 -1 roll put % X[n] + Rows 2 sub -1 0 { % for loop to calculate X[i] + /xi exch def % current index + A xi get % i-th row + /Axi exch def + /sum 0 def + Cols 2 sub -1 xi 1 add { + /n exch def + /sum sum Axi n get X n get mul add def + } for + Axi Cols 1 sub get % b=Axi[Cols-1] + sum sub % b-sum + Axi xi get div % b-sum / Axi[xi] + X xi 3 -1 roll put % X[xi] + } for + X + end +} def +% +/c@_0 2.515517 def +/c@_1 0.802853 def +/c@_2 0.010328 def +/d@_1 1.432788 def +/d@_2 0.189269 def +/d@_3 0.001308 def +/norminv { + 5 dict begin + neg 1 add 1 exch div ln 2 mul sqrt + /t exch def + /t2 t dup mul def + /t3 t2 t mul def + c@_0 c@_1 t mul add c@_2 t2 mul add 1 d@_1 t mul add + d@_2 t2 mul add d@_3 t3 mul add div neg t add + end +} def +%end{norminv Michael Sharpe} +% +% +% END pst-math.pro + +%%EndProcSet +%%BeginProcSet: pstricks-add.pro 0 0 +%% $Id: pstricks-add.pro 37 2014-07-22 13:52:12Z herbert $ +%% +% PostScript prologue for pstricks-add.tex. +% +% Version 0.23, 2009/12/17 +% +/tx@addDict 410 dict def tx@addDict begin +%% +realtime srand % set random generator +% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +/GTriangle { + gsave + /mtrx CM def + /colorA ED /colorB ED /colorC ED % save the colors + /yA ED /xA ED % save the origin + xA yA translate + rotate % \psk@gangle + /yB ED /xB ED /yC ED /xC ED % save other coordinates + /ds [ % save data in a array + 0 0 0 colorA aload pop % fd x y xr xg xb + 0 xB xA sub yB yA sub colorB aload pop + 0 xC xA sub yC yA sub colorC aload pop +% 1 xC xB add yB colorA aload pop % for use with 4 points ABCD + ] def + newpath + << + /ShadingType 4 % single Gouraud + /ColorSpace [ /DeviceRGB ] + /DataSource ds + >> + shfill + closepath + mtrx + setmatrix grestore} def +% +/RGBtoCMYK { % on stack r g b -> C M Y K + 3 dict begin + /Blue ED /Green ED /Red ED + 1 Red sub % Cyan + 1 Green sub % Magenta + 1 Blue sub % Yellow + 0 % Black + end +} def +% +/CMYKtoGRAY { % on stack c m y k -> gray + exch 0.11 mul add + exch 0.59 mul add + exch 0.3 mul add + dup 1 gt { pop 1 } if + neg 1 add +} def +% +/RGBtoGRAY { % on stack r g b -> gray + 0.11 mul + exch 0.59 mul add + exch 0.3 mul add +} def +% +/HSBtoGRAY { + 6 dict begin + /b exch def + /s exch def 6 mul dup cvi dup + /i exch def sub + /f exch def + /F [[0 1 f sub 1][f 0 1][1 0 1 f sub][1 f 0][1 f sub 1 0][0 1 f][0 1 1]] def + F i get { s mul neg 1 add b mul} forall + 0.11 mul + exch 0.59 mul add + exch 0.3 mul add + end +} def +% +%% convertisseur longueur d'onde ->R,G,B Manuel Luque +%% lambda max=780 nanometres +%% lambda min=380 nanometres +%% adaptation de : +%% http://www.physics.sfasu.edu/astro/color.html +%% www.efg2.com/lab +% +/Gamma 0.8 def +/calculateRGB { + lambda 379 le {/Red 0 def /Green 0 def /Blue 0 def} if + lambda 781 ge {/Red 0 def /Green 0 def /Blue 0 def} if + lambda 380 ge {lambda 439 le { + /R {lambda 440 sub neg 440 380 sub div} def + /Red R factor mul Gamma exp def + /G 0 def + /Green G factor mul Gamma exp def + /B 1 def + /Blue B factor mul Gamma exp def} if + } if + lambda 440 ge { lambda 489 le { + /G {lambda 440 sub 490 440 sub div} def + /Green G factor mul Gamma exp def + /R 0 def /Red 0 def + /B 1 def + /Blue B factor mul Gamma exp def } if + } if + lambda 490 ge {lambda 509 le { + /B {lambda 510 sub neg 510 490 sub div} def + /Blue B factor mul Gamma exp def + /R 0 def /Red 0 def + /G 1 def + /Green G factor mul Gamma exp def } if + } if + lambda 510 ge {lambda 579 le { + /R {lambda 510 sub 580 510 sub div } def + /Red R factor mul Gamma exp def + /Blue 0 def + /G 1 def + /Green G factor mul Gamma exp def } if + } if + lambda 580 ge {lambda 644 le { + /G {lambda 645 sub neg 645 580 sub div } def + /Green G factor mul Gamma exp def + /Blue 0 def + /R 1 def + /Red R factor mul Gamma exp def } if + } if + lambda 645 ge { lambda 780 le { + /Red 1 factor mul Gamma exp def + /Blue 0 def + /Green 0 def } if + } if +} def +% +/factor { + lambda 380 ge {lambda 419 le { 0.3 0.7 lambda 380 sub mul 420 380 sub div add} if } if + lambda 420 ge {lambda 700 le { 1 } if } if + lambda 701 ge {lambda 780 le { 0.3 0.7 780 lambda sub mul 780 700 sub div add} if } if +} def +% +/wavelengthToRGB { % the wavelength in nm must be on top of the stack + cvi /lambda exch def % no floating point here + calculateRGB +} def % now the colors are saved in Red Green Blue +% +/wavelengthToGRAY { % the wavelength in nm must be on top of the stack + cvi /lambda exch def % no floating point here + calculateRGB + Red Green Blue RGBtoGRAY +} def % now the gray color is on the stack +% +/wavelengthToCMYK { % the wavelength in nm must be on top of the stack + cvi /lambda exch def % no floating point here + gsave + calculateRGB Red Green Blue RGBtoCMYK + /Black ED /Yellow ED /Magenta ED /Cyan ED + grestore +} def % now the colors are saved in Cyan Magenta Yellow Black +% +/axfill { + 8 dict begin + /xw exch def /nl exch def + /C1 exch def /y1 exch def/x1 exch def + /C0 exch def /y0 exch def/x0 exch def + << /ShadingType 2 + /ColorSpace /DeviceRGB + /Coords [ x0 y0 x1 y1 ] + /EmulatorHints [ xw 2 div dup ] + /Function << + /FunctionType 2 + /Domain [0 1] + /C0 C0 + /C1 C1 + /N 1 + >> + >> shfill + end +} bind def +% +%/amplHand {.8} def +%/dtHand 2 def +/varHand { rand sin amplHand mul add } def +/MovetoByHand { moveto } def +%/MovetoByHand { /y0 ED /x0 ED x0 y0 moveto } def +/LinetoByHand { 4 dict begin + /y1 ED /x1 ED + currentpoint /y0 ED /x0 ED + x0 x1 sub dup mul y0 y1 sub dup mul add sqrt /dEnd ED + 0 dtHand dEnd { dup + x1 x0 sub mul dEnd div x0 add varHand exch + y1 y0 sub mul dEnd div y0 add varHand lineto + } for +% /x0 x1 def /y0 y1 def + end +} def +% +end +% +% END pstricks-add.pro + +%%EndProcSet +%%BeginProcSet: pst-coil.pro 0 0 +%!ps +% PostScript prologue for pst-coil.tex. +% Version 1.07, 2015/05/13 +% For distribution, see pstricks.tex. +% +/tx@CoilDict 40 dict def tx@CoilDict begin +/CoilLoop { + /t ED + t sin AspectSin mul t 180 div AspectCos mul add t cos lineto } def +/Coil { + /Inc ED + dup sin /AspectSin ED + cos /AspectCos ED + /ArmB ED + /ArmA ED + /h ED /w ED /y1 ED /x1 ED /y0 ED /x0 ED + x0 y0 translate y1 y0 sub x1 x0 sub 2 copy Pyth + /TotalLength ED + Atan rotate + /BeginAngle ArmA AspectCos Div w h mul Div 360 mul def + /EndAngle TotalLength ArmB sub AspectCos Div + w h mul Div 360 mul def + 1 0 0 0 ArrowA ArmA 0 lineto + /mtrx CM def + w h mul 2 Div w 2 Div scale BeginAngle Inc + EndAngle { CoilLoop } for + EndAngle CoilLoop mtrx setmatrix TotalLength ArmB sub 0 lineto CP +% DG/SR modification 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b(b)p Fq(g)293 3478 +y Fz(par)31 b(la)f(\034gure)h(suiv)-5 b(an)m(te)31 b(:)p +0 TeXcolorgray 0 TeXcolorgray 1391 3731 a +tx@Dict begin CP CP translate 0.6 0.6 scale NET end + 1391 3731 a +1391 3495 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1391 3495 a @beginspecial @setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 1.8 SLW 0 setgray 0.0 0.0 2 copy moveto 12.80365 +0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave 1 setgray + 1. .setopacityalpha fill grestore gsave 1.8 SLW 0 setgray 1. .setopacityalpha + 0 setlinejoin 0 setlinecap stroke grestore end + +@endspecial +1391 3495 a +tx@Dict begin tx@NodeDict begin {0.0 0.0 12.80365 } false /N@A 11 +{InitCnode } /NodeScale {} def NewNode end end + 1391 3495 a 1391 3495 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 1391 3495 a 1391 +3495 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@A known { -12.80365 +0. /N@A load 0 GetEdge 0.0 0. AddOffset /N@A load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 1391 3495 a 1391 3495 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 1391 3495 a 0 TeXcolorgray +1391 3495 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1391 3495 a 1391 3495 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1391 3495 a 1391 +3495 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 1391 3495 a 1391 3495 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 1391 3495 a 1391 3495 +a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1391 3495 a 1391 3495 a +tx@Dict begin ps@rot RotBegin end + 1391 3495 a 1391 3495 a +tx@Dict begin RotEnd end + 1391 +3495 a 1391 3495 a +tx@Dict begin PutEnd end + 1391 3495 a 1391 3495 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1391 3495 +a 1391 3495 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1391 3495 a 0 TeXcolorgray 1391 3495 a +tx@Dict begin PutEnd end + +1391 3495 a 1391 3495 a +tx@Dict begin PutEnd end + 1391 3495 a 1391 3495 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Aw 10 {InitPnode +} /NodeScale {} def NewNode end end + 1391 +3495 a 1391 3495 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Ae 10 {InitPnode +} /NodeScale {} def NewNode end end + 1391 3495 a 1391 3495 a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@An 10 {InitPnode +} /NodeScale {} def NewNode end end + 1391 3495 +a 1391 3495 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@As 10 {InitPnode +} /NodeScale {} def NewNode end end + 1391 3495 a 1391 3495 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Anw +10 {InitPnode } /NodeScale {} def NewNode end end + 1391 3495 a 1391 +3495 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Ane +10 {InitPnode } /NodeScale {} def NewNode end end + 1391 3495 a 1391 3495 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Asw +10 {InitPnode } /NodeScale {} def NewNode end end + 1391 3495 a 1391 3495 +a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Ase +10 {InitPnode } /NodeScale {} def NewNode end end + 1391 3495 a 1391 3495 a +tx@Dict begin PutEnd end + 1391 3495 a 1391 3495 a +tx@Dict begin { 85.35823 0.0 } PutCoor PutBegin end + 1391 +3495 a @beginspecial @setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 1.8 SLW 0 setgray 0.0 0.0 2 copy moveto 12.80365 +0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave 1 setgray + 1. .setopacityalpha fill grestore gsave 1.8 SLW 0 setgray 1. .setopacityalpha + 0 setlinejoin 0 setlinecap stroke grestore end + +@endspecial 1391 3495 +a +tx@Dict begin tx@NodeDict begin {0.0 0.0 12.80365 } false /N@B 11 +{InitCnode } /NodeScale {} def NewNode end end + 1391 3495 a 1391 3495 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 1391 3495 a 1391 3495 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@B known { -12.80365 +0. /N@B load 0 GetEdge 0.0 0. AddOffset /N@B load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 1391 +3495 a 1391 3495 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 1391 3495 a 0 TeXcolorgray 1391 3495 +a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1391 3495 a 1391 3495 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1391 3495 a 1391 3495 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 1391 +3495 a 1391 3495 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 1391 3495 a 1391 3495 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1391 3495 +a 1391 3495 a +tx@Dict begin ps@rot RotBegin end + 1391 3495 a 1391 3495 a +tx@Dict begin RotEnd end + 1391 3495 a 1391 +3495 a +tx@Dict begin PutEnd end + 1391 3495 a 1391 3495 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1391 3495 a 1391 3495 +a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1391 3495 a 0 TeXcolorgray 1391 3495 a +tx@Dict begin PutEnd end + 1391 3495 a +1391 3495 a +tx@Dict begin PutEnd end + 1391 3495 a 1391 3495 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Bw 10 {InitPnode +} /NodeScale {} def NewNode end end + 1391 3495 a 1391 +3495 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Be 10 {InitPnode +} /NodeScale {} def NewNode end end + 1391 3495 a 1391 3495 a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@Bn 10 {InitPnode +} /NodeScale {} def NewNode end end + 1391 3495 a 1391 3495 +a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@Bs 10 {InitPnode +} /NodeScale {} def NewNode end end + 1391 3495 a 1391 3495 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Bnw +10 {InitPnode } /NodeScale {} def NewNode end end + 1391 3495 a 1391 3495 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Bne +10 {InitPnode } /NodeScale {} def NewNode end end + 1391 +3495 a 1391 3495 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Bsw +10 {InitPnode } /NodeScale {} def NewNode end end + 1391 3495 a 1391 3495 a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Bse +10 {InitPnode } /NodeScale {} def NewNode end end + 1391 3495 +a 1391 3495 a +tx@Dict begin PutEnd end + 1391 3495 a 1391 3495 a +tx@Dict begin { 170.71646 0.0 } PutCoor PutBegin end + 1391 3495 a @beginspecial +@setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 1.8 SLW 0 setgray 0.0 0.0 2 copy moveto 12.80365 +0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave 1 setgray + 1. .setopacityalpha fill grestore gsave 1.8 SLW 0 setgray 1. .setopacityalpha + 0 setlinejoin 0 setlinecap stroke grestore end + +@endspecial 1391 3495 a +tx@Dict begin tx@NodeDict begin {0.0 0.0 12.80365 } false /N@C 11 +{InitCnode } /NodeScale {} def NewNode end end + 1391 3495 a 1391 +3495 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 1391 3495 a 1391 3495 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@C known { -12.80365 +0. /N@C load 0 GetEdge 0.0 0. AddOffset /N@C load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 1391 3495 a 1391 3495 +a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 1391 3495 a 0 TeXcolorgray 1391 3495 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1391 3495 a +1391 3495 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1391 3495 a 1391 3495 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 1391 3495 a 1391 +3495 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 1391 3495 a 1391 3495 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1391 3495 a 1391 3495 +a +tx@Dict begin ps@rot RotBegin end + 1391 3495 a 1391 3495 a +tx@Dict begin RotEnd end + 1391 3495 a 1391 3495 a +tx@Dict begin PutEnd end + 1391 +3495 a 1391 3495 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1391 3495 a 1391 3495 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1391 3495 +a 0 TeXcolorgray 1391 3495 a +tx@Dict begin PutEnd end + 1391 3495 a 1391 3495 a +tx@Dict begin PutEnd end + +1391 3495 a 1391 3495 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Cw 10 {InitPnode +} /NodeScale {} def NewNode end end + 1391 3495 a 1391 3495 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Ce 10 {InitPnode +} /NodeScale {} def NewNode end end + 1391 +3495 a 1391 3495 a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@Cn 10 {InitPnode +} /NodeScale {} def NewNode end end + 1391 3495 a 1391 3495 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@Cs 10 {InitPnode +} /NodeScale {} def NewNode end end + 1391 3495 +a 1391 3495 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Cnw +10 {InitPnode } /NodeScale {} def NewNode end end + 1391 3495 a 1391 3495 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Cne +10 {InitPnode } /NodeScale {} def NewNode end end + 1391 3495 a 1391 +3495 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Csw +10 {InitPnode } /NodeScale {} def NewNode end end + 1391 3495 a 1391 3495 a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Cse +10 {InitPnode } /NodeScale {} def NewNode end end + 1391 3495 a 1391 3495 +a +tx@Dict begin PutEnd end + 1391 3495 a 1391 3495 a +tx@Dict begin { 256.07469 0.0 } PutCoor PutBegin end + 1391 3495 a @beginspecial @setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 2.87999 SLW 0 setgray 0.0 0.0 2 copy moveto +12.80365 0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave +1 setgray 1. .setopacityalpha fill grestore gsave 2.87999 SLW 0 + setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap stroke + grestore gsave 0.71999 SLW 1 setgray 1. .setopacityalpha stroke +grestore end + + +@endspecial 1391 3495 a +tx@Dict begin tx@NodeDict begin {0.0 0.0 13.34364 } false /N@D 11 +{InitCnode } /NodeScale {} def NewNode end end + 1391 3495 a 1391 3495 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 1391 +3495 a 1391 3495 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@D known { -12.80365 +0. /N@D load 0 GetEdge 0.0 0. AddOffset /N@D load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 1391 3495 a 1391 3495 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 1391 3495 +a 0 TeXcolorgray 1391 3495 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1391 3495 a 1391 3495 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + +1391 3495 a 1391 3495 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 1391 3495 a 1391 3495 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 1391 +3495 a 1391 3495 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1391 3495 a 1391 3495 a +tx@Dict begin ps@rot RotBegin end + 1391 3495 +a 1391 3495 a +tx@Dict begin RotEnd end + 1391 3495 a 1391 3495 a +tx@Dict begin PutEnd end + 1391 3495 a 1391 +3495 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1391 3495 a 1391 3495 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1391 3495 a 0 TeXcolorgray +1391 3495 a +tx@Dict begin PutEnd end + 1391 3495 a 1391 3495 a +tx@Dict begin PutEnd end + 1391 3495 a 1391 +3495 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Dw 10 {InitPnode +} /NodeScale {} def NewNode end end + 1391 3495 a 1391 3495 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@De 10 {InitPnode +} /NodeScale {} def NewNode end end + 1391 3495 a 1391 3495 +a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@Dn 10 {InitPnode +} /NodeScale {} def NewNode end end + 1391 3495 a 1391 3495 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@Ds 10 {InitPnode +} /NodeScale {} def NewNode end end + 1391 3495 a 1391 3495 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Dnw +10 {InitPnode } /NodeScale {} def NewNode end end + 1391 +3495 a 1391 3495 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Dne +10 {InitPnode } /NodeScale {} def NewNode end end + 1391 3495 a 1391 3495 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Dsw +10 {InitPnode } /NodeScale {} def NewNode end end + 1391 3495 +a 1391 3495 a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Dse +10 {InitPnode } /NodeScale {} def NewNode end end + 1391 3495 a 1391 3495 a +tx@Dict begin PutEnd end + 1391 3495 a 1391 +3495 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@Aw /N@A InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 1391 3495 a 1391 3495 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@A /N@B InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 1391 3495 a 1391 3495 +a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 add } def /ps@rot +{ } def grestore end + 1391 3495 a 1391 3495 a +tx@Dict begin tx@NodeDict begin /t 0.45 def LPut end PutBegin end + 1391 3495 a 1391 3495 a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 9.83968 8.01474 +0.0 NAngle 90 add Uput exch pop add a PtoC h1 add exch w1 add exch +} PutCoor PutBegin end + 1391 +3495 a 0 TeXcolorgray 1350 3528 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1350 3528 a 1350 3528 +a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1350 3528 a Fm(a)1350 3528 y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1350 3528 a 1350 3528 +a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1350 3528 a 0 TeXcolorgray 1391 3495 a +tx@Dict begin PutEnd end + 1391 3495 a +1391 3495 a +tx@Dict begin PutEnd end + 1391 3495 a 1391 3495 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@B /N@C InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 1391 3495 a 1391 +3495 a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 add } def /ps@rot +{ } def grestore end + 1391 3495 a 1391 3495 a +tx@Dict begin tx@NodeDict begin /t 0.45 def LPut end PutBegin end + 1391 3495 a 1391 3495 +a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 7.9889 12.92703 +0.0 NAngle 90 add Uput exch pop add a PtoC h1 add exch w1 add exch +} PutCoor PutBegin end + 1391 3495 a 0 TeXcolorgray 1358 3548 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1358 3548 a +1358 3548 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1358 3548 a Fm(b)1358 3548 y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1358 3548 a +1358 3548 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1358 3548 a 0 TeXcolorgray 1391 3495 a +tx@Dict begin PutEnd end + 1391 +3495 a 1391 3495 a +tx@Dict begin PutEnd end + 1391 3495 a 1391 3495 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@C /N@D InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 1391 3495 +a 1391 3495 a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 add } def /ps@rot +{ } def grestore end + 1391 3495 a 1391 3495 a +tx@Dict begin tx@NodeDict begin /t 0.45 def LPut end PutBegin end + 1391 3495 a 1391 +3495 a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 9.83968 8.01474 +0.0 NAngle 90 add Uput exch pop add a PtoC h1 add exch w1 add exch +} PutCoor PutBegin end + 1391 3495 a 0 TeXcolorgray 1350 3528 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1350 3528 +a 1350 3528 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1350 3528 a Fm(a)1350 3528 y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1350 3528 +a 1350 3528 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1350 3528 a 0 TeXcolorgray 1391 3495 a +tx@Dict begin PutEnd end + +1391 3495 a 1391 3495 a +tx@Dict begin PutEnd end + 1391 3495 a 1391 3731 a +tx@Dict begin CP CP translate 1 0.6 div 1 0.6 div scale NET end + 1391 +3731 a 293 4027 a Fz(Relev)m(ons)c(qu'il)h(est)f(p)s(ossible)g(de)h +(consid\351rer)g(l'unique)h(mot)e(construit)i(par)f(cet)f(automate)i +(comme)293 4139 y(un)h(langage)i(qu'on)f(p)s(eut)f(d\351\034nir)h +(math\351matiquemen)m(t)j(tr\350s)c(simplemen)m(t)i(par)f(:)1548 +4350 y Fm(L)25 b Ft(=)g Fq(f)p Fm(m)h Fq(2)f Ft(\006)2034 +4312 y Fl(\003)2088 4350 y Fq(j)15 b Fm(m)25 b Ft(=)g +Fm(aba)p Fq(g)434 4562 y Fz(o\371)31 b(la)f(barre)h(v)m(erticale)h +Fq(j)e Fz(signi\034e)h(\020tel)g(que\021.)434 4704 y(\311videmmen)m(t,) +d(l'ob)5 b(jectif)26 b(est)g(de)g(p)s(ermettre)h(la)f(construction)h +(de)f(langages)g(plus)g(\351tendus.)h(On)293 4817 y(p)s(eut)j(par)h +(exemple)g(construire)g(le)g(langage)g(constitu\351)h(d'une)f +(s\351quence)f(de)h(ab)f(ainsi)h(:)p 0 TeXcolorgray 0 +TeXcolorgray 1817 5070 a +tx@Dict begin CP CP translate 0.6 0.6 scale NET end + 1817 5070 a 1817 5070 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1817 +5070 a @beginspecial @setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 2.87999 SLW 0 setgray 0.0 0.0 2 copy moveto +12.80365 0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave +1 setgray 1. .setopacityalpha fill grestore gsave 2.87999 SLW 0 + setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap stroke + grestore gsave 0.71999 SLW 1 setgray 1. .setopacityalpha stroke +grestore end + +@endspecial 1817 5070 +a +tx@Dict begin tx@NodeDict begin {0.0 0.0 13.34364 } false /N@A 11 +{InitCnode } /NodeScale {} def NewNode end end + 1817 5070 a 1817 5070 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 1817 5070 a 1817 5070 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@A known { -12.80365 +0. /N@A load 0 GetEdge 0.0 0. AddOffset /N@A load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 1817 +5070 a 1817 5070 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 1817 5070 a 0 TeXcolorgray 1817 5070 +a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1817 5070 a 1817 5070 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1817 5070 a 1817 5070 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 1817 +5070 a 1817 5070 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 1817 5070 a 1817 5070 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1817 5070 +a 1817 5070 a +tx@Dict begin ps@rot RotBegin end + 1817 5070 a 1817 5070 a +tx@Dict begin RotEnd end + 1817 5070 a 1817 +5070 a +tx@Dict begin PutEnd end + 1817 5070 a 1817 5070 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1817 5070 a 1817 5070 +a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1817 5070 a 0 TeXcolorgray 1817 5070 a +tx@Dict begin PutEnd end + 1817 5070 a +1817 5070 a +tx@Dict begin PutEnd end + 1817 5070 a 1817 5070 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Aw 10 {InitPnode +} /NodeScale {} def NewNode end end + 1817 5070 a 1817 +5070 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Ae 10 {InitPnode +} /NodeScale {} def NewNode end end + 1817 5070 a 1817 5070 a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@An 10 {InitPnode +} /NodeScale {} def NewNode end end + 1817 5070 a 1817 5070 +a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@As 10 {InitPnode +} /NodeScale {} def NewNode end end + 1817 5070 a 1817 5070 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Anw +10 {InitPnode } /NodeScale {} def NewNode end end + 1817 5070 a 1817 5070 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Ane +10 {InitPnode } /NodeScale {} def NewNode end end + 1817 +5070 a 1817 5070 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Asw +10 {InitPnode } /NodeScale {} def NewNode end end + 1817 5070 a 1817 5070 a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Ase +10 {InitPnode } /NodeScale {} def NewNode end end + 1817 5070 +a 1817 5070 a +tx@Dict begin PutEnd end + 1817 5070 a 1817 5070 a +tx@Dict begin { 85.35823 0.0 } PutCoor PutBegin end + 1817 5070 a @beginspecial +@setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 1.8 SLW 0 setgray 0.0 0.0 2 copy moveto 12.80365 +0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave 1 setgray + 1. .setopacityalpha fill grestore gsave 1.8 SLW 0 setgray 1. .setopacityalpha + 0 setlinejoin 0 setlinecap stroke grestore end + +@endspecial 1817 5070 a +tx@Dict begin tx@NodeDict begin {0.0 0.0 12.80365 } false /N@B 11 +{InitCnode } /NodeScale {} def NewNode end end + 1817 5070 a 1817 +5070 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 1817 5070 a 1817 5070 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@B known { -12.80365 +0. /N@B load 0 GetEdge 0.0 0. AddOffset /N@B load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 1817 5070 a 1817 5070 +a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 1817 5070 a 0 TeXcolorgray 1817 5070 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1817 5070 a +1817 5070 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1817 5070 a 1817 5070 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 1817 5070 a 1817 +5070 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 1817 5070 a 1817 5070 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1817 5070 a 1817 5070 +a +tx@Dict begin ps@rot RotBegin end + 1817 5070 a 1817 5070 a +tx@Dict begin RotEnd end + 1817 5070 a 1817 5070 a +tx@Dict begin PutEnd end + 1817 +5070 a 1817 5070 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1817 5070 a 1817 5070 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1817 5070 +a 0 TeXcolorgray 1817 5070 a +tx@Dict begin PutEnd end + 1817 5070 a 1817 5070 a +tx@Dict begin PutEnd end + +1817 5070 a 1817 5070 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Bw 10 {InitPnode +} /NodeScale {} def NewNode end end + 1817 5070 a 1817 5070 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Be 10 {InitPnode +} /NodeScale {} def NewNode end end + 1817 +5070 a 1817 5070 a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@Bn 10 {InitPnode +} /NodeScale {} def NewNode end end + 1817 5070 a 1817 5070 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@Bs 10 {InitPnode +} /NodeScale {} def NewNode end end + 1817 5070 +a 1817 5070 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Bnw +10 {InitPnode } /NodeScale {} def NewNode end end + 1817 5070 a 1817 5070 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Bne +10 {InitPnode } /NodeScale {} def NewNode end end + 1817 5070 a 1817 +5070 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Bsw +10 {InitPnode } /NodeScale {} def NewNode end end + 1817 5070 a 1817 5070 a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Bse +10 {InitPnode } /NodeScale {} def NewNode end end + 1817 5070 a 1817 5070 +a +tx@Dict begin PutEnd end + 1817 5070 a 1817 5070 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@Aw /N@A InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 1817 5070 a 1817 5070 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 1.0 1.0 neg 0.0 0.0 0 0 /N@A /N@B InitNC { yB +yA sub xB xA sub Atan dup 15. add /AngleA exch def 15. sub 180 add +/AngleB exch def 0.8 0.8 GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub +Pyth 2 div dup 3 -1 roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def +/ArmTypeB 0 def GetArmA GetArmB xA2 yA2 xA1 yA1 2 copy /y0 ED /x0 ED +tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict begin ArrowB end /y3 +ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /cx x1 x0 sub 3 mul def /cy y1 +y0 sub 3 mul def /bx x2 x1 sub 3 mul cx sub def /by y2 y1 sub 3 mul +cy sub def /ax x3 x0 sub cx sub bx sub def /ay y3 y0 sub cy sub by +sub def /getValues { ax t0 3 exp mul bx t0 t0 mul mul add cx t0 mul +add x0 add ay t0 3 exp mul by t0 t0 mul mul add cy t0 mul add y0 add +ax t 3 exp mul bx t t mul mul add cx t mul add x0 add ay t 3 exp mul +by t t mul mul add cy t mul add y0 add } def /getdL { getValues 3 -1 +roll sub 3 1 roll sub Pyth } def /CurveLength { /u 0 def /du 0.01 def +0 100 { /t0 u def /u u du add def /t u def getdL add } repeat } def +/GetArrowPos { /ende 0.5 1 gt {ArrowPos}{ArrowPos CurveLength mul} +ifelse def /u 0 def /du 0.01 def /sum 0 def { /t0 u def /u u du add +def /t u def /sum getdL sum add def sum ende gt {exit} if } loop u +} def /dArrowPos 1. 1 gt {1.0 1. 1.0 add div }{ 0.5 } ifelse def /ArrowPos +0. def /loopNo 0.5 1 gt {CurveLength 0.5 div cvi }{ 1. } ifelse +def loopNo cvi { /ArrowPos ArrowPos dArrowPos add def /t GetArrowPos +def /t0 t 0.95 mul def getValues ArrowInside pop pop pop pop } repeat +x1 y1 x2 y2 x3 y3 curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 +] cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines +} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } if end + gsave 1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 +setlinecap stroke grestore grestore end + 1817 +5070 a 1817 5070 a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 add } def /ps@rot +{ } def grestore end + 1817 5070 a 1817 5070 a +tx@Dict begin tx@NodeDict begin /t 0.40 def LPut end PutBegin end + 1817 5070 +a 1817 5070 a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 9.83968 8.01474 +0.0 NAngle 90 add Uput exch pop add a PtoC h1 add exch w1 add exch +} PutCoor PutBegin end + 1817 5070 a 0 TeXcolorgray 1776 5103 a +tx@Dict begin CP CP translate 1. 1. scale NET end + +1776 5103 a 1776 5103 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1776 5103 a Fm(a)1776 5103 y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + +1776 5103 a 1776 5103 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1776 5103 a 0 TeXcolorgray 1817 +5070 a +tx@Dict begin PutEnd end + 1817 5070 a 1817 5070 a +tx@Dict begin PutEnd end + 1817 5070 a 1817 5070 +a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 1.0 1.0 neg 0.0 0.0 0 0 /N@B /N@A InitNC { yB +yA sub xB xA sub Atan dup 15. add /AngleA exch def 15. sub 180 add +/AngleB exch def 0.8 0.8 GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub +Pyth 2 div dup 3 -1 roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def +/ArmTypeB 0 def GetArmA GetArmB xA2 yA2 xA1 yA1 2 copy /y0 ED /x0 ED +tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict begin ArrowB end /y3 +ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /cx x1 x0 sub 3 mul def /cy y1 +y0 sub 3 mul def /bx x2 x1 sub 3 mul cx sub def /by y2 y1 sub 3 mul +cy sub def /ax x3 x0 sub cx sub bx sub def /ay y3 y0 sub cy sub by +sub def /getValues { ax t0 3 exp mul bx t0 t0 mul mul add cx t0 mul +add x0 add ay t0 3 exp mul by t0 t0 mul mul add cy t0 mul add y0 add +ax t 3 exp mul bx t t mul mul add cx t mul add x0 add ay t 3 exp mul +by t t mul mul add cy t mul add y0 add } def /getdL { getValues 3 -1 +roll sub 3 1 roll sub Pyth } def /CurveLength { /u 0 def /du 0.01 def +0 100 { /t0 u def /u u du add def /t u def getdL add } repeat } def +/GetArrowPos { /ende 0.5 1 gt {ArrowPos}{ArrowPos CurveLength mul} +ifelse def /u 0 def /du 0.01 def /sum 0 def { /t0 u def /u u du add +def /t u def /sum getdL sum add def sum ende gt {exit} if } loop u +} def /dArrowPos 1. 1 gt {1.0 1. 1.0 add div }{ 0.5 } ifelse def /ArrowPos +0. def /loopNo 0.5 1 gt {CurveLength 0.5 div cvi }{ 1. } ifelse +def loopNo cvi { /ArrowPos ArrowPos dArrowPos add def /t GetArrowPos +def /t0 t 0.95 mul def getValues ArrowInside pop pop pop pop } repeat +x1 y1 x2 y2 x3 y3 curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 +] cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines +} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } if end + gsave 1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 +setlinecap stroke grestore grestore end + 1817 5070 a 1817 5070 a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 add } def /ps@rot +{ } def grestore end + 1817 5070 a 1817 5070 a +tx@Dict begin tx@NodeDict begin /t 0.40 def LPut end PutBegin end + 1817 +5070 a 1817 5070 a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 7.9889 12.92703 +0.0 NAngle 90 add Uput exch pop add a PtoC h1 add exch w1 add exch +} PutCoor PutBegin end + 1817 5070 a 0 TeXcolorgray 1784 5124 +a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1784 5124 a 1784 5124 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1784 5124 a Fm(b)1784 5124 +y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1784 5124 a 1784 5124 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1784 5124 a 0 TeXcolorgray +1817 5070 a +tx@Dict begin PutEnd end + 1817 5070 a 1817 5070 a +tx@Dict begin PutEnd end + 1817 5070 a 1817 +5070 a +tx@Dict begin CP CP translate 1 0.6 div 1 0.6 div scale NET end + 1817 5070 a 0 TeXcolorgray 0 TeXcolorgray eop +end +%%Page: 7 7 +TeXDict begin 7 6 bop 0 TeXcolorgray 0 TeXcolorgray -5 +14 a Fu(T)-8 b(ABLE)32 b(DES)f(MA)-8 b(TI\310RES)2399 +b Fz(7)p -5 51 3473 4 v 0 TeXcolorgray 136 347 a(Cet)31 +b(exemple)g(p)s(ermet)g(d'engendrer)h(une)f(in\034nit\351)g(de)g(mots)f +(du)h(t)m(yp)s(e)f(ab,)h(abab,)g(ababab,)h(.)15 b(.)g(.)136 +485 y(P)m(ar)39 b(ailleurs,)g(la)f(pr\351sence)h(d'un)f(\351tat)h +(initial,)g(sim)m(ultan\351men)m(t)i(\034nal,)d(implique)h(l'existence) +-5 598 y(dans)k(le)f(langage)i(d'un)f(mot)g(vide)g(not\351)g +Fm(\017)p Fz(.)f(On)h(\351crira)g(donc)g(math\351matiquemen)m(t)j(le)c +(langage)-5 711 y(ainsi)31 b(:)905 825 y Fm(L)25 b Ft(=)g +Fq(f)p Fm(m)g Fq(2)g Ft(\006)1390 787 y Fl(\003)1444 +825 y Fq(j)15 b Fm(m)26 b Ft(=)f Fm(\017;)15 b(ab;)g(abab;)g(ababab;)g +(:)g(:)g(:)s Fq(g)136 993 y Fz(Un)31 b(dernier)g(exemple)g(mon)m(tre)h +(qu'il)f(est)f(p)s(ossible)g(de)h(g\351n\351rer)g(des)f(langages)h +(bien)g(plus)g(com-)-5 1106 y(plexes.)g(Sans)f(dire)h(imm\351diatemen)m +(t)i(ce)e(don)m(t)g(il)f(s'agit,)h(v)m(oici)g(l'automate)h(:)p +0 TeXcolorgray 0 TeXcolorgray 1306 1446 a +tx@Dict begin CP CP translate 0.6 0.6 scale NET end + 1306 1446 a +1306 1564 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1306 1564 a @beginspecial @setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 1.8 SLW 0 setgray 0.0 0.0 2 copy moveto 12.80365 +0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave 1 setgray + 1. .setopacityalpha fill grestore gsave 1.8 SLW 0 setgray 1. .setopacityalpha + 0 setlinejoin 0 setlinecap stroke grestore end + +@endspecial +1306 1564 a +tx@Dict begin tx@NodeDict begin {0.0 0.0 12.80365 } false /N@A 11 +{InitCnode } /NodeScale {} def NewNode end end + 1306 1564 a 1306 1564 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 1306 1564 a 1306 +1564 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@A known { -12.80365 +0. /N@A load 0 GetEdge 0.0 0. AddOffset /N@A load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 1306 1564 a 1306 1564 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 1306 1564 a 0 TeXcolorgray +1306 1564 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1306 1564 a 1306 1564 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1306 1564 a 1306 +1564 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 1306 1564 a 1306 1564 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 1306 1564 a 1306 1564 +a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1306 1564 a 1306 1564 a +tx@Dict begin ps@rot RotBegin end + 1306 1564 a 1306 1564 a +tx@Dict begin RotEnd end + 1306 +1564 a 1306 1564 a +tx@Dict begin PutEnd end + 1306 1564 a 1306 1564 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1306 1564 +a 1306 1564 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1306 1564 a 0 TeXcolorgray 1306 1564 a +tx@Dict begin PutEnd end + +1306 1564 a 1306 1564 a +tx@Dict begin PutEnd end + 1306 1564 a 1306 1564 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Aw 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 +1564 a 1306 1564 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Ae 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 1564 a 1306 1564 a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@An 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 1564 +a 1306 1564 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@As 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 1564 a 1306 1564 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Anw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 1564 a 1306 +1564 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Ane +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 1564 a 1306 1564 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Asw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 1564 a 1306 1564 +a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Ase +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 1564 a 1306 1564 a +tx@Dict begin PutEnd end + 1306 1564 a 1306 1564 a +tx@Dict begin { 85.35823 0.0 } PutCoor PutBegin end + 1306 +1564 a @beginspecial @setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 1.8 SLW 0 setgray 0.0 0.0 2 copy moveto 12.80365 +0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave 1 setgray + 1. .setopacityalpha fill grestore gsave 1.8 SLW 0 setgray 1. .setopacityalpha + 0 setlinejoin 0 setlinecap stroke grestore end + +@endspecial 1306 1564 +a +tx@Dict begin tx@NodeDict begin {0.0 0.0 12.80365 } false /N@B 11 +{InitCnode } /NodeScale {} def NewNode end end + 1306 1564 a 1306 1564 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 1306 1564 a 1306 1564 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@B known { -12.80365 +0. /N@B load 0 GetEdge 0.0 0. AddOffset /N@B load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 1306 +1564 a 1306 1564 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 1306 1564 a 0 TeXcolorgray 1306 1564 +a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1306 1564 a 1306 1564 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1306 1564 a 1306 1564 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 1306 +1564 a 1306 1564 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 1306 1564 a 1306 1564 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1306 1564 +a 1306 1564 a +tx@Dict begin ps@rot RotBegin end + 1306 1564 a 1306 1564 a +tx@Dict begin RotEnd end + 1306 1564 a 1306 +1564 a +tx@Dict begin PutEnd end + 1306 1564 a 1306 1564 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1306 1564 a 1306 1564 +a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1306 1564 a 0 TeXcolorgray 1306 1564 a +tx@Dict begin PutEnd end + 1306 1564 a +1306 1564 a +tx@Dict begin PutEnd end + 1306 1564 a 1306 1564 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Bw 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 1564 a 1306 +1564 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Be 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 1564 a 1306 1564 a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@Bn 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 1564 a 1306 1564 +a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@Bs 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 1564 a 1306 1564 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Bnw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 1564 a 1306 1564 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Bne +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 +1564 a 1306 1564 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Bsw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 1564 a 1306 1564 a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Bse +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 1564 +a 1306 1564 a +tx@Dict begin PutEnd end + 1306 1564 a 1306 1564 a +tx@Dict begin { 170.71646 0.0 } PutCoor PutBegin end + 1306 1564 a @beginspecial +@setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 2.87999 SLW 0 setgray 0.0 0.0 2 copy moveto +12.80365 0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave +1 setgray 1. .setopacityalpha fill grestore gsave 2.87999 SLW 0 + setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap stroke + grestore gsave 0.71999 SLW 1 setgray 1. .setopacityalpha stroke +grestore end + +@endspecial 1306 1564 a +tx@Dict begin tx@NodeDict begin {0.0 0.0 13.34364 } false /N@C 11 +{InitCnode } /NodeScale {} def NewNode end end + 1306 1564 a 1306 +1564 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 1306 1564 a 1306 1564 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@C known { -12.80365 +0. /N@C load 0 GetEdge 0.0 0. AddOffset /N@C load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 1306 1564 a 1306 1564 +a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 1306 1564 a 0 TeXcolorgray 1306 1564 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1306 1564 a +1306 1564 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1306 1564 a 1306 1564 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 1306 1564 a 1306 +1564 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 1306 1564 a 1306 1564 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1306 1564 a 1306 1564 +a +tx@Dict begin ps@rot RotBegin end + 1306 1564 a 1306 1564 a +tx@Dict begin RotEnd end + 1306 1564 a 1306 1564 a +tx@Dict begin PutEnd end + 1306 +1564 a 1306 1564 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1306 1564 a 1306 1564 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1306 1564 +a 0 TeXcolorgray 1306 1564 a +tx@Dict begin PutEnd end + 1306 1564 a 1306 1564 a +tx@Dict begin PutEnd end + +1306 1564 a 1306 1564 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Cw 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 1564 a 1306 1564 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Ce 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 +1564 a 1306 1564 a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@Cn 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 1564 a 1306 1564 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@Cs 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 1564 +a 1306 1564 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Cnw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 1564 a 1306 1564 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Cne +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 1564 a 1306 +1564 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Csw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 1564 a 1306 1564 a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Cse +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 1564 a 1306 1564 +a +tx@Dict begin PutEnd end + 1306 1564 a 1306 1564 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@Aw /N@A InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 1306 1564 a 1306 1564 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@A /N@B InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 1306 +1564 a 1306 1564 a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 add } def /ps@rot +{ } def grestore end + 1306 1564 a 1306 1564 a +tx@Dict begin tx@NodeDict begin /t 0.45 def LPut end PutBegin end + 1306 1564 +a 1306 1564 a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 9.83968 8.01474 +0.0 NAngle 90 add Uput exch pop add a PtoC h1 add exch w1 add exch +} PutCoor PutBegin end + 1306 1564 a 0 TeXcolorgray 1265 1598 a +tx@Dict begin CP CP translate 1. 1. scale NET end + +1265 1598 a 1265 1598 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1265 1598 a Fm(a)1265 1598 y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + +1265 1598 a 1265 1598 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1265 1598 a 0 TeXcolorgray 1306 +1564 a +tx@Dict begin PutEnd end + 1306 1564 a 1306 1564 a +tx@Dict begin PutEnd end + 1306 1564 a 1306 1564 +a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 1.0 1.0 neg 0.0 0.0 0 0 /N@B /N@C InitNC { yB +yA sub xB xA sub Atan dup 15. add /AngleA exch def 15. sub 180 add +/AngleB exch def 0.8 0.8 GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub +Pyth 2 div dup 3 -1 roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def +/ArmTypeB 0 def GetArmA GetArmB xA2 yA2 xA1 yA1 2 copy /y0 ED /x0 ED +tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict begin ArrowB end /y3 +ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /cx x1 x0 sub 3 mul def /cy y1 +y0 sub 3 mul def /bx x2 x1 sub 3 mul cx sub def /by y2 y1 sub 3 mul +cy sub def /ax x3 x0 sub cx sub bx sub def /ay y3 y0 sub cy sub by +sub def /getValues { ax t0 3 exp mul bx t0 t0 mul mul add cx t0 mul +add x0 add ay t0 3 exp mul by t0 t0 mul mul add cy t0 mul add y0 add +ax t 3 exp mul bx t t mul mul add cx t mul add x0 add ay t 3 exp mul +by t t mul mul add cy t mul add y0 add } def /getdL { getValues 3 -1 +roll sub 3 1 roll sub Pyth } def /CurveLength { /u 0 def /du 0.01 def +0 100 { /t0 u def /u u du add def /t u def getdL add } repeat } def +/GetArrowPos { /ende 0.5 1 gt {ArrowPos}{ArrowPos CurveLength mul} +ifelse def /u 0 def /du 0.01 def /sum 0 def { /t0 u def /u u du add +def /t u def /sum getdL sum add def sum ende gt {exit} if } loop u +} def /dArrowPos 1. 1 gt {1.0 1. 1.0 add div }{ 0.5 } ifelse def /ArrowPos +0. def /loopNo 0.5 1 gt {CurveLength 0.5 div cvi }{ 1. } ifelse +def loopNo cvi { /ArrowPos ArrowPos dArrowPos add def /t GetArrowPos +def /t0 t 0.95 mul def getValues ArrowInside pop pop pop pop } repeat +x1 y1 x2 y2 x3 y3 curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 +] cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines +} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } if end + gsave 1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 +setlinecap stroke grestore grestore end + 1306 1564 a 1306 1564 a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 add } def /ps@rot +{ } def grestore end + 1306 1564 a 1306 1564 a +tx@Dict begin tx@NodeDict begin /t 0.40 def LPut end PutBegin end + 1306 +1564 a 1306 1564 a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 7.9889 12.92703 +0.0 NAngle 90 add Uput exch pop add a PtoC h1 add exch w1 add exch +} PutCoor PutBegin end + 1306 1564 a 0 TeXcolorgray 1273 1618 +a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1273 1618 a 1273 1618 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1273 1618 a Fm(b)1273 1618 +y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1273 1618 a 1273 1618 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1273 1618 a 0 TeXcolorgray +1306 1564 a +tx@Dict begin PutEnd end + 1306 1564 a 1306 1564 a +tx@Dict begin PutEnd end + 1306 1564 a 1306 +1564 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 1.0 1.0 neg 0.0 0.0 0 0 /N@C /N@B InitNC { yB +yA sub xB xA sub Atan dup 15. add /AngleA exch def 15. sub 180 add +/AngleB exch def 0.8 0.8 GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub +Pyth 2 div dup 3 -1 roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def +/ArmTypeB 0 def GetArmA GetArmB xA2 yA2 xA1 yA1 2 copy /y0 ED /x0 ED +tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict begin ArrowB end /y3 +ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /cx x1 x0 sub 3 mul def /cy y1 +y0 sub 3 mul def /bx x2 x1 sub 3 mul cx sub def /by y2 y1 sub 3 mul +cy sub def /ax x3 x0 sub cx sub bx sub def /ay y3 y0 sub cy sub by +sub def /getValues { ax t0 3 exp mul bx t0 t0 mul mul add cx t0 mul +add x0 add ay t0 3 exp mul by t0 t0 mul mul add cy t0 mul add y0 add +ax t 3 exp mul bx t t mul mul add cx t mul add x0 add ay t 3 exp mul +by t t mul mul add cy t mul add y0 add } def /getdL { getValues 3 -1 +roll sub 3 1 roll sub Pyth } def /CurveLength { /u 0 def /du 0.01 def +0 100 { /t0 u def /u u du add def /t u def getdL add } repeat } def +/GetArrowPos { /ende 0.5 1 gt {ArrowPos}{ArrowPos CurveLength mul} +ifelse def /u 0 def /du 0.01 def /sum 0 def { /t0 u def /u u du add +def /t u def /sum getdL sum add def sum ende gt {exit} if } loop u +} def /dArrowPos 1. 1 gt {1.0 1. 1.0 add div }{ 0.5 } ifelse def /ArrowPos +0. def /loopNo 0.5 1 gt {CurveLength 0.5 div cvi }{ 1. } ifelse +def loopNo cvi { /ArrowPos ArrowPos dArrowPos add def /t GetArrowPos +def /t0 t 0.95 mul def getValues ArrowInside pop pop pop pop } repeat +x1 y1 x2 y2 x3 y3 curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 +] cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines +} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } if end + gsave 1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 +setlinecap stroke grestore grestore end + 1306 1564 a 1306 1564 a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 add } def /ps@rot +{ } def grestore end + 1306 1564 a 1306 1564 +a +tx@Dict begin tx@NodeDict begin /t 0.40 def LPut end PutBegin end + 1306 1564 a 1306 1564 a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 7.9889 12.92703 +0.0 NAngle 90 add Uput exch pop add a PtoC h1 add exch w1 add exch +} PutCoor PutBegin end + 1306 1564 a 0 TeXcolorgray +1273 1618 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1273 1618 a 1273 1618 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1273 1618 a Fm(b)1273 +1618 y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1273 1618 a 1273 1618 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1273 1618 a 0 TeXcolorgray +1306 1564 a +tx@Dict begin PutEnd end + 1306 1564 a 1306 1564 a +tx@Dict begin PutEnd end + 1306 1564 a 1306 +1564 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@A /N@A InitNC { /AngleA +120. def /AngleB 60. def 7. 7. GetEdgeA GetEdgeB xA1 xB1 sub yA1 +yB1 sub Pyth 2 div dup 3 -1 roll mul /ArmA ED mul /ArmB ED /ArmTypeA +0 def /ArmTypeB 0 def GetArmA GetArmB xA2 yA2 xA1 yA1 2 copy /y0 ED +/x0 ED tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict begin ArrowB +end /y3 ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /cx x1 x0 sub 3 mul def +/cy y1 y0 sub 3 mul def /bx x2 x1 sub 3 mul cx sub def /by y2 y1 sub +3 mul cy sub def /ax x3 x0 sub cx sub bx sub def /ay y3 y0 sub cy sub +by sub def /getValues { ax t0 3 exp mul bx t0 t0 mul mul add cx t0 +mul add x0 add ay t0 3 exp mul by t0 t0 mul mul add cy t0 mul add y0 +add ax t 3 exp mul bx t t mul mul add cx t mul add x0 add ay t 3 exp +mul by t t mul mul add cy t mul add y0 add } def /getdL { getValues +3 -1 roll sub 3 1 roll sub Pyth } def /CurveLength { /u 0 def /du 0.01 +def 0 100 { /t0 u def /u u du add def /t u def getdL add } repeat } +def /GetArrowPos { /ende 0.5 1 gt {ArrowPos}{ArrowPos CurveLength +mul} ifelse def /u 0 def /du 0.01 def /sum 0 def { /t0 u def /u u du +add def /t u def /sum getdL sum add def sum ende gt {exit} if } loop +u } def /dArrowPos 1. 1 gt {1.0 1. 1.0 add div }{ 0.5 } ifelse def +/ArrowPos 0. def /loopNo 0.5 1 gt {CurveLength 0.5 div cvi }{ 1. +} ifelse def loopNo cvi { /ArrowPos ArrowPos dArrowPos add def /t GetArrowPos +def /t0 t 0.95 mul def getValues ArrowInside pop pop pop pop } repeat +x1 y1 x2 y2 x3 y3 curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 +] cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines +} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } if end + gsave 1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 +setlinecap stroke grestore grestore end + 1306 1564 a 1306 1564 a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 add } def /ps@rot +{ } def grestore end + 1306 1564 a 1306 1564 +a +tx@Dict begin tx@NodeDict begin /t 0.25 def LPut end PutBegin end + 1306 1564 a 1306 1564 a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 9.83968 8.01474 +0.0 NAngle 90 add Uput exch pop add a PtoC h1 add exch w1 add exch +} PutCoor PutBegin end + 1306 1564 a 0 TeXcolorgray +1265 1598 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1265 1598 a 1265 1598 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1265 1598 a Fm(a)1265 +1598 y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1265 1598 a 1265 1598 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1265 1598 a 0 TeXcolorgray +1306 1564 a +tx@Dict begin PutEnd end + 1306 1564 a 1306 1564 a +tx@Dict begin PutEnd end + 1306 1564 a 1306 +1446 a +tx@Dict begin CP CP translate 1 0.6 div 1 0.6 div scale NET end + 1306 1446 a 136 1831 a Fz(Le)d(langage)f(reconn)m(u)i(est)d +(constitu\351)i(de)f(mots)g(comprenan)m(t)i(en)e(premier)g(lieu)h(au)e +(moins)h(un)g(a,)-5 1944 y(suivi)i(par)h(un)g(nom)m(bre)g(impair)h(de)e +(b.)h(On)f(p)s(eut)h(aussi)f(\351crire)h(:)596 2149 y +Fm(L)25 b Ft(=)g Fq(f)p Fm(m)h Fq(2)f Ft(\006)1082 2112 +y Fl(\003)1136 2149 y Fq(j)15 b Fm(m)25 b Ft(=)g Fm(a)1425 +2112 y Fs(n)1472 2149 y Fm(b)1511 2112 y Fs(m)1578 2149 +y Fm(;)30 b(n)25 b Fq(2)g Fj(N)1865 2112 y Fl(\003)1919 +2149 y Fz(et)h Fm(m)f Ft(=)g(2)c Fq(\001)f Fm(k)k Ft(+)19 +b(1)p Fm(;)31 b(k)e Fq(2)c Fj(N)p Fq(g)-5 2355 y Fz(o\371)33 +b Fj(N)189 2322 y Fl(\003)261 2355 y Fz(est)g(l'ensem)m(ble)h(des)f +(nom)m(bres)h(naturels)f(\(en)m(tiers)h(p)s(ositifs\))e(sans)h(le)f +(z\351ro)i(et)f Fj(N)f Fz(celui)h(a)m(v)m(ec)-5 2467 +y(le)e(z\351ro.)136 2631 y(En)e(conclusion,)h(on)e(p)s(eut)g +(d\351\034nir)h(un)g(automate)g(de)g(mani\350re)g(g\351n\351rale)h +(comme)f(un)f(ensem)m(ble)-5 2744 y(constitu\351)p 0 +TeXcolorgray 136 2857 a(\026)p 0 TeXcolorgray 46 w(d'un)j(alphab)s(et)g +Ft(\006)p Fz(,)p 0 TeXcolorgray 136 2970 a(\026)p 0 TeXcolorgray +46 w(d'un)g(ensem)m(ble)h(\034ni)f(d'\351tat,)g(g\351n\351ralemen)m(t)i +(not\351)e Fm(E)5 b Fz(,)p 0 TeXcolorgray 136 3083 a(\026)p +0 TeXcolorgray 46 w(d'un)31 b(ensem)m(ble)h(d'\351tat)f(initiaux)g +Fm(I)37 b Fz(tel)31 b(que)f Fm(I)i Fq(\022)25 b Fm(E)5 +b Fz(,)p 0 TeXcolorgray 136 3196 a(\026)p 0 TeXcolorgray +46 w(d'un)31 b(ensem)m(ble)h(d'\351tat)f(\034naux)g Fm(F)43 +b Fz(tel)31 b(que)f Fm(F)38 b Fq(\022)25 b Fm(E)36 b +Fz(et)p 0 TeXcolorgray 136 3309 a(\026)p 0 TeXcolorgray +46 w(d'une)31 b(fonction)g(de)g(transition)1551 3423 +y Fm(\016)e Ft(:)d Fm(E)f Fq(\002)20 b Ft(\006)25 b Fq(\000)-16 +b(!)26 b Fm(E)136 3591 y Fz(O\371)31 b(le)g(signe)f Fq(\022)g +Fz(signi\034e)h(\020appartien)m(t)h(\340\021.)136 3729 +y(Le)39 b(dernier)g(p)s(oin)m(t)f(de)g(cette)h(d\351\034nition)g +(m\351rite)g(des)f(explications.)h(La)f(repr\351sen)m(tation)i(gra-)-5 +3842 y(phique)22 b(des)f(automates)h(donn\351e)g(ci-dessus)e(met)i +(clairemen)m(t)h(en)e(\351vidence)h(les)f(transitions)g(\(\035\350c)m +(hes\))-5 3955 y(en)m(tre)i(\351tats.)f(Celles-ci)h(doiv)m(en)m(t)g +(aussi)f(trouv)m(er)g(une)g(expression)g(math\351matique.)i(C'est)e(la) +g(fonction)-5 4068 y(de)31 b(transition.)136 4207 y(Reprenons)37 +b(le)g(dernier)h(exemple)f(d'automate)i(donn\351)e(ci-dessus.)g +(Consid\351rons)g(c)m(hacun)i(des)-5 4320 y(p)s(oin)m(ts)31 +b(p)s(ermettan)m(t)h(de)f(le)f(d\351\034nir)h(math\351matiquemen)m(t.) +136 4433 y(L'alphab)s(et)h(tout)f(d'ab)s(ord)g(est)f(clairemen)m(t)j +(donn\351)e(par)f Ft(\006)25 b(=)g Fq(f)p Fm(a;)15 b(b)p +Fq(g)p Fz(.)136 4546 y(L'ensem)m(ble)41 b(des)d(\351tats)g(est)g +Fm(E)44 b Ft(=)38 b Fq(f)p Fm(e)1460 4560 y Fr(1)1500 +4546 y Fm(;)15 b(e)1582 4560 y Fr(2)1622 4546 y Fm(;)g(e)1704 +4560 y Fr(3)1745 4546 y Fq(g)38 b Fz(et)g(on)h(p)s(eut)f(en)g(donner)i +(une)e(repr\351sen)m(tation)-5 4659 y(l\351g\350remen)m(t)33 +b(plus)d(compl\350te)i(ainsi)e(:)p 0 TeXcolorgray 0 TeXcolorgray +1306 4999 a +tx@Dict begin CP CP translate 0.6 0.6 scale NET end + 1306 4999 a 1306 5117 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1306 5117 a @beginspecial +@setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 1.8 SLW 0 setgray 0.0 0.0 2 copy moveto 12.80365 +0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave 1 setgray + 1. .setopacityalpha fill grestore gsave 1.8 SLW 0 setgray 1. .setopacityalpha + 0 setlinejoin 0 setlinecap stroke grestore end + +@endspecial 1306 5117 a +tx@Dict begin tx@NodeDict begin {0.0 0.0 12.80365 } false /N@A 11 +{InitCnode } /NodeScale {} def NewNode end end + 1306 5117 a 1306 +5117 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 1306 5117 a 1306 5117 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@A known { -12.80365 +0. /N@A load 0 GetEdge 0.0 0. AddOffset /N@A load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 1306 5117 a 1306 5117 +a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 1306 5117 a 0 TeXcolorgray 1306 5117 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1306 5117 a +1306 5117 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1306 5117 a 1306 5117 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 1306 5117 a 1306 +5117 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 1306 5117 a 1306 5117 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1306 5117 a 1306 5117 +a +tx@Dict begin ps@rot RotBegin end + 1306 5117 a 1265 5130 a Fm(e)1307 5144 y Fr(1)1306 +5117 y +tx@Dict begin RotEnd end + 1306 5117 a 1306 5117 a +tx@Dict begin PutEnd end + 1306 5117 a 1306 5117 +a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1306 5117 a 1306 5117 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1306 5117 a 0 TeXcolorgray +1306 5117 a +tx@Dict begin PutEnd end + 1306 5117 a 1306 5117 a +tx@Dict begin PutEnd end + 1306 5117 a 1306 +5117 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Aw 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 5117 a 1306 5117 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Ae 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 5117 a 1306 5117 +a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@An 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 5117 a 1306 5117 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@As 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 5117 a 1306 5117 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Anw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 +5117 a 1306 5117 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Ane +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 5117 a 1306 5117 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Asw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 5117 +a 1306 5117 a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Ase +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 5117 a 1306 5117 a +tx@Dict begin PutEnd end + 1306 5117 a 1306 +5117 a +tx@Dict begin { 85.35823 0.0 } PutCoor PutBegin end + 1306 5117 a @beginspecial @setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 1.8 SLW 0 setgray 0.0 0.0 2 copy moveto 12.80365 +0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave 1 setgray + 1. .setopacityalpha fill grestore gsave 1.8 SLW 0 setgray 1. .setopacityalpha + 0 setlinejoin 0 setlinecap stroke grestore end + +@endspecial +1306 5117 a +tx@Dict begin tx@NodeDict begin {0.0 0.0 12.80365 } false /N@B 11 +{InitCnode } /NodeScale {} def NewNode end end + 1306 5117 a 1306 5117 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 1306 5117 a 1306 +5117 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@B known { -12.80365 +0. /N@B load 0 GetEdge 0.0 0. AddOffset /N@B load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 1306 5117 a 1306 5117 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 1306 5117 a 0 TeXcolorgray +1306 5117 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1306 5117 a 1306 5117 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1306 5117 a 1306 +5117 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 1306 5117 a 1306 5117 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 1306 5117 a 1306 5117 +a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1306 5117 a 1306 5117 a +tx@Dict begin ps@rot RotBegin end + 1306 5117 a 1265 5130 a Fm(e)1307 +5144 y Fr(2)1306 5117 y +tx@Dict begin RotEnd end + 1306 5117 a 1306 5117 a +tx@Dict begin PutEnd end + 1306 +5117 a 1306 5117 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1306 5117 a 1306 5117 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1306 5117 +a 0 TeXcolorgray 1306 5117 a +tx@Dict begin PutEnd end + 1306 5117 a 1306 5117 a +tx@Dict begin PutEnd end + +1306 5117 a 1306 5117 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Bw 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 5117 a 1306 5117 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Be 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 +5117 a 1306 5117 a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@Bn 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 5117 a 1306 5117 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@Bs 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 5117 +a 1306 5117 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Bnw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 5117 a 1306 5117 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Bne +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 5117 a 1306 +5117 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Bsw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 5117 a 1306 5117 a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Bse +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 5117 a 1306 5117 +a +tx@Dict begin PutEnd end + 1306 5117 a 1306 5117 a +tx@Dict begin { 170.71646 0.0 } PutCoor PutBegin end + 1306 5117 a @beginspecial @setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 2.87999 SLW 0 setgray 0.0 0.0 2 copy moveto +12.80365 0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave +1 setgray 1. .setopacityalpha fill grestore gsave 2.87999 SLW 0 + setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap stroke + grestore gsave 0.71999 SLW 1 setgray 1. .setopacityalpha stroke +grestore end + + +@endspecial 1306 5117 a +tx@Dict begin tx@NodeDict begin {0.0 0.0 13.34364 } false /N@C 11 +{InitCnode } /NodeScale {} def NewNode end end + 1306 5117 a 1306 5117 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 1306 +5117 a 1306 5117 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@C known { -12.80365 +0. /N@C load 0 GetEdge 0.0 0. AddOffset /N@C load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 1306 5117 a 1306 5117 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 1306 5117 +a 0 TeXcolorgray 1306 5117 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1306 5117 a 1306 5117 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + +1306 5117 a 1306 5117 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 1306 5117 a 1306 5117 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 1306 +5117 a 1306 5117 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1306 5117 a 1306 5117 a +tx@Dict begin ps@rot RotBegin end + 1306 5117 +a 1265 5130 a Fm(e)1307 5144 y Fr(3)1306 5117 y +tx@Dict begin RotEnd end + 1306 +5117 a 1306 5117 a +tx@Dict begin PutEnd end + 1306 5117 a 1306 5117 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1306 5117 +a 1306 5117 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1306 5117 a 0 TeXcolorgray 1306 5117 a +tx@Dict begin PutEnd end + +1306 5117 a 1306 5117 a +tx@Dict begin PutEnd end + 1306 5117 a 1306 5117 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Cw 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 +5117 a 1306 5117 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Ce 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 5117 a 1306 5117 a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@Cn 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 5117 +a 1306 5117 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@Cs 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 5117 a 1306 5117 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Cnw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 5117 a 1306 +5117 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Cne +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 5117 a 1306 5117 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Csw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 5117 a 1306 5117 +a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Cse +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 5117 a 1306 5117 a +tx@Dict begin PutEnd end + 1306 5117 a 1306 5117 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@Aw /N@A InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 1306 +5117 a 1306 5117 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@A /N@B InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 1306 5117 a 1306 5117 a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 add } def /ps@rot +{ } def grestore end + 1306 5117 +a 1306 5117 a +tx@Dict begin tx@NodeDict begin /t 0.45 def LPut end PutBegin end + 1306 5117 a 1306 5117 a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 9.83968 8.01474 +0.0 NAngle 90 add Uput exch pop add a PtoC h1 add exch w1 add exch +} PutCoor PutBegin end + 1306 5117 a 0 TeXcolorgray +1265 5151 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1265 5151 a 1265 5151 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1265 5151 a Fm(a)1265 +5151 y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1265 5151 a 1265 5151 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1265 5151 a 0 TeXcolorgray +1306 5117 a +tx@Dict begin PutEnd end + 1306 5117 a 1306 5117 a +tx@Dict begin PutEnd end + 1306 5117 a 1306 +5117 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 1.0 1.0 neg 0.0 0.0 0 0 /N@B /N@C InitNC { yB +yA sub xB xA sub Atan dup 15. add /AngleA exch def 15. sub 180 add +/AngleB exch def 0.8 0.8 GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub +Pyth 2 div dup 3 -1 roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def +/ArmTypeB 0 def GetArmA GetArmB xA2 yA2 xA1 yA1 2 copy /y0 ED /x0 ED +tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict begin ArrowB end /y3 +ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /cx x1 x0 sub 3 mul def /cy y1 +y0 sub 3 mul def /bx x2 x1 sub 3 mul cx sub def /by y2 y1 sub 3 mul +cy sub def /ax x3 x0 sub cx sub bx sub def /ay y3 y0 sub cy sub by +sub def /getValues { ax t0 3 exp mul bx t0 t0 mul mul add cx t0 mul +add x0 add ay t0 3 exp mul by t0 t0 mul mul add cy t0 mul add y0 add +ax t 3 exp mul bx t t mul mul add cx t mul add x0 add ay t 3 exp mul +by t t mul mul add cy t mul add y0 add } def /getdL { getValues 3 -1 +roll sub 3 1 roll sub Pyth } def /CurveLength { /u 0 def /du 0.01 def +0 100 { /t0 u def /u u du add def /t u def getdL add } repeat } def +/GetArrowPos { /ende 0.5 1 gt {ArrowPos}{ArrowPos CurveLength mul} +ifelse def /u 0 def /du 0.01 def /sum 0 def { /t0 u def /u u du add +def /t u def /sum getdL sum add def sum ende gt {exit} if } loop u +} def /dArrowPos 1. 1 gt {1.0 1. 1.0 add div }{ 0.5 } ifelse def /ArrowPos +0. def /loopNo 0.5 1 gt {CurveLength 0.5 div cvi }{ 1. } ifelse +def loopNo cvi { /ArrowPos ArrowPos dArrowPos add def /t GetArrowPos +def /t0 t 0.95 mul def getValues ArrowInside pop pop pop pop } repeat +x1 y1 x2 y2 x3 y3 curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 +] cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines +} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } if end + gsave 1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 +setlinecap stroke grestore grestore end + 1306 5117 a 1306 5117 a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 add } def /ps@rot +{ } def grestore end + 1306 5117 a 1306 5117 +a +tx@Dict begin tx@NodeDict begin /t 0.40 def LPut end PutBegin end + 1306 5117 a 1306 5117 a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 7.9889 12.92703 +0.0 NAngle 90 add Uput exch pop add a PtoC h1 add exch w1 add exch +} PutCoor PutBegin end + 1306 5117 a 0 TeXcolorgray +1273 5171 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1273 5171 a 1273 5171 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1273 5171 a Fm(b)1273 +5171 y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1273 5171 a 1273 5171 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1273 5171 a 0 TeXcolorgray +1306 5117 a +tx@Dict begin PutEnd end + 1306 5117 a 1306 5117 a +tx@Dict begin PutEnd end + 1306 5117 a 1306 +5117 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 1.0 1.0 neg 0.0 0.0 0 0 /N@C /N@B InitNC { yB +yA sub xB xA sub Atan dup 15. add /AngleA exch def 15. sub 180 add +/AngleB exch def 0.8 0.8 GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub +Pyth 2 div dup 3 -1 roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def +/ArmTypeB 0 def GetArmA GetArmB xA2 yA2 xA1 yA1 2 copy /y0 ED /x0 ED +tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict begin ArrowB end /y3 +ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /cx x1 x0 sub 3 mul def /cy y1 +y0 sub 3 mul def /bx x2 x1 sub 3 mul cx sub def /by y2 y1 sub 3 mul +cy sub def /ax x3 x0 sub cx sub bx sub def /ay y3 y0 sub cy sub by +sub def /getValues { ax t0 3 exp mul bx t0 t0 mul mul add cx t0 mul +add x0 add ay t0 3 exp mul by t0 t0 mul mul add cy t0 mul add y0 add +ax t 3 exp mul bx t t mul mul add cx t mul add x0 add ay t 3 exp mul +by t t mul mul add cy t mul add y0 add } def /getdL { getValues 3 -1 +roll sub 3 1 roll sub Pyth } def /CurveLength { /u 0 def /du 0.01 def +0 100 { /t0 u def /u u du add def /t u def getdL add } repeat } def +/GetArrowPos { /ende 0.5 1 gt {ArrowPos}{ArrowPos CurveLength mul} +ifelse def /u 0 def /du 0.01 def /sum 0 def { /t0 u def /u u du add +def /t u def /sum getdL sum add def sum ende gt {exit} if } loop u +} def /dArrowPos 1. 1 gt {1.0 1. 1.0 add div }{ 0.5 } ifelse def /ArrowPos +0. def /loopNo 0.5 1 gt {CurveLength 0.5 div cvi }{ 1. } ifelse +def loopNo cvi { /ArrowPos ArrowPos dArrowPos add def /t GetArrowPos +def /t0 t 0.95 mul def getValues ArrowInside pop pop pop pop } repeat +x1 y1 x2 y2 x3 y3 curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 +] cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines +} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } if end + gsave 1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 +setlinecap stroke grestore grestore end + 1306 5117 a 1306 5117 a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 add } def /ps@rot +{ } def grestore end + 1306 5117 a 1306 5117 +a +tx@Dict begin tx@NodeDict begin /t 0.40 def LPut end PutBegin end + 1306 5117 a 1306 5117 a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 7.9889 12.92703 +0.0 NAngle 90 add Uput exch pop add a PtoC h1 add exch w1 add exch +} PutCoor PutBegin end + 1306 5117 a 0 TeXcolorgray +1273 5171 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1273 5171 a 1273 5171 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1273 5171 a Fm(b)1273 +5171 y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1273 5171 a 1273 5171 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1273 5171 a 0 TeXcolorgray +1306 5117 a +tx@Dict begin PutEnd end + 1306 5117 a 1306 5117 a +tx@Dict begin PutEnd end + 1306 5117 a 1306 +5117 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@A /N@A InitNC { /AngleA +120. def /AngleB 60. def 7. 7. GetEdgeA GetEdgeB xA1 xB1 sub yA1 +yB1 sub Pyth 2 div dup 3 -1 roll mul /ArmA ED mul /ArmB ED /ArmTypeA +0 def /ArmTypeB 0 def GetArmA GetArmB xA2 yA2 xA1 yA1 2 copy /y0 ED +/x0 ED tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict begin ArrowB +end /y3 ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /cx x1 x0 sub 3 mul def +/cy y1 y0 sub 3 mul def /bx x2 x1 sub 3 mul cx sub def /by y2 y1 sub +3 mul cy sub def /ax x3 x0 sub cx sub bx sub def /ay y3 y0 sub cy sub +by sub def /getValues { ax t0 3 exp mul bx t0 t0 mul mul add cx t0 +mul add x0 add ay t0 3 exp mul by t0 t0 mul mul add cy t0 mul add y0 +add ax t 3 exp mul bx t t mul mul add cx t mul add x0 add ay t 3 exp +mul by t t mul mul add cy t mul add y0 add } def /getdL { getValues +3 -1 roll sub 3 1 roll sub Pyth } def /CurveLength { /u 0 def /du 0.01 +def 0 100 { /t0 u def /u u du add def /t u def getdL add } repeat } +def /GetArrowPos { /ende 0.5 1 gt {ArrowPos}{ArrowPos CurveLength +mul} ifelse def /u 0 def /du 0.01 def /sum 0 def { /t0 u 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round sub dup 0 rmoveto 0 360 arc closepath gsave 1 setgray + 1. .setopacityalpha fill grestore gsave 1.8 SLW 0 setgray 1. .setopacityalpha + 0 setlinejoin 0 setlinecap stroke grestore end + +@endspecial 1746 397 +a +tx@Dict begin tx@NodeDict begin {0.0 0.0 12.80365 } false /N@B 11 +{InitCnode } /NodeScale {} def NewNode end end + 1746 397 a 1746 397 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 1746 397 a 1746 397 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@B known { -12.80365 +0. /N@B load 0 GetEdge 0.0 0. AddOffset /N@B load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 1746 397 +a 1746 397 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 1746 397 a 0 TeXcolorgray 1746 397 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1746 +397 a 1746 397 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1746 397 a 1746 397 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 1746 397 a 1746 +397 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 1746 397 a 1746 397 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1746 397 a 1746 397 a +tx@Dict begin ps@rot RotBegin end + 1746 +397 a 1705 410 a Fm(e)1747 424 y Fr(1)1746 397 y +tx@Dict begin RotEnd end + 1746 +397 a 1746 397 a +tx@Dict begin PutEnd end + 1746 397 a 1746 397 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1746 397 a 1746 +397 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1746 397 a 0 TeXcolorgray 1746 397 a +tx@Dict begin PutEnd end + 1746 397 a +1746 397 a +tx@Dict begin PutEnd end + 1746 397 a 1746 397 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Bw 10 {InitPnode +} /NodeScale {} def NewNode end end + 1746 397 a 1746 397 +a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Be 10 {InitPnode +} /NodeScale {} def NewNode end end + 1746 397 a 1746 397 a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@Bn 10 {InitPnode +} /NodeScale {} def NewNode end end + 1746 397 a 1746 397 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@Bs 10 {InitPnode +} /NodeScale {} def NewNode end end + 1746 397 +a 1746 397 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Bnw +10 {InitPnode } /NodeScale {} def NewNode end end + 1746 397 a 1746 397 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Bne +10 {InitPnode } /NodeScale {} def NewNode end end + 1746 397 a 1746 397 +a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Bsw +10 {InitPnode } /NodeScale {} def NewNode end end + 1746 397 a 1746 397 a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Bse +10 {InitPnode } /NodeScale {} def NewNode end end + 1746 397 a 1746 397 a +tx@Dict begin PutEnd end + 1746 397 +a 1746 397 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1746 397 a @beginspecial @setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 2.87999 SLW 0 setgray 0.0 0.0 2 copy moveto +12.80365 0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave +1 setgray 1. .setopacityalpha fill grestore gsave 2.87999 SLW 0 + setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap stroke + grestore gsave 0.71999 SLW 1 setgray 1. .setopacityalpha stroke +grestore end + +@endspecial +1746 397 a +tx@Dict begin tx@NodeDict begin {0.0 0.0 13.34364 } false /N@A 11 +{InitCnode } /NodeScale {} def NewNode end end + 1746 397 a 1746 397 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 1746 397 a 1746 397 +a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@A known { -12.80365 +0. /N@A load 0 GetEdge 0.0 0. AddOffset /N@A load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 1746 397 a 1746 397 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 1746 397 a 0 TeXcolorgray 1746 +397 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1746 397 a 1746 397 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1746 397 a 1746 397 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 1746 +397 a 1746 397 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 1746 397 a 1746 397 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1746 397 a 1746 +397 a +tx@Dict begin ps@rot RotBegin end + 1746 397 a 1705 410 a Fm(e)1747 424 y Fr(0)1746 +397 y +tx@Dict begin RotEnd end + 1746 397 a 1746 397 a +tx@Dict begin PutEnd end + 1746 397 a 1746 397 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1746 +397 a 1746 397 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1746 397 a 0 TeXcolorgray 1746 397 a +tx@Dict begin PutEnd end + +1746 397 a 1746 397 a +tx@Dict begin PutEnd end + 1746 397 a 1746 397 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Aw 10 {InitPnode +} /NodeScale {} def NewNode end end + 1746 397 +a 1746 397 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Ae 10 {InitPnode +} /NodeScale {} def NewNode end end + 1746 397 a 1746 397 a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@An 10 {InitPnode +} /NodeScale {} def NewNode end end + 1746 397 a 1746 397 +a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@As 10 {InitPnode +} /NodeScale {} def NewNode end end + 1746 397 a 1746 397 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Anw +10 {InitPnode } /NodeScale {} def NewNode end end + 1746 397 a 1746 397 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Ane +10 {InitPnode } /NodeScale {} def NewNode end end + 1746 397 +a 1746 397 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Asw +10 {InitPnode } /NodeScale {} def NewNode end end + 1746 397 a 1746 397 a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Ase +10 {InitPnode } /NodeScale {} def NewNode end end + 1746 397 a 1746 397 +a +tx@Dict begin PutEnd end + 1746 397 a 1746 397 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@Aw /N@A InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 1746 397 a 1746 397 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 1.0 1.0 neg 0.0 0.0 0 0 /N@A /N@B InitNC { yB +yA sub xB xA sub Atan dup 15. add /AngleA exch def 15. sub 180 add +/AngleB exch def 0.8 0.8 GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub +Pyth 2 div dup 3 -1 roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def +/ArmTypeB 0 def GetArmA GetArmB xA2 yA2 xA1 yA1 2 copy /y0 ED /x0 ED +tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict begin ArrowB end /y3 +ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /cx x1 x0 sub 3 mul def /cy y1 +y0 sub 3 mul def /bx x2 x1 sub 3 mul cx sub def /by y2 y1 sub 3 mul +cy sub def /ax x3 x0 sub cx sub bx sub def /ay y3 y0 sub cy sub by +sub def /getValues { ax t0 3 exp mul bx t0 t0 mul mul add cx t0 mul +add x0 add ay t0 3 exp mul by t0 t0 mul mul add cy t0 mul add y0 add +ax t 3 exp mul bx t t mul mul add cx t mul add x0 add ay t 3 exp mul +by t t mul mul add cy t mul add y0 add } def /getdL { getValues 3 -1 +roll sub 3 1 roll sub Pyth } def /CurveLength { /u 0 def /du 0.01 def +0 100 { /t0 u def /u u du add def /t u def getdL add } repeat } def +/GetArrowPos { /ende 0.5 1 gt {ArrowPos}{ArrowPos CurveLength mul} +ifelse def /u 0 def /du 0.01 def /sum 0 def { /t0 u def /u u du add +def /t u def /sum getdL sum add def sum ende gt {exit} if } loop u +} def /dArrowPos 1. 1 gt {1.0 1. 1.0 add div }{ 0.5 } ifelse def /ArrowPos +0. def /loopNo 0.5 1 gt {CurveLength 0.5 div cvi }{ 1. } ifelse +def loopNo cvi { /ArrowPos ArrowPos dArrowPos add def /t GetArrowPos +def /t0 t 0.95 mul def getValues ArrowInside pop pop pop pop } repeat +x1 y1 x2 y2 x3 y3 curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 +] cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines +} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } if end + gsave 1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 +setlinecap stroke grestore grestore end + 1746 397 +a 1746 397 a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 add } def /ps@rot +{ } def grestore end + 1746 397 a 1746 397 a +tx@Dict begin tx@NodeDict begin /t 0.40 def LPut end PutBegin end + 1746 397 a 1746 397 +a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 7.9889 12.92703 +0.0 NAngle 90 add Uput exch pop add a PtoC h1 add exch w1 add exch +} PutCoor PutBegin end + 1746 397 a 0 TeXcolorgray 1713 451 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1713 451 a 1713 +451 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1713 451 a Fm(b)1713 451 y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1713 451 a 1713 451 +a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1713 451 a 0 TeXcolorgray 1746 397 a +tx@Dict begin PutEnd end + 1746 397 a 1746 +397 a +tx@Dict begin PutEnd end + 1746 397 a 1746 397 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 1.0 1.0 neg 0.0 0.0 0 0 /N@B /N@A InitNC { yB +yA sub xB xA sub Atan dup 15. add /AngleA exch def 15. sub 180 add +/AngleB exch def 0.8 0.8 GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub +Pyth 2 div dup 3 -1 roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def +/ArmTypeB 0 def GetArmA GetArmB xA2 yA2 xA1 yA1 2 copy /y0 ED /x0 ED +tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict begin ArrowB end /y3 +ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /cx x1 x0 sub 3 mul def /cy y1 +y0 sub 3 mul def /bx x2 x1 sub 3 mul cx sub def /by y2 y1 sub 3 mul +cy sub def /ax x3 x0 sub cx sub bx sub def /ay y3 y0 sub cy sub by +sub def /getValues { ax t0 3 exp mul bx t0 t0 mul mul add cx t0 mul +add x0 add ay t0 3 exp mul by t0 t0 mul mul add cy t0 mul add y0 add +ax t 3 exp mul bx t t mul mul add cx t mul add x0 add ay t 3 exp mul +by t t mul mul add cy t mul add y0 add } def /getdL { getValues 3 -1 +roll sub 3 1 roll sub Pyth } def /CurveLength { /u 0 def /du 0.01 def +0 100 { /t0 u def /u u du add def /t u def getdL add } repeat } def +/GetArrowPos { /ende 0.5 1 gt {ArrowPos}{ArrowPos CurveLength mul} +ifelse def /u 0 def /du 0.01 def /sum 0 def { /t0 u def /u u du add +def /t u def /sum getdL sum add def sum ende gt {exit} if } loop u +} def /dArrowPos 1. 1 gt {1.0 1. 1.0 add div }{ 0.5 } ifelse def /ArrowPos +0. def /loopNo 0.5 1 gt {CurveLength 0.5 div cvi }{ 1. } ifelse +def loopNo cvi { /ArrowPos ArrowPos dArrowPos add def /t GetArrowPos +def /t0 t 0.95 mul def getValues ArrowInside pop pop pop pop } repeat +x1 y1 x2 y2 x3 y3 curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 +] cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines +} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } if end + gsave 1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 +setlinecap stroke grestore grestore end + 1746 397 a 1746 397 a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 add } def /ps@rot +{ } def grestore end + 1746 +397 a 1746 397 a +tx@Dict begin tx@NodeDict begin /t 0.40 def LPut end PutBegin end + 1746 397 a 1746 397 a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 9.83968 8.01474 +0.0 NAngle 90 add Uput exch pop add a PtoC h1 add exch w1 add exch +} PutCoor PutBegin end + 1746 397 a 0 TeXcolorgray +1705 430 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1705 430 a 1705 430 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1705 430 a Fm(a)1705 +430 y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1705 430 a 1705 430 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1705 430 a 0 TeXcolorgray +1746 397 a +tx@Dict begin PutEnd end + 1746 397 a 1746 397 a +tx@Dict begin PutEnd end + 1746 397 a 1746 397 +a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@B /N@B InitNC { /AngleA +30. def /AngleB -30. def 7. 7. GetEdgeA GetEdgeB xA1 xB1 sub yA1 +yB1 sub Pyth 2 div dup 3 -1 roll mul /ArmA ED mul /ArmB ED /ArmTypeA +0 def /ArmTypeB 0 def GetArmA GetArmB xA2 yA2 xA1 yA1 2 copy /y0 ED +/x0 ED tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict begin ArrowB +end /y3 ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /cx x1 x0 sub 3 mul def +/cy y1 y0 sub 3 mul def /bx x2 x1 sub 3 mul cx sub def /by y2 y1 sub +3 mul cy sub def /ax x3 x0 sub cx sub bx sub def /ay y3 y0 sub cy sub +by sub def /getValues { ax t0 3 exp mul bx t0 t0 mul mul add cx t0 +mul add x0 add ay t0 3 exp mul by t0 t0 mul mul add cy t0 mul add y0 +add ax t 3 exp mul bx t t mul mul add cx t mul add x0 add ay t 3 exp +mul by t t mul mul add cy t mul add y0 add } def /getdL { getValues +3 -1 roll sub 3 1 roll sub Pyth } def /CurveLength { /u 0 def /du 0.01 +def 0 100 { /t0 u def /u u du add def /t u def getdL add } repeat } +def /GetArrowPos { /ende 0.5 1 gt {ArrowPos}{ArrowPos CurveLength +mul} ifelse def /u 0 def /du 0.01 def /sum 0 def { /t0 u 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setgray 0.0 0.0 2 copy moveto 12.80365 +0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave 1 setgray + 1. .setopacityalpha fill grestore gsave 1.8 SLW 0 setgray 1. .setopacityalpha + 0 setlinejoin 0 setlinecap stroke grestore end + +@endspecial +1306 3295 a +tx@Dict begin tx@NodeDict begin {0.0 0.0 12.80365 } false /N@A 11 +{InitCnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 1306 3295 a 1306 +3295 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@A known { -12.80365 +0. /N@A load 0 GetEdge 0.0 0. AddOffset /N@A load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 1306 3295 a 1306 3295 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 1306 3295 a 0 TeXcolorgray +1306 3295 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1306 3295 a 1306 3295 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1306 3295 a 1306 +3295 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 1306 3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 1306 3295 a 1306 3295 +a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1306 3295 a 1306 3295 a +tx@Dict begin ps@rot RotBegin end + 1306 3295 a 1265 3308 a Fm(e)1307 +3322 y Fr(1)1306 3295 y +tx@Dict begin RotEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 +3295 a 1306 3295 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1306 3295 a 1306 3295 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1306 3295 +a 0 TeXcolorgray 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin PutEnd end + +1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Aw 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Ae 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 +3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@An 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@As 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 +a 1306 3295 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Anw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Ane +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 +3295 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Asw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Ase +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 +a +tx@Dict begin PutEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin { 56.90549 28.45274 } PutCoor PutBegin end + 1306 3295 a @beginspecial @setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 1.8 SLW 0 setgray 0.0 0.0 2 copy moveto 12.80365 +0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave 1 setgray + 1. .setopacityalpha fill grestore gsave 1.8 SLW 0 setgray 1. .setopacityalpha + 0 setlinejoin 0 setlinecap stroke grestore end + + +@endspecial 1306 3295 a +tx@Dict begin tx@NodeDict begin {0.0 0.0 12.80365 } false /N@B 11 +{InitCnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 1306 +3295 a 1306 3295 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@B known { -12.80365 +0. /N@B load 0 GetEdge 0.0 0. AddOffset /N@B load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 1306 3295 a 1306 3295 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 1306 3295 +a 0 TeXcolorgray 1306 3295 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1306 3295 a 1306 3295 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + +1306 3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 1306 3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 1306 +3295 a 1306 3295 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1306 3295 a 1306 3295 a +tx@Dict begin ps@rot RotBegin end + 1306 3295 +a 1265 3308 a Fm(e)1307 3322 y Fr(2)1306 3295 y +tx@Dict begin RotEnd end + 1306 +3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1306 3295 +a 1306 3295 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1306 3295 a 0 TeXcolorgray 1306 3295 a +tx@Dict begin PutEnd end + +1306 3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Bw 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 +3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Be 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@Bn 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 +a 1306 3295 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@Bs 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Bnw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 +3295 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Bne +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Bsw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 +a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Bse +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin { 113.81097 28.45274 } PutCoor PutBegin end + 1306 +3295 a @beginspecial @setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 1.8 SLW 0 setgray 0.0 0.0 2 copy moveto 12.80365 +0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave 1 setgray + 1. .setopacityalpha fill grestore gsave 1.8 SLW 0 setgray 1. .setopacityalpha + 0 setlinejoin 0 setlinecap stroke grestore end + +@endspecial 1306 3295 +a +tx@Dict begin tx@NodeDict begin {0.0 0.0 12.80365 } false /N@C 11 +{InitCnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 1306 3295 a 1306 3295 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@C known { -12.80365 +0. /N@C load 0 GetEdge 0.0 0. AddOffset /N@C load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 1306 +3295 a 1306 3295 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 1306 3295 a 0 TeXcolorgray 1306 3295 +a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1306 3295 a 1306 3295 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1306 3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 1306 +3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 1306 3295 a 1306 3295 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1306 3295 +a 1306 3295 a +tx@Dict begin ps@rot RotBegin end + 1306 3295 a 1265 3308 a Fm(e)1307 3322 +y Fr(3)1306 3295 y +tx@Dict begin RotEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 +a 1306 3295 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1306 3295 a 1306 3295 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1306 3295 a 0 TeXcolorgray +1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 +3295 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Cw 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Ce 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 +a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@Cn 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@Cs 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Cnw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 +3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Cne +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Csw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 +a 1306 3295 a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Cse +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 +3295 a +tx@Dict begin { 113.81097 85.35823 } PutCoor PutBegin end + 1306 3295 a @beginspecial @setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 1.8 SLW 0 setgray 0.0 0.0 2 copy moveto 12.80365 +0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave 1 setgray + 1. .setopacityalpha fill grestore gsave 1.8 SLW 0 setgray 1. .setopacityalpha + 0 setlinejoin 0 setlinecap stroke grestore end + +@endspecial +1306 3295 a +tx@Dict begin tx@NodeDict begin {0.0 0.0 12.80365 } false /N@E 11 +{InitCnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 1306 3295 a 1306 +3295 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@E known { -12.80365 +0. /N@E load 0 GetEdge 0.0 0. AddOffset /N@E load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 1306 3295 a 1306 3295 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 1306 3295 a 0 TeXcolorgray +1306 3295 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1306 3295 a 1306 3295 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1306 3295 a 1306 +3295 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 1306 3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 1306 3295 a 1306 3295 +a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1306 3295 a 1306 3295 a +tx@Dict begin ps@rot RotBegin end + 1306 3295 a 1265 3308 a Fm(e)1307 +3322 y Fr(5)1306 3295 y +tx@Dict begin RotEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 +3295 a 1306 3295 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1306 3295 a 1306 3295 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1306 3295 +a 0 TeXcolorgray 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin PutEnd end + +1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Ew 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Ee 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 +3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@En 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@Es 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 +a 1306 3295 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Enw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Ene +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 +3295 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Esw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Ese +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 +a +tx@Dict begin PutEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin { 56.90549 -28.45274 } PutCoor PutBegin end + 1306 3295 a @beginspecial @setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 1.8 SLW 0 setgray 0.0 0.0 2 copy moveto 12.80365 +0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave 1 setgray + 1. .setopacityalpha fill grestore gsave 1.8 SLW 0 setgray 1. .setopacityalpha + 0 setlinejoin 0 setlinecap stroke grestore end + + +@endspecial 1306 3295 a +tx@Dict begin tx@NodeDict begin {0.0 0.0 12.80365 } false /N@G 11 +{InitCnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 1306 +3295 a 1306 3295 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@G known { -12.80365 +0. /N@G load 0 GetEdge 0.0 0. AddOffset /N@G load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 1306 3295 a 1306 3295 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 1306 3295 +a 0 TeXcolorgray 1306 3295 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1306 3295 a 1306 3295 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + +1306 3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 1306 3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 1306 +3295 a 1306 3295 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1306 3295 a 1306 3295 a +tx@Dict begin ps@rot RotBegin end + 1306 3295 +a 1265 3308 a Fm(e)1307 3322 y Fr(8)1306 3295 y +tx@Dict begin RotEnd end + 1306 +3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1306 3295 +a 1306 3295 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1306 3295 a 0 TeXcolorgray 1306 3295 a +tx@Dict begin PutEnd end + +1306 3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Gw 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 +3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Ge 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@Gn 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 +a 1306 3295 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@Gs 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Gnw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 +3295 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Gne +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Gsw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 +a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Gse +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin { 113.81097 -28.45274 } PutCoor PutBegin end + 1306 +3295 a @beginspecial @setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 1.8 SLW 0 setgray 0.0 0.0 2 copy moveto 12.80365 +0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave 1 setgray + 1. .setopacityalpha fill grestore gsave 1.8 SLW 0 setgray 1. .setopacityalpha + 0 setlinejoin 0 setlinecap stroke grestore end + +@endspecial 1306 3295 +a +tx@Dict begin tx@NodeDict begin {0.0 0.0 12.80365 } false /N@H 11 +{InitCnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 1306 3295 a 1306 3295 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@H known { -12.80365 +0. /N@H load 0 GetEdge 0.0 0. AddOffset /N@H load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 1306 +3295 a 1306 3295 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 1306 3295 a 0 TeXcolorgray 1306 3295 +a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1306 3295 a 1306 3295 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1306 3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 1306 +3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 1306 3295 a 1306 3295 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1306 3295 +a 1306 3295 a +tx@Dict begin ps@rot RotBegin end + 1306 3295 a 1265 3308 a Fm(e)1307 3322 +y Fr(9)1306 3295 y +tx@Dict begin RotEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 +a 1306 3295 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1306 3295 a 1306 3295 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1306 3295 a 0 TeXcolorgray +1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 +3295 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Hw 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@He 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 +a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@Hn 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@Hs 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Hnw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 +3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Hne +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Hsw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 +a 1306 3295 a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Hse +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 +3295 a +tx@Dict begin { 170.71646 -28.45274 } PutCoor PutBegin end + 1306 3295 a @beginspecial @setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 2.87999 SLW 0 setgray 0.0 0.0 2 copy moveto +12.80365 0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave +1 setgray 1. .setopacityalpha fill grestore gsave 2.87999 SLW 0 + setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap stroke + grestore gsave 0.71999 SLW 1 setgray 1. .setopacityalpha stroke +grestore end + +@endspecial +1306 3295 a +tx@Dict begin tx@NodeDict begin {0.0 0.0 13.34364 } false /N@I 11 +{InitCnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 1306 3295 a 1306 +3295 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@I known { -12.80365 +0. /N@I load 0 GetEdge 0.0 0. AddOffset /N@I load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 1306 3295 a 1306 3295 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 1306 3295 a 0 TeXcolorgray +1306 3295 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1306 3295 a 1306 3295 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1306 3295 a 1306 +3295 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 1306 3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 1306 3295 a 1306 3295 +a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1306 3295 a 1306 3295 a +tx@Dict begin ps@rot RotBegin end + 1306 3295 a 1247 3308 a Fm(e)1289 +3322 y Fr(10)1306 3295 y +tx@Dict begin RotEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 +3295 a 1306 3295 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1306 3295 a 1306 3295 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1306 3295 +a 0 TeXcolorgray 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin PutEnd end + +1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Iw 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Ie 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 +3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@In 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@Is 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 +a 1306 3295 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Inw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Ine +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 +3295 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Isw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Ise +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 +a +tx@Dict begin PutEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin { 170.71646 28.45274 } PutCoor PutBegin end + 1306 3295 a @beginspecial @setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 2.87999 SLW 0 setgray 0.0 0.0 2 copy moveto +12.80365 0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave +1 setgray 1. .setopacityalpha fill grestore gsave 2.87999 SLW 0 + setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap stroke + grestore gsave 0.71999 SLW 1 setgray 1. .setopacityalpha stroke +grestore end + + +@endspecial 1306 3295 a +tx@Dict begin tx@NodeDict begin {0.0 0.0 13.34364 } false /N@D 11 +{InitCnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 1306 +3295 a 1306 3295 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@D known { -12.80365 +0. /N@D load 0 GetEdge 0.0 0. AddOffset /N@D load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 1306 3295 a 1306 3295 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 1306 3295 +a 0 TeXcolorgray 1306 3295 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1306 3295 a 1306 3295 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + +1306 3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 1306 3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 1306 +3295 a 1306 3295 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1306 3295 a 1306 3295 a +tx@Dict begin ps@rot RotBegin end + 1306 3295 +a 1265 3308 a Fm(e)1307 3322 y Fr(4)1306 3295 y +tx@Dict begin RotEnd end + 1306 +3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1306 3295 +a 1306 3295 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1306 3295 a 0 TeXcolorgray 1306 3295 a +tx@Dict begin PutEnd end + +1306 3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Dw 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 +3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@De 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@Dn 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 +a 1306 3295 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@Ds 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Dnw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 +3295 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Dne +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Dsw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 +a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Dse +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin { 170.71646 85.35823 } PutCoor PutBegin end + 1306 +3295 a @beginspecial @setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 2.87999 SLW 0 setgray 0.0 0.0 2 copy moveto +12.80365 0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave +1 setgray 1. .setopacityalpha fill grestore gsave 2.87999 SLW 0 + setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap stroke + grestore gsave 0.71999 SLW 1 setgray 1. .setopacityalpha stroke +grestore end + +@endspecial 1306 3295 +a +tx@Dict begin tx@NodeDict begin {0.0 0.0 13.34364 } false /N@F 11 +{InitCnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 1306 3295 a 1306 3295 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@F known { -12.80365 +0. /N@F load 0 GetEdge 0.0 0. AddOffset /N@F load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 1306 +3295 a 1306 3295 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 1306 3295 a 0 TeXcolorgray 1306 3295 +a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1306 3295 a 1306 3295 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1306 3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 1306 +3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 1306 3295 a 1306 3295 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1306 3295 +a 1306 3295 a +tx@Dict begin ps@rot RotBegin end + 1306 3295 a 1265 3308 a Fm(e)1307 3322 +y Fr(6)1306 3295 y +tx@Dict begin RotEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 +a 1306 3295 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1306 3295 a 1306 3295 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1306 3295 a 0 TeXcolorgray +1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 +3295 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Fw 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Fe 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 +a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@Fn 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@Fs 10 {InitPnode +} /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Fnw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 +3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Fne +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Fsw +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 +a 1306 3295 a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Fse +10 {InitPnode } /NodeScale {} def NewNode end end + 1306 3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 +3295 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@Aw /N@A InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 1306 3295 a 1306 3295 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@A /N@B InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 1306 3295 a 1306 3295 +a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 add } def /ps@rot +{ } def grestore end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin /t 0.45 def LPut end PutBegin end + 1306 3295 a 1306 3295 a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 9.83968 8.01474 +0.0 NAngle 90 add Uput exch pop add a PtoC h1 add exch w1 add exch +} PutCoor PutBegin end + 1306 +3295 a 0 TeXcolorgray 1265 3328 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1265 3328 a 1265 3328 +a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1265 3328 a Fm(a)1265 3328 y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1265 3328 a 1265 3328 +a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1265 3328 a 0 TeXcolorgray 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a +1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@B /N@C InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 1306 3295 a 1306 +3295 a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 add } def /ps@rot +{ } def grestore end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin /t 0.45 def LPut end PutBegin end + 1306 3295 a 1306 3295 +a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 9.83968 8.01474 +0.0 NAngle 90 add Uput exch pop add a PtoC h1 add exch w1 add exch +} PutCoor PutBegin end + 1306 3295 a 0 TeXcolorgray 1265 3328 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1265 3328 a +1265 3328 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1265 3328 a Fm(a)1265 3328 y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1265 3328 a +1265 3328 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1265 3328 a 0 TeXcolorgray 1306 3295 a +tx@Dict begin PutEnd end + 1306 +3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@C /N@D InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 1306 3295 +a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 add } def /ps@rot +{ } def grestore end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin /t 0.45 def LPut end PutBegin end + 1306 3295 a 1306 +3295 a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 7.9889 12.92703 +0.0 NAngle 90 add Uput exch pop add a PtoC h1 add exch w1 add exch +} PutCoor PutBegin end + 1306 3295 a 0 TeXcolorgray 1273 3348 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1273 3348 +a 1273 3348 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1273 3348 a Fm(b)1273 3348 y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1273 3348 +a 1273 3348 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1273 3348 a 0 TeXcolorgray 1306 3295 a +tx@Dict begin PutEnd end + +1306 3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@B /N@E InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 1306 +3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 add } def /ps@rot +{ } def grestore end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin /t 0.45 def LPut end PutBegin end + 1306 3295 +a 1306 3295 a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 7.9889 12.92703 +0.0 NAngle 90 add Uput exch pop add a PtoC h1 add exch w1 add exch +} PutCoor PutBegin end + 1306 3295 a 0 TeXcolorgray 1273 3348 a +tx@Dict begin CP CP translate 1. 1. scale NET end + +1273 3348 a 1273 3348 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1273 3348 a Fm(b)1273 3348 y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + +1273 3348 a 1273 3348 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1273 3348 a 0 TeXcolorgray 1306 +3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 3295 +a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@E /N@F InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 1306 3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 add } def /ps@rot +{ } def grestore end + 1306 3295 a 1306 3295 a +tx@Dict begin tx@NodeDict begin /t 0.45 def LPut end PutBegin end + 1306 +3295 a 1306 3295 a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 7.9889 12.92703 +0.0 NAngle 90 add Uput exch pop add a PtoC h1 add exch w1 add exch +} PutCoor PutBegin end + 1306 3295 a 0 TeXcolorgray 1273 3348 +a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1273 3348 a 1273 3348 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1273 3348 a Fm(b)1273 3348 +y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1273 3348 a 1273 3348 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1273 3348 a 0 TeXcolorgray +1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 +3295 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@A /N@G InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 1306 3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 add } def /ps@rot +{ } def grestore end + 1306 3295 a 1306 3295 +a +tx@Dict begin tx@NodeDict begin /t 0.45 def LPut end PutBegin end + 1306 3295 a 1306 3295 a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 7.9889 12.92703 +0.0 NAngle 90 add Uput exch pop add a PtoC h1 add exch w1 add exch +} PutCoor PutBegin end + 1306 3295 a 0 TeXcolorgray +1273 3348 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1273 3348 a 1273 3348 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1273 3348 a Fm(b)1273 +3348 y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1273 3348 a 1273 3348 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1273 3348 a 0 TeXcolorgray +1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 +3295 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@G /N@H InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 1306 3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 add } def /ps@rot +{ } def grestore end + 1306 3295 a 1306 3295 +a +tx@Dict begin tx@NodeDict begin /t 0.45 def LPut end PutBegin end + 1306 3295 a 1306 3295 a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 7.9889 12.92703 +0.0 NAngle 90 add Uput exch pop add a PtoC h1 add exch w1 add exch +} PutCoor PutBegin end + 1306 3295 a 0 TeXcolorgray +1273 3348 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1273 3348 a 1273 3348 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1273 3348 a Fm(b)1273 +3348 y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1273 3348 a 1273 3348 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1273 3348 a 0 TeXcolorgray +1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 +3295 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@H /N@I InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 1306 3295 a 1306 3295 a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 add } def /ps@rot +{ } def grestore end + 1306 3295 a 1306 3295 +a +tx@Dict begin tx@NodeDict begin /t 0.45 def LPut end PutBegin end + 1306 3295 a 1306 3295 a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 7.9889 12.92703 +0.0 NAngle 90 add Uput exch pop add a PtoC h1 add exch w1 add exch +} PutCoor PutBegin end + 1306 3295 a 0 TeXcolorgray +1273 3348 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1273 3348 a 1273 3348 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1273 3348 a Fm(b)1273 +3348 y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1273 3348 a 1273 3348 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1273 3348 a 0 TeXcolorgray +1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 3295 a +tx@Dict begin PutEnd end + 1306 3295 a 1306 +3058 a +tx@Dict begin CP CP translate 1 0.6 div 1 0.6 div scale NET end + 1306 3058 a 0 TeXcolorgray -13 3647 a Fv(4)p 0 +TeXcolorgray 46 w Fz(Le)k(probl\350me)i(p)s(eut)e(sem)m(bler)i +(complexe)f(car)f(l'\351nonc\351)i(du)e(mot)h(allum\351)g(fait)g +(imm\351diatemen)m(t)-5 3760 y(p)s(enser)27 b(\340)h(\351tein)m(t.)g +(Naturellemen)m(t,)i(on)e(p)s(ense)f(\340)g(un)g(\351tat)h(allum\351)g +(et)g(\340)f(un)g(autre)h(\351tein)m(t.)h(Comme)f(il)-5 +3873 y(y)h(a)h(trois)f(couleurs,)h(on)g(s'imagine)g(alors)g(six)e +(\351tats)i(et)g(leur)f(repr\351sen)m(tation)j(a)m(v)m(ec)f(des)e +(transitions)-5 3986 y(du)i(t)m(yp)s(e)f(\020s'allume\021)39 +b(ou)31 b(\020s'\351tein)m(t\021)39 b(ne)30 b(v)-5 b(a)30 +b(pas)h(de)f(soit.)136 4099 y(P)m(our)35 b(s'en)e(sortir)h(simplemen)m +(t,)h(il)e(faut)g(au)h(con)m(traire)h(p)s(enser)e(\020quel)g(est)g(la)h +(couleur)g(du)f(feu\021.)-5 4211 y(Cela)d(supp)s(ose)e(qu'on)h(ne)g +(consid\350re)h(que)f(l'\351tat)g(allum\351)h(du)f(feu.)g(On)g(p)s(eut) +g(alors)g(dire)g(rouge,)g(puis)-5 4324 y(jaune,)k(puis)f(v)m(ert,)h +(puis)g(rouge,)g(.)15 b(.)g(.)33 b(On)g(p)s(ourrait)g(aussi)f(dire,)h +(c'est)f(le)h(rouge)g(qui)f(s'allume,)h(puis)-5 4437 +y(c'est)h(le)f(jaune)h(qui)f(s'allume,)h(puis)g(.)15 +b(.)g(.)33 b(Cela)h(nous)g(p)s(ermet)g(alors)f(de)h(repr\351sen)m(ter)h +(la)e(succession)-5 4550 y(des)d(\351tats)h(allum\351)g(du)g(feu)f(par) +h(:)p 0 TeXcolorgray 0 TeXcolorgray 952 4928 a +tx@Dict begin CP CP translate 0.6 0.6 scale NET end + 952 4928 +a 952 4692 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 952 4692 a @beginspecial @setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 1.8 SLW 0 setgray 0.0 0.0 2 copy moveto 12.80365 +0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave 1 setgray + 1. .setopacityalpha fill grestore gsave 1.8 SLW 0 setgray 1. .setopacityalpha + 0 setlinejoin 0 setlinecap stroke grestore end + +@endspecial +952 4692 a +tx@Dict begin tx@NodeDict begin {0.0 0.0 12.80365 } false /N@A 11 +{InitCnode } /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 952 4692 a 952 4692 +a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@A known { -12.80365 +0. /N@A load 0 GetEdge 0.0 0. AddOffset /N@A load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 952 4692 a 952 4692 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 952 4692 a 0 TeXcolorgray 952 +4692 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 952 4692 a 952 4692 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 952 4692 a 952 4692 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 952 +4692 a 952 4692 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 952 4692 a 952 4692 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 952 4692 a 952 +4692 a +tx@Dict begin ps@rot RotBegin end + 952 4692 a 930 4712 a Fm(r)952 4692 y +tx@Dict begin RotEnd end + 952 4692 +a 952 4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 4692 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 952 4692 a 952 4692 +a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 952 4692 a 0 TeXcolorgray 952 4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 +4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Aw 10 {InitPnode +} /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Ae 10 {InitPnode +} /NodeScale {} def NewNode end end + 952 +4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@An 10 {InitPnode +} /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@As 10 {InitPnode +} /NodeScale {} def NewNode end end + 952 4692 a 952 +4692 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Anw +10 {InitPnode } /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Ane +10 {InitPnode } /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Asw +10 {InitPnode } /NodeScale {} def NewNode end end + 952 +4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Ase +10 {InitPnode } /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 +4692 a +tx@Dict begin { 113.81097 0.0 } PutCoor PutBegin end + 952 4692 a @beginspecial @setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 1.8 SLW 0 setgray 0.0 0.0 2 copy moveto 12.80365 +0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave 1 setgray + 1. .setopacityalpha fill grestore gsave 1.8 SLW 0 setgray 1. .setopacityalpha + 0 setlinejoin 0 setlinecap stroke grestore end + +@endspecial +952 4692 a +tx@Dict begin tx@NodeDict begin {0.0 0.0 12.80365 } false /N@B 11 +{InitCnode } /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 952 4692 a 952 4692 +a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@B known { -12.80365 +0. /N@B load 0 GetEdge 0.0 0. AddOffset /N@B load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 952 4692 a 952 4692 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 952 4692 a 0 TeXcolorgray 952 +4692 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 952 4692 a 952 4692 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 952 4692 a 952 4692 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 952 +4692 a 952 4692 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 952 4692 a 952 4692 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 952 4692 a 952 +4692 a +tx@Dict begin ps@rot RotBegin end + 952 4692 a 931 4713 a Fm(j)952 4692 y +tx@Dict begin RotEnd end + 952 4692 +a 952 4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 4692 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 952 4692 a 952 4692 +a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 952 4692 a 0 TeXcolorgray 952 4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 +4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Bw 10 {InitPnode +} /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Be 10 {InitPnode +} /NodeScale {} def NewNode end end + 952 +4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@Bn 10 {InitPnode +} /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@Bs 10 {InitPnode +} /NodeScale {} def NewNode end end + 952 4692 a 952 +4692 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Bnw +10 {InitPnode } /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Bne +10 {InitPnode } /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Bsw +10 {InitPnode } /NodeScale {} def NewNode end end + 952 +4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Bse +10 {InitPnode } /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 +4692 a +tx@Dict begin { 56.90549 -56.90549 } PutCoor PutBegin end + 952 4692 a @beginspecial @setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 1.8 SLW 0 setgray 0.0 0.0 2 copy moveto 12.80365 +0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave 1 setgray + 1. .setopacityalpha fill grestore gsave 1.8 SLW 0 setgray 1. .setopacityalpha + 0 setlinejoin 0 setlinecap stroke grestore end + +@endspecial +952 4692 a +tx@Dict begin tx@NodeDict begin {0.0 0.0 12.80365 } false /N@C 11 +{InitCnode } /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 952 4692 a 952 4692 +a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@C known { -12.80365 +0. /N@C load 0 GetEdge 0.0 0. AddOffset /N@C load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 952 4692 a 952 4692 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 952 4692 a 0 TeXcolorgray 952 +4692 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 952 4692 a 952 4692 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 952 4692 a 952 4692 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 952 +4692 a 952 4692 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 952 4692 a 952 4692 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 952 4692 a 952 +4692 a +tx@Dict begin ps@rot RotBegin end + 952 4692 a 928 4712 a Fm(v)952 4692 y +tx@Dict begin RotEnd end + 952 4692 +a 952 4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 4692 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 952 4692 a 952 4692 +a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 952 4692 a 0 TeXcolorgray 952 4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 +4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Cw 10 {InitPnode +} /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Ce 10 {InitPnode +} /NodeScale {} def NewNode end end + 952 +4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@Cn 10 {InitPnode +} /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@Cs 10 {InitPnode +} /NodeScale {} def NewNode end end + 952 4692 a 952 +4692 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Cnw +10 {InitPnode } /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Cne +10 {InitPnode } /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Csw +10 {InitPnode } /NodeScale {} def NewNode end end + 952 +4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Cse +10 {InitPnode } /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 +4692 a +tx@Dict begin { 199.1692 0.0 } PutCoor PutBegin end + 952 4692 a @beginspecial @setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 1.8 SLW 0 setgray 0.0 0.0 2 copy moveto 12.80365 +0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave 1 setgray + 1. .setopacityalpha fill grestore gsave 1.8 SLW 0 setgray 1. .setopacityalpha + 0 setlinejoin 0 setlinecap stroke grestore end + +@endspecial +952 4692 a +tx@Dict begin tx@NodeDict begin {0.0 0.0 12.80365 } false /N@D 11 +{InitCnode } /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 952 4692 a 952 4692 +a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@D known { -12.80365 +0. /N@D load 0 GetEdge 0.0 0. AddOffset /N@D load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 952 4692 a 952 4692 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 952 4692 a 0 TeXcolorgray 952 +4692 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 952 4692 a 952 4692 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 952 4692 a 952 4692 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 952 +4692 a 952 4692 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 952 4692 a 952 4692 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 952 4692 a 952 +4692 a +tx@Dict begin ps@rot RotBegin end + 952 4692 a 952 4692 a +tx@Dict begin RotEnd end + 952 4692 a 952 4692 a +tx@Dict begin PutEnd end + 952 +4692 a 952 4692 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 952 4692 a 952 4692 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 952 4692 a 0 +TeXcolorgray 952 4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 4692 a +tx@Dict begin PutEnd end + 952 4692 +a 952 4692 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Dw 10 {InitPnode +} /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@De 10 {InitPnode +} /NodeScale {} def NewNode end end + 952 4692 a 952 4692 +a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@Dn 10 {InitPnode +} /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@Ds 10 {InitPnode +} /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Dnw +10 {InitPnode } /NodeScale {} def NewNode end end + 952 4692 +a 952 4692 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Dne +10 {InitPnode } /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Dsw +10 {InitPnode } /NodeScale {} def NewNode end end + 952 4692 a 952 4692 +a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Dse +10 {InitPnode } /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 4692 a +tx@Dict begin { 312.98018 0.0 } PutCoor PutBegin end + 952 4692 +a @beginspecial @setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 1.8 SLW 0 setgray 0.0 0.0 2 copy moveto 12.80365 +0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave 1 setgray + 1. .setopacityalpha fill grestore gsave 1.8 SLW 0 setgray 1. .setopacityalpha + 0 setlinejoin 0 setlinecap stroke grestore end + +@endspecial 952 4692 a +tx@Dict begin tx@NodeDict begin {0.0 0.0 12.80365 } false /N@E 11 +{InitCnode } /NodeScale {} def NewNode end end + 952 +4692 a 952 4692 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 952 4692 a 952 4692 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@E known { -12.80365 +0. /N@E load 0 GetEdge 0.0 0. AddOffset /N@E load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 952 4692 a 952 +4692 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 952 4692 a 0 TeXcolorgray 952 4692 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 952 4692 +a 952 4692 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 952 4692 a 952 4692 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 952 4692 a 952 4692 +a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 952 4692 a 952 4692 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 952 4692 a 952 4692 a +tx@Dict begin ps@rot RotBegin end + 952 4692 +a 952 4692 a +tx@Dict begin RotEnd end + 952 4692 a 952 4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 4692 +a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 952 4692 a 952 4692 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 952 4692 a 0 TeXcolorgray 952 +4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Ew 10 {InitPnode +} /NodeScale {} def NewNode end end + 952 +4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Ee 10 {InitPnode +} /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@En 10 {InitPnode +} /NodeScale {} def NewNode end end + 952 4692 a 952 +4692 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@Es 10 {InitPnode +} /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Enw +10 {InitPnode } /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Ene +10 {InitPnode } /NodeScale {} def NewNode end end + 952 +4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Esw +10 {InitPnode } /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Ese +10 {InitPnode } /NodeScale {} def NewNode end end + 952 4692 a 952 +4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 4692 a +tx@Dict begin { 256.07469 -56.90549 } PutCoor PutBegin end + 952 4692 a @beginspecial +@setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 1.8 SLW 0 setgray 0.0 0.0 2 copy moveto 12.80365 +0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave 1 setgray + 1. .setopacityalpha fill grestore gsave 1.8 SLW 0 setgray 1. .setopacityalpha + 0 setlinejoin 0 setlinecap stroke grestore end + +@endspecial 952 4692 a +tx@Dict begin tx@NodeDict begin {0.0 0.0 12.80365 } false /N@F 11 +{InitCnode } /NodeScale {} def NewNode end end + 952 4692 a 952 4692 +a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 952 4692 a 952 4692 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@F known { -12.80365 +0. /N@F load 0 GetEdge 0.0 0. AddOffset /N@F load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 952 4692 a 952 4692 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 952 4692 +a 0 TeXcolorgray 952 4692 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 952 4692 a 952 4692 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 952 +4692 a 952 4692 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 952 4692 a 952 4692 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 952 4692 a 952 +4692 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 952 4692 a 952 4692 a +tx@Dict begin ps@rot RotBegin end + 952 4692 a 952 4692 a +tx@Dict begin RotEnd end + 952 +4692 a 952 4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 4692 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 952 4692 a 952 +4692 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 952 4692 a 0 TeXcolorgray 952 4692 a +tx@Dict begin PutEnd end + 952 4692 +a 952 4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Fw 10 {InitPnode +} /NodeScale {} def NewNode end end + 952 4692 a 952 4692 +a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Fe 10 {InitPnode +} /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@Fn 10 {InitPnode +} /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@Fs 10 {InitPnode +} /NodeScale {} def NewNode end end + 952 4692 +a 952 4692 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Fnw +10 {InitPnode } /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Fne +10 {InitPnode } /NodeScale {} def NewNode end end + 952 4692 a 952 4692 +a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Fsw +10 {InitPnode } /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Fse +10 {InitPnode } /NodeScale {} def NewNode end end + 952 4692 a 952 4692 a +tx@Dict begin PutEnd end + 952 4692 +a 952 4692 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@Aw /N@A InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 952 4692 a 952 4692 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@Dw /N@D InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 952 4692 a 952 4692 +a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@A /N@B InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 952 4692 a 952 4692 a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 add } def /ps@rot +{ } def grestore end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin /t 0.45 def LPut end PutBegin end + 952 4692 +a 952 4692 a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 0.0 0.0 0.0 +NAngle 90 add Uput exch pop add a PtoC h1 add exch w1 add exch } PutCoor +PutBegin end + 952 4692 a 0 TeXcolorgray 952 4692 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 952 +4692 a 952 4692 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 952 4692 a 952 4692 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 952 4692 a 952 +4692 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 952 4692 a 0 TeXcolorgray 952 4692 a +tx@Dict begin PutEnd end + 952 4692 +a 952 4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 4692 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@B /N@C InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 952 4692 a 952 4692 +a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 sub } def /ps@rot +{ } def grestore end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin /t 0.45 def LPut end PutBegin end + 952 4692 a 952 4692 a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 0.0 0.0 0.0 +NAngle 90 sub Uput exch pop add a PtoC h1 add exch w1 add exch } PutCoor +PutBegin end + 952 4692 +a 0 TeXcolorgray 952 4692 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 952 4692 a 952 4692 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 952 +4692 a 952 4692 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 952 4692 a 952 4692 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 952 4692 a 0 +TeXcolorgray 952 4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 4692 a +tx@Dict begin PutEnd end + 952 4692 +a 952 4692 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@C /N@A InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 952 4692 a 952 4692 a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 sub } def /ps@rot +{ } def grestore end + 952 4692 a 952 4692 +a +tx@Dict begin tx@NodeDict begin /t 0.45 def LPut end PutBegin end + 952 4692 a 952 4692 a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 0.0 0.0 0.0 +NAngle 90 sub Uput exch pop add a PtoC h1 add exch w1 add exch } PutCoor +PutBegin end + 952 4692 a 0 TeXcolorgray 952 +4692 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 952 4692 a 952 4692 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 952 4692 a 952 4692 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 952 +4692 a 952 4692 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 952 4692 a 0 TeXcolorgray 952 4692 +a +tx@Dict begin PutEnd end + 952 4692 a 952 4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 4692 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@D /N@E InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 952 4692 +a 952 4692 a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 add } def /ps@rot +{ } def grestore end + 952 4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin /t 0.45 def LPut end PutBegin end + 952 4692 a 952 4692 +a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 8.91537 8.01474 +0.0 NAngle 90 add Uput exch pop add a PtoC h1 add exch w1 add exch +} PutCoor PutBegin end + 952 4692 a 0 TeXcolorgray 915 4725 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 915 4725 a 915 +4725 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 915 4725 a Fm(r)915 4725 y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 915 4725 a 915 4725 +a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 915 4725 a 0 TeXcolorgray 952 4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 +4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 4692 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@E /N@F InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 952 4692 a 952 4692 a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 sub } def /ps@rot +{ } def grestore end + 952 +4692 a 952 4692 a +tx@Dict begin tx@NodeDict begin /t 0.45 def LPut end PutBegin end + 952 4692 a 952 4692 a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 8.73131 12.27701 +3.61957 NAngle 90 sub Uput exch pop add a PtoC h1 add exch w1 add +exch } PutCoor PutBegin end + 952 4692 a 0 +TeXcolorgray 916 4728 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 916 4728 a 916 4728 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 916 4728 +a Fm(j)916 4728 y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 916 4728 a 916 4728 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 916 4728 a 0 +TeXcolorgray 952 4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 4692 a +tx@Dict begin PutEnd end + 952 4692 +a 952 4692 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@F /N@D InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 952 4692 a 952 4692 a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 sub } def /ps@rot +{ } def grestore end + 952 4692 a 952 4692 +a +tx@Dict begin tx@NodeDict begin /t 0.45 def LPut end PutBegin end + 952 4692 a 952 4692 a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 9.69098 8.01474 +0.0 NAngle 90 sub Uput exch pop add a PtoC h1 add exch w1 add exch +} PutCoor PutBegin end + 952 4692 a 0 TeXcolorgray 912 +4725 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 912 4725 a 912 4725 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 912 4725 a Fm(v)912 4725 +y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 912 4725 a 912 4725 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 912 4725 a 0 TeXcolorgray 952 +4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 4692 a +tx@Dict begin PutEnd end + 952 4692 a 952 4928 a +tx@Dict begin CP CP translate 1 0.6 div 1 0.6 div scale NET end + 952 +4928 a 0 TeXcolorgray 0 TeXcolorgray eop end +%%Page: 26 26 +TeXDict begin 26 25 bop 0 TeXcolorgray 0 TeXcolorgray +293 14 a Fz(26)2353 b Fu(T)-8 b(ABLE)32 b(DES)f(MA)-8 +b(TI\310RES)p 293 51 3473 4 v 0 TeXcolorgray 293 347 +a Fz(En)35 b(particulier,)i(l'automate)f(ci-dessus)f(p)s(ermet)g(la)g +(reconnaissance)h(de)f(la)g(c)m(ha\356ne)h(de)f(caract\350re)293 +459 y(rjvrjvrjv.)15 b(.)g(.)28 b(qui)h(est)g(une)h(repr\351sen)m +(tation)h(de)f(la)f(succession)g(des)g(couleurs)h(\(allum\351es\))g(du) +g(feu)f(de)293 572 y(circulation.)p 0 TeXcolorgray 285 +785 a Fv(5)p 0 TeXcolorgray 45 w Fz(On)39 b(a)f(vu)h(qu'il)f(y)g(a)h +(64)f(p)s(ossibilit\351s)g(p)s(our)h(former)g(un)f(acide)i(amin\351)f +(constitu\351)g(d'un)g(triplet)293 898 y(c)m(hoisi)25 +b(parmi)g(quatre)f(n)m(ucl\351otides.)i(Il)e(p)s(eut)g(sem)m(bler)h +(alors)g(que)f(l'automate)i(corresp)s(ondan)m(t)f(doiv)m(e)293 +1011 y(\352tre)41 b(tr\350s)g(complexe.)g(Mais)f(au)h(con)m(traire,)i +(en)e(raison)g(du)g(fait)f(qu'on)h(p)s(eut)g(mettre)g(n'imp)s(orte)293 +1124 y(quel)31 b(n)m(ucl\351otide)j(\340)d(c)m(haque)i(place)f(parmi)g +(les)f(trois)h(disp)s(onibles)f(nous)h(p)s(ermet)g(de)f(constituer)i +(un)293 1237 y(automate)f(particuli\350remen)m(t)h(simple)e(:)p +0 TeXcolorgray 0 TeXcolorgray 1462 1479 a +tx@Dict begin CP CP translate 0.6 0.6 scale NET end + 1462 1479 a +1462 1479 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1462 1479 a @beginspecial @setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 1.8 SLW 0 setgray 0.0 0.0 2 copy moveto 12.80365 +0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave 1 setgray + 1. .setopacityalpha fill grestore gsave 1.8 SLW 0 setgray 1. .setopacityalpha + 0 setlinejoin 0 setlinecap stroke grestore end + +@endspecial +1462 1479 a +tx@Dict begin tx@NodeDict begin {0.0 0.0 12.80365 } false /N@A 11 +{InitCnode } /NodeScale {} def NewNode end end + 1462 1479 a 1462 1479 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 1462 1479 a 1462 +1479 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@A known { -12.80365 +0. /N@A load 0 GetEdge 0.0 0. AddOffset /N@A load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 1462 1479 a 1462 1479 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 1462 1479 a 0 TeXcolorgray +1462 1479 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1462 1479 a 1462 1479 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1462 1479 a 1462 +1479 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 1462 1479 a 1462 1479 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 1462 1479 a 1462 1479 +a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1462 1479 a 1462 1479 a +tx@Dict begin ps@rot RotBegin end + 1462 1479 a 1462 1479 a +tx@Dict begin RotEnd end + 1462 +1479 a 1462 1479 a +tx@Dict begin PutEnd end + 1462 1479 a 1462 1479 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1462 1479 +a 1462 1479 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1462 1479 a 0 TeXcolorgray 1462 1479 a +tx@Dict begin PutEnd end + +1462 1479 a 1462 1479 a +tx@Dict begin PutEnd end + 1462 1479 a 1462 1479 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Aw 10 {InitPnode +} /NodeScale {} def NewNode end end + 1462 +1479 a 1462 1479 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Ae 10 {InitPnode +} /NodeScale {} def NewNode end end + 1462 1479 a 1462 1479 a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@An 10 {InitPnode +} /NodeScale {} def NewNode end end + 1462 1479 +a 1462 1479 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@As 10 {InitPnode +} /NodeScale {} def NewNode end end + 1462 1479 a 1462 1479 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Anw +10 {InitPnode } /NodeScale {} def NewNode end end + 1462 1479 a 1462 +1479 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Ane +10 {InitPnode } /NodeScale {} def NewNode end end + 1462 1479 a 1462 1479 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Asw +10 {InitPnode } /NodeScale {} def NewNode end end + 1462 1479 a 1462 1479 +a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Ase +10 {InitPnode } /NodeScale {} def NewNode end end + 1462 1479 a 1462 1479 a +tx@Dict begin PutEnd end + 1462 1479 a 1462 1479 a +tx@Dict begin { 113.81097 0.0 } PutCoor PutBegin end + 1462 +1479 a @beginspecial @setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 1.8 SLW 0 setgray 0.0 0.0 2 copy moveto 12.80365 +0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave 1 setgray + 1. .setopacityalpha fill grestore gsave 1.8 SLW 0 setgray 1. .setopacityalpha + 0 setlinejoin 0 setlinecap stroke grestore end + +@endspecial 1462 1479 +a +tx@Dict begin tx@NodeDict begin {0.0 0.0 12.80365 } false /N@B 11 +{InitCnode } /NodeScale {} def NewNode end end + 1462 1479 a 1462 1479 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 1462 1479 a 1462 1479 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@B known { -12.80365 +0. /N@B load 0 GetEdge 0.0 0. AddOffset /N@B load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 1462 +1479 a 1462 1479 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 1462 1479 a 0 TeXcolorgray 1462 1479 +a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1462 1479 a 1462 1479 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1462 1479 a 1462 1479 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 1462 +1479 a 1462 1479 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 1462 1479 a 1462 1479 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1462 1479 +a 1462 1479 a +tx@Dict begin ps@rot RotBegin end + 1462 1479 a 1462 1479 a +tx@Dict begin RotEnd end + 1462 1479 a 1462 +1479 a +tx@Dict begin PutEnd end + 1462 1479 a 1462 1479 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1462 1479 a 1462 1479 +a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1462 1479 a 0 TeXcolorgray 1462 1479 a +tx@Dict begin PutEnd end + 1462 1479 a +1462 1479 a +tx@Dict begin PutEnd end + 1462 1479 a 1462 1479 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Bw 10 {InitPnode +} /NodeScale {} def NewNode end end + 1462 1479 a 1462 +1479 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Be 10 {InitPnode +} /NodeScale {} def NewNode end end + 1462 1479 a 1462 1479 a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@Bn 10 {InitPnode +} /NodeScale {} def NewNode end end + 1462 1479 a 1462 1479 +a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@Bs 10 {InitPnode +} /NodeScale {} def NewNode end end + 1462 1479 a 1462 1479 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Bnw +10 {InitPnode } /NodeScale {} def NewNode end end + 1462 1479 a 1462 1479 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Bne +10 {InitPnode } /NodeScale {} def NewNode end end + 1462 +1479 a 1462 1479 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Bsw +10 {InitPnode } /NodeScale {} def NewNode end end + 1462 1479 a 1462 1479 a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Bse +10 {InitPnode } /NodeScale {} def NewNode end end + 1462 1479 +a 1462 1479 a +tx@Dict begin PutEnd end + 1462 1479 a 1462 1479 a +tx@Dict begin { 227.62195 0.0 } PutCoor PutBegin end + 1462 1479 a @beginspecial +@setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 1.8 SLW 0 setgray 0.0 0.0 2 copy moveto 12.80365 +0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave 1 setgray + 1. .setopacityalpha fill grestore gsave 1.8 SLW 0 setgray 1. .setopacityalpha + 0 setlinejoin 0 setlinecap stroke grestore end + +@endspecial 1462 1479 a +tx@Dict begin tx@NodeDict begin {0.0 0.0 12.80365 } false /N@C 11 +{InitCnode } /NodeScale {} def NewNode end end + 1462 1479 a 1462 +1479 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 1462 1479 a 1462 1479 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@C known { -12.80365 +0. /N@C load 0 GetEdge 0.0 0. AddOffset /N@C load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 1462 1479 a 1462 1479 +a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 1462 1479 a 0 TeXcolorgray 1462 1479 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1462 1479 a +1462 1479 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1462 1479 a 1462 1479 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 1462 1479 a 1462 +1479 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 1462 1479 a 1462 1479 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1462 1479 a 1462 1479 +a +tx@Dict begin ps@rot RotBegin end + 1462 1479 a 1462 1479 a +tx@Dict begin RotEnd end + 1462 1479 a 1462 1479 a +tx@Dict begin PutEnd end + 1462 +1479 a 1462 1479 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1462 1479 a 1462 1479 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1462 1479 +a 0 TeXcolorgray 1462 1479 a +tx@Dict begin PutEnd end + 1462 1479 a 1462 1479 a +tx@Dict begin PutEnd end + +1462 1479 a 1462 1479 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Cw 10 {InitPnode +} /NodeScale {} def NewNode end end + 1462 1479 a 1462 1479 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@Ce 10 {InitPnode +} /NodeScale {} def NewNode end end + 1462 +1479 a 1462 1479 a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@Cn 10 {InitPnode +} /NodeScale {} def NewNode end end + 1462 1479 a 1462 1479 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@Cs 10 {InitPnode +} /NodeScale {} def NewNode end end + 1462 1479 +a 1462 1479 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Cnw +10 {InitPnode } /NodeScale {} def NewNode end end + 1462 1479 a 1462 1479 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Cne +10 {InitPnode } /NodeScale {} def NewNode end end + 1462 1479 a 1462 +1479 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Csw +10 {InitPnode } /NodeScale {} def NewNode end end + 1462 1479 a 1462 1479 a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Cse +10 {InitPnode } /NodeScale {} def NewNode end end + 1462 1479 a 1462 1479 +a +tx@Dict begin PutEnd end + 1462 1479 a 1462 1479 a +tx@Dict begin { 341.43292 0.0 } PutCoor PutBegin end + 1462 1479 a @beginspecial @setspecial + tx@Dict begin STP newpath /ArrowA { moveto } def /ArrowB { } def +/ArrowInside { } def 2.87999 SLW 0 setgray 0.0 0.0 2 copy moveto +12.80365 0 CLW mul round sub dup 0 rmoveto 0 360 arc closepath gsave +1 setgray 1. .setopacityalpha fill grestore gsave 2.87999 SLW 0 + setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap stroke + grestore gsave 0.71999 SLW 1 setgray 1. .setopacityalpha stroke +grestore end + + +@endspecial 1462 1479 a +tx@Dict begin tx@NodeDict begin {0.0 0.0 13.34364 } false /N@D 11 +{InitCnode } /NodeScale {} def NewNode end end + 1462 1479 a 1462 1479 a +tx@Dict begin gsave STV CP T /ps@refangle 0. def grestore end + 1462 +1479 a 1462 1479 a +tx@Dict begin { tx@NodeDict begin tx@NodeDict /N@D known { -12.80365 +0. /N@D load 0 GetEdge 0.0 0. AddOffset /N@D load GetCenter 3 -1 roll +add 3 1 roll add exch } { CP } ifelse end } PutCoor PutBegin end + 1462 1479 a 1462 1479 a +tx@Dict begin { 0.0 0.0 0.0 0.0 0. Uput UUput } PutCoor PutBegin +end + 1462 1479 +a 0 TeXcolorgray 1462 1479 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1462 1479 a 1462 1479 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + +1462 1479 a 1462 1479 a +tx@Dict begin gsave STV CP T /ps@rot 0 def grestore end + 1462 1479 a 1462 1479 a +tx@Dict begin gsave STV CP T /ps@rot mark RAngle /ps@a ED cleartomark +ps@a neg def grestore end + 1462 +1479 a 1462 1479 a +tx@Dict begin { 0.0 0.0 } PutCoor PutBegin end + 1462 1479 a 1462 1479 a +tx@Dict begin ps@rot RotBegin end + 1462 1479 +a 1462 1479 a +tx@Dict begin RotEnd end + 1462 1479 a 1462 1479 a +tx@Dict begin PutEnd end + 1462 1479 a 1462 +1479 a +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1462 1479 a 1462 1479 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1462 1479 a 0 TeXcolorgray +1462 1479 a +tx@Dict begin PutEnd end + 1462 1479 a 1462 1479 a +tx@Dict begin PutEnd end + 1462 1479 a 1462 +1479 a +tx@Dict begin tx@NodeDict begin {-38.41095 0.0 } false /N@Dw 10 {InitPnode +} /NodeScale {} def NewNode end end + 1462 1479 a 1462 1479 a +tx@Dict begin tx@NodeDict begin {38.41095 0.0 } false /N@De 10 {InitPnode +} /NodeScale {} def NewNode end end + 1462 1479 a 1462 1479 +a +tx@Dict begin tx@NodeDict begin {0.0 38.41095 } false /N@Dn 10 {InitPnode +} /NodeScale {} def NewNode end end + 1462 1479 a 1462 1479 a +tx@Dict begin tx@NodeDict begin {0.0 -38.41095 } false /N@Ds 10 {InitPnode +} /NodeScale {} def NewNode end end + 1462 1479 a 1462 1479 a +tx@Dict begin tx@NodeDict begin {-27.54047 27.54047 } false /N@Dnw +10 {InitPnode } /NodeScale {} def NewNode end end + 1462 +1479 a 1462 1479 a +tx@Dict begin tx@NodeDict begin {27.54047 27.54047 } false /N@Dne +10 {InitPnode } /NodeScale {} def NewNode end end + 1462 1479 a 1462 1479 a +tx@Dict begin tx@NodeDict begin {-27.54047 -27.54047 } false /N@Dsw +10 {InitPnode } /NodeScale {} def NewNode end end + 1462 1479 +a 1462 1479 a +tx@Dict begin tx@NodeDict begin {27.54047 -27.54047 } false /N@Dse +10 {InitPnode } /NodeScale {} def NewNode end end + 1462 1479 a 1462 1479 a +tx@Dict begin PutEnd end + 1462 1479 a 1462 +1479 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@Aw /N@A InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 1462 1479 a 1462 1479 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@A /N@B InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 1462 1479 a 1462 1479 +a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 add } def /ps@rot +{ } def grestore end + 1462 1479 a 1462 1479 a +tx@Dict begin tx@NodeDict begin /t 0.45 def LPut end PutBegin end + 1462 1479 a 1462 1479 a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 72.98738 13.96121 +4.65372 NAngle 90 add Uput exch pop add a PtoC h1 add exch w1 add +exch } PutCoor PutBegin end + 1462 +1479 a 0 TeXcolorgray 1159 1518 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1159 1518 a 1159 1518 +a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1159 1518 a Fm(C)7 b Fq(j)p Fm(G)p Fq(j)p Fm(T)13 b +Fq(j)p Fm(U)1159 1518 y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1159 1518 a 1159 1518 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1159 +1518 a 0 TeXcolorgray 1462 1479 a +tx@Dict begin PutEnd end + 1462 1479 a 1462 1479 +a +tx@Dict begin PutEnd end + 1462 1479 a 1462 1479 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@B /N@C InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 1462 1479 a 1462 1479 a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 sub } def /ps@rot +{ } def grestore end + 1462 +1479 a 1462 1479 a +tx@Dict begin tx@NodeDict begin /t 0.45 def LPut end PutBegin end + 1462 1479 a 1462 1479 a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 72.98738 13.96121 +4.65372 NAngle 90 sub Uput exch pop add a PtoC h1 add exch w1 add +exch } PutCoor PutBegin end + 1462 1479 +a 0 TeXcolorgray 1159 1518 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1159 1518 a 1159 1518 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + +1159 1518 a Fm(C)7 b Fq(j)p Fm(G)p Fq(j)p Fm(T)13 b Fq(j)p +Fm(U)1159 1518 y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1159 1518 a 1159 1518 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1159 1518 a +0 TeXcolorgray 1462 1479 a +tx@Dict begin PutEnd end + 1462 1479 a 1462 1479 a +tx@Dict begin PutEnd end + 1462 +1479 a 1462 1479 a +tx@Dict begin gsave STV newpath /ArrowA { moveto } def /ArrowB { BeginArrow +1. 1. scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul +dup /h ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 +0 L w h L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore +grestore 0 h a sub moveto EndArrow } def /ArrowInside { } def 1.0 +SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. + scale false 0.1 1.4 5.0 0. CLW mul add dup 2 div /w ED mul dup /h +ED mul /a ED { 0 h T 1 -1 scale } if gsave w neg h moveto 0 0 L w h +L w neg a neg rlineto gsave 1. .setopacityalpha fill grestore grestore +0 h a sub moveto EndArrow } def /ArrowInside { } def /NCLW CLW +def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@C /N@D InitNC { NCCoor +tx@Dict begin ArrowA CP 4 2 roll ArrowB 4 copy /y2 ED /x2 ED /y1 ED +/x1 ED x1 y1 0.5 1 gt { /Alpha y2 y1 sub x2 x1 sub atan def /ArrowPos +0. def /Length x2 x1 sub y2 y1 sub Pyth def /dArrowPos 0.5 abs def +{/ArrowPos ArrowPos dArrowPos add def ArrowPos Length gt { exit } if +x1 Alpha cos ArrowPos mul add y1 Alpha sin ArrowPos mul add ArrowInside +pop pop } loop }{/ArrowPos 0. def /dArrowPos 1. 1 gt {1.0 1. 1.0 add +div }{ 0.5 } ifelse def 1. cvi { /ArrowPos ArrowPos dArrowPos add +def x2 x1 sub ArrowPos mul x1 add y2 y1 sub ArrowPos mul y1 add ArrowInside +pop pop } repeat } ifelse pop pop lineto pop pop end } if end gsave +1.0 SLW 0 setgray 1. .setopacityalpha 0 setlinejoin 0 setlinecap +stroke grestore grestore end + 1462 1479 a 1462 1479 a +tx@Dict begin gsave STV CP T /ps@refangle {NAngle 90 sub } def /ps@rot +{ } def grestore end + 1462 1479 +a 1462 1479 a +tx@Dict begin tx@NodeDict begin /t 0.45 def LPut end PutBegin end + 1462 1479 a 1462 1479 a +tx@Dict begin { 3.5 tx@Dict /NCLW known { NCLW add } if 72.98738 13.96121 +4.65372 NAngle 90 sub Uput exch pop add a PtoC h1 add exch w1 add +exch } PutCoor PutBegin end + 1462 1479 a 0 TeXcolorgray +1159 1518 a +tx@Dict begin CP CP translate 1. 1. scale NET end + 1159 1518 a 1159 1518 a +tx@Dict begin CP CP translate 1.7 1.7 scale NET end + 1159 1518 a Fm(C)7 +b Fq(j)p Fm(G)p Fq(j)p Fm(T)13 b Fq(j)p Fm(U)1159 1518 +y +tx@Dict begin CP CP translate 1 1.7 div 1 1.7 div scale NET end + 1159 1518 a 1159 1518 a +tx@Dict begin CP CP translate 1 1. div 1 1. div scale NET end + 1159 1518 a 0 TeXcolorgray +1462 1479 a +tx@Dict begin PutEnd end + 1462 1479 a 1462 1479 a +tx@Dict begin PutEnd end + 1462 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0000000..389096e --- /dev/null +++ b/Lsystem.tex @@ -0,0 +1,1285 @@ +\documentclass[11pt,a4paper,titlepage,twoside]{article} + +\input{Preambule.tex} % Toutes les instructions du préambule sont dans un fichier séparé qui est utilisé pour tous les cours. + % Il faut le mettre à l'extérieur du dossier dans lequel se trouve le cours --> ../Preambule.tex + +\usepackage{geometry} % La seule présence de ce paquet élargit les marges correctement + +\lstloadlanguages{python,TeX,XML} % chargement des languages nécessaires (ici un seul) +%\lstset{language=TeX} % language choisi par défaut + +\setlength{\headheight}{22.6pt} % pour élargir un peu l'entête +%\setcounter{tocdepth}{5} % pour définir la profondeur de la table des matières +%\renewcommand{\thefootnote}{\alph{footnote}} % pour mettre les notes de bas de page en alphabétique + +%\usepackage{etex} +%\usepackage[all,2cell]{xy} +%\UseAllTwocells +%\usepackage{xymtex} % pour la représentation de molécules chimiques +%\let\urlorig\url % pour mettre des deux points dans les url, passage en langue anglaise +%\renewcommand{\url}[1]{% +% \begin{otherlanguage}{english}\urlorig{#1}\end{otherlanguage}% +%} +%\usepackage[colorlinks=true,linkcolor=black,urlcolor=blue,pagebackref,frenchlinks]{hyperref} +\usepackage{eurosym} +\usepackage{textcomp} +\usepackage{auto-pst-pdf} +\usepackage{vaucanson-g} +%\usepackage{algorithm,algorithmique} +\usepackage[french,boxruled]{algorithm2e} + +\setlength{\headheight}{13.6pt} % spécifie la hauteur de l'entête pour qu'elle ne couvre pas le titre + +\setcounter{topnumber}{3} % nombre maximal de flottant en haut d'une page (par défaut 2) +\setcounter{bottomnumber}{2} % nombre maximal de flottant en bas d'une page (par défaut 1) +%\setcounter{totalnumber}{3} % nombre maximal de flottant sur une page (par défaut 3) +\renewcommand{\topfraction}{0.9} % proportion maximale de la page qui peut recevoir des flottants en haut de la page (par défaut 0,7) +\renewcommand{\textfraction}{0.1} % proportion minimale d'une page qui reçoit du texte (sans les pages de flottants) +\renewcommand{\floatpagefraction}{0.8} % proportion minimale de flottants sur une page pour qu'elle devienne une page uniquement de flottants +\renewcommand{\bottomfraction}{0.5} % proportion maximale de la page qui peut recevoir des flottants en bas de la page (par défaut 0,3) + +\usepackage{answers} % pour utiliser le module answers +\Newassociation{sol}{Solution}{ans} % définition de l'étiquette Solution du module answers +\newtheorem{exc}{Exercice} % définition de l'étiquette Exercice du module answers +\newenvironment{ex}{\begin{exc}\normalfont}{\end{exc}} % cela (\normalfont) annule l'italique des énoncés des exercices + +\graphicspath{{./Images/}{./Introduction/Images/}} + +%\fancyhead[RE,LO]{Lycée Blaise Cendrars\\La Chaux-de-Fonds} +\fancyhead[LE,RO]{\thepage} +\lfoot{} \cfoot{} \rfoot{} +\pagestyle{fancy} + +%\usepackage[firstpage]{draftwatermark} +%\SetWatermarkLightness{0.8} +%\SetWatermarkAngle{60} +%\SetWatermarkScale{0.8} +%\SetWatermarkFontSize{5cm} +%\SetWatermarkText{DOCUMENT À LAISSER AU LYCÉE} + +\begin{document} + +%\frontmatter +\newcommand\HRule{\noindent\rule{\linewidth}{1.5pt}} +\begin{titlepage} + \vspace*{\stretch{1}} + \HRule + \begin{flushright} + \LARGE Option complémentaire \\ + \LARGE Informatique \\ + \textsc{Langages formels et L-system} + \end{flushright} + \HRule\\ +% \textsc{Document à laisser au lycée} + \vspace*{\stretch{2}} + \begin{center} + \today + \end{center} +\end{titlepage} + +\section*{Copyright} +\addcontentsline{toc}{section}{Copyright} +Copyright (c) 2011 Guyot Vincent + +\medskip +Permission vous est donnée de copier, distribuer et/ou modifier ce document selon les termes de la Licence GNU Free Documentation License, Version 1.1 ou ultérieure publiée par la Free Software Foundation. + +\medskip +\begin{tabbing}\small +\hspace*{1cm}\=\hspace*{1cm}\=\kill +\>Avec les sections inaltérables suivantes :\\ +\>\>\textit{Pas de section inaltérable}\\ +\>avec le texte de première page de couverture suivant :\\ +\>\>\textit{Option complémentaire \textsc{Informatique}}\\ +\>avec le texte de dernière page de couverture suivant :\\ +\>\>\textit{Pas de texte de texte de dernière page de couverture}\\ +\end{tabbing} + +Une copie transparente de ce document se trouve à l'adresse suivante : +www.cvgg.org +\begin{itemize} +\item Lsystem.tex : copie transparente, +\item Lsystem.dvi : copie opaque au format dvi. +\end{itemize} + +\bigskip +Normalement une copie de la licence GFDL doit se trouver avec le présent document. Comme il est relativement court et comme on trouve facilement le texte de la GFDL sur internet, elle n'est pas reproduite ici pour ne pas utiliser inutilement du papier. +\thispagestyle{empty} + +\newpage + +\section*{Avertissement} +\addcontentsline{toc}{section}{Avertissement} + +\thispagestyle{empty} + +\tableofcontents{} +\addcontentsline{toc}{section}{Table des matières} +\thispagestyle{empty} + +\newpage + +%\mainmatter + +\Opensolutionfile{ans}[Solutions] % ouvre le fichier Solutions.tex qui va contenir les solutions + +\section{Introduction} +Ce cours s'adresse à des élèves dont les connaissances en informatiques se limitent à l'utilisation de logiciels et à des connaissances de base (boucles et fonctions) en programmation. + +\smallskip +Il ne s'adresse pas spécifiquement à des élèves non mathématiciens, car ceux-ci peuvent aussi y trouver de l'intérêt, mais surtout à des personnes qui ne se destinent ni à l'informatique, ni au mathématiques. C'est pourquoi, l'accent à été mis sur la plus grande clarté possible dans l'exposé des notions qui relèvent des mathématiques et sur les exemples. Il constitue à mon avis une approche adaptée pour des personnes intéressées par la biologie et par les langues. + +\smallskip +Les objectifs sont multiples. +\begin{description} + \item[Le premier] d'entre eux est de montrer aux élèves que l'analyse formelle est un préalable nécessaire à la compréhension des langages, en particulier informatiques. + \item[Le deuxième] est de présenter un exemple de la diversité des modes de représentation de ces langages. + \item[Le troisième] est d'illustrer le lien qui existe entre les langages et certains processus naturels, lien qui montre l'importance de l'étude des formalismes présentés. + \item[Le dernier] objectif est finalement de permettre à l'élève d'utiliser des connaissances très sommaires de programmation pour analyser les processus mécaniquement dans des proportion inaccessibles à l'homme. +\end{description} + +\section{Langage formel} +Rendre compte de la notion de langage formel le plus simplement possible n'est pas chose facile. Il se trouve que c'est la rigueur du langage mathématique qui permet d'en exprimer au mieux l'idée. Nous allons donc l'utiliser le plus simplement possible étant donné les difficultés qu'il peut représenter. Il sera néanmoins bien souvent plus une aide qu'un obstacle. + +Tout langage est au départ constitué d'un alphabet. Il faut considérer le terme d'alphabet au sens large du terme comme un simple ensemble de signes ou lettres, même non alphabétiques au sens commun du terme. Ainsi, les ensembles suivants notés \(\Sigma_i\) constituent des exemples d'alphabet : +\begin{align*} + \Sigma_1 &= \{\text{\officialeuro, \maltese, \textleaf, \checkmark, 1, 4, \textvisiblespace}\}\\ + \Sigma_2 &= \{\text{\textlnot, \textdollaroldstyle, \textcircled{a}, \pounds}\}\\ + \Sigma_3 &= \{\text{a, b, \textvisiblespace, }\epsilon\} +\end{align*} +A partir de l'alphabet, on peut alors définir ce qu'est un mot. Il s'agit simplement d'une suite finie de lettres prises dans l'alphabet. Ainsi, on peut former les mots suivants sur l'alphabet \(\Sigma_1\) : \hspace{1cm}\officialeuro\officialeuro\maltese\officialeuro\officialeuro\checkmark\hspace{1cm} ou \hspace{1cm}\checkmark14\maltese\maltese\hspace{1cm} Pour l'alphabet \(\Sigma_3\), on pourrait avoir : \hspace{1cm}aababbab\hspace{1cm} ou \hspace{1cm}aabbbba \hspace{1cm}mais pas \hspace{1cm}aab\textlnot + +Si m est un mot sur un alphabet \(\Sigma\), on écrira : +\[\text{m}\in\Sigma^*\] +pour dire que le mot m appartient à l'ensemble \(\Sigma^*\) des mot possibles formé sur l'alphabet \(\Sigma\). Finalement, on définit \(\epsilon\) comme un mot vide. La raison pour laquelle on fait cela sera vue plus tard. + +\medskip +Maintenant qu'on a des mots, le problème est de pouvoir les utiliser. Si on considère les mots suivants de \(\Sigma_3\) : +\[\text{abaab\textvisiblespace\hspace{1cm}}\text{aba\textvisiblespace\hspace{1cm}}\text{aabaaba\textvisiblespace}\] +il serait possible de former queque chose qui ressemble à des phrases en plaçant ces mots les uns à côté des autres. Cette opération se nomme la concaténation et on la note par un point. Ainsi, on va formellement définir la concaténation ainsi : +\[\text{Soit m et n }\in \Sigma^*\text{\hspace{0.3cm} Alors, m.n=mn}\] +On peut ainsi tenter de reproduire l'apparence formelle d'un langage par concaténation des mots ci-dessus (en s'imaginant un espace invisible) : +\[\text{abaab\textvisiblespace}.\text{aba\textvisiblespace}.\text{aabaaba\textvisiblespace}=\text{abaab aba aabaaba}\] +Certes, le résultat ne correspond qu'en apparence à un langage. Mais, l'alphabet est très restreint. + +\medskip +De manière naturelle, on va donc définir la notion de langage comme un sous-ensemble de l'ensemble de tous les mots d'un alphabet donné. Un langage est donc un ensemble de mots. On peut donc faire l'union, l'intersection, \dots de deux langages. + +\medskip +L'exemple classique est le langage des expressions arithmétiques. Il est défini à partir de l'alphabet suivant : +\[\Sigma = \{0,1,2,3,4,5,6,7,8,9,+,-,*,/\}\] +Le langage des expressions arithmétiques s'écrit alors : +\[L(\text{expr. arithm.})=\{\dots ,342, 34+4, 45+34, 2=4, \dots\}\] +Relevez qu'il ne s'agit pas de l'ensemble de tous les mots formé sur l'alphabet, mais d'un sous ensemble qui ne contient pas l'expression 34+++43==, par exemple. Évidemment, il faut spécifier comment on réalise le choix de chaque expression. + +\begin{ex} + Sur l'alphabet des chiffres binaires (que vous noterez précisément), donnez le langage \(L\) des nombres à trois chiffres. + \begin{sol} + L'alphabet des chiffres binaires est : \(\Sigma=\{0,1\}\) et le langage des nombres binaires à trois chiffres est : + \[L=\{000,001,010,011,100,101,110,111\}\] + \end{sol} +\end{ex} + +\begin{ex}\label{codons} + Pour la molécule d'ADN, définissez un alphabet de ses composants et déterminez quelques éléments du langage des \emph{codons}, acide aminé constitué de triplets de nucléotides. Combien peut-il y en avoir si leur arrangement tient compte de l'ordre et qu'il peut y avoir des répétitions. + + Indication : les nucléotides sont les briques de base de l'ADN. + \begin{sol} + En considérant les quatre nucléotides composant l'ADN, la thymine (T), la cytosine (C), l'adénine (A) et la guanine (G), on peut définir l'alphabet nécessaire ainsi : + \[\Sigma=\{A,C,G,T\}\] + Les codons sont quant à eux des ensemble de trois nucléotides, pris parmi les quatre disponibles dans l'alphabet. Le langage ainsi formé est : + \[L=\{AAC,ACT,ACT,CTG,GTC, \dots\}\] + et la combinatoire donne pour un arrangement avec répétition de trois éléments pris parmi quatre : + \[\overline{A_3^4}=4^3=64\;\text{possibilités}\] + \end{sol} +\end{ex} + +\bigskip +La génération ou la vérification des expressions valables pour un langage donné peut se faire de différentes manières. Nous allons présenter dans la section suivante les moyens qui nous permettrons de décrire simplement des langages particuliers. On peut utiliser pour cela soit un automate soit une expression régulière. + +\section{Langage rationnel} + +\subsection{Les automates} + +Une manière particulièrement compréhensible de tester l'appartenance d'une expression à un langage est d'utiliser un automate. Il s'agit d'une sorte de machine symbolique capable de générer un ensemble de mots. On la suit dans la construction de ceux-ci selon des flèches qui mènent d'un état à l'autre par la concaténation successive de certaines mots de l'alphabet. Pour des raison de compréhension, on va commencer par donner un exemple d'automate à travers une représentation graphique qui fait correspondre des ronds aux états et des flèches aux éléments de l'alphabet. Un état initial est un rond sur lequel arrive une flèche vide et un état final est constitué d'un double cercle. Ainsi, on représente un automate reconnaissant le mot : +\[aba\;\text{ sur l'alphabet }\;\Sigma=\{a,b\}\] +par la figure suivante : +\begin{center} +\begin{postscript} +\VCDraw{% +\begin{VCPicture}{(0,-2)(9,0)} +% states +\State{(0,0)}{A} \State{(3,0)}{B} \State{(6,0)}{C} \FinalState{(9,0)}{D} +% initial--final +\Initial{A} +% transitions +\EdgeL{A}{B}{a} \EdgeL{B}{C}{b} \EdgeL{C}{D}{a} +% +\end{VCPicture}} +\end{postscript} +\end{center} +Relevons qu'il est possible de considérer l'unique mot construit par cet automate comme un langage qu'on peut définir mathématiquement très simplement par : +\[L=\{m\in\Sigma^*\,|\,m=aba\}\] + +où la barre verticale \(|\) signifie ``tel que''. + +\smallskip +Évidemment, l'objectif est de permettre la construction de langages plus étendus. On peut par exemple construire le langage constitué d'une séquence de ab ainsi : +\begin{center} +\begin{postscript} +\VCDraw{% +\begin{VCPicture}{(0,-1)(3,1)} +% states +\FinalState{(0,0)}{A} \State{(3,0)}{B} +% initial--final +\Initial{A} +% transitions +\ArcL{A}{B}{a} \ArcL{B}{A}{b} +% +\end{VCPicture}} +\end{postscript} +\end{center} + +Cet exemple permet d'engendrer une infinité de mots du type ab, abab, ababab, \dots + +\smallskip +Par ailleurs, la présence d'un état initial, simultanément final, implique l'existence dans le langage d'un mot vide noté \(\epsilon\). On écrira donc mathématiquement le langage ainsi : +\[L=\{m\in\Sigma^*\,|\,m=\epsilon,ab,abab,ababab,\dots\}\] + +Un dernier exemple montre qu'il est possible de générer des langages bien plus complexes. Sans dire immédiatement ce dont il s'agit, voici l'automate : + +\begin{center} +\begin{postscript} +\VCDraw{% +\begin{VCPicture}{(0,-1)(6,2)} +% states +\State{(0,0)}{A} \State{(3,0)}{B} \StateLineDouble \State{(6,0)}{C} +% initial--final +\Initial{A} +% transitions +\EdgeL{A}{B}{a} \ArcL{B}{C}{b} \ArcL{C}{B}{b} +\LoopN{A}{a} +% +\end{VCPicture}} +\end{postscript} +\end{center} + +Le langage reconnu est constitué de mots comprenant en premier lieu au moins un a, suivi par un nombre impair de b. On peut aussi écrire : +\[L=\{m\in\Sigma^*\,|\,m=a^nb^m,\,n\in\mathbb{N}^*\,\text{et}\;m=2\cdot k+1,\,k\in\mathbb{N}\}\] +où \(\mathbb{N}^*\) est l'ensemble des nombres naturels (entiers positifs) sans le zéro et \(\mathbb{N}\) celui avec le zéro. + +\medskip +En conclusion, on peut définir un automate de manière générale comme un ensemble constitué +\begin{itemize} + \item d'un alphabet \(\Sigma\), + \item d'un ensemble fini d'état, généralement noté \(E\), + \item d'un ensemble d'état initiaux \(I\) tel que \(I\subseteq E\), + \item d'un ensemble d'état finaux \(F\) tel que \(F\subseteq E\) et + \item d'une fonction de transition \[\delta:E\times\Sigma\longrightarrow E\] +\end{itemize} + +Où le signe \(\subseteq\) signifie ``appartient à''. + +\smallskip +Le dernier point de cette définition mérite des explications. La représentation graphique des automates donnée ci-dessus met clairement en évidence les transitions (flèches) entre états. Celles-ci doivent aussi trouver une expression mathématique. C'est la fonction de transition. + +\smallskip +Reprenons le dernier exemple d'automate donné ci-dessus. Considérons chacun des points permettant de le définir mathématiquement. + +L'alphabet tout d'abord est clairement donné par \(\Sigma=\{a,b\}\). + +L'ensemble des états est \(E=\{e_1,e_2,e_3\}\) et on peut en donner une représentation légèrement plus complète ainsi : + +\begin{center} +\begin{postscript} +\VCDraw{% +\begin{VCPicture}{(0,-1)(6,2)} +% states +\State[e_1]{(0,0)}{A} \State[e_2]{(3,0)}{B} \StateLineDouble \State[e_3]{(6,0)}{C} +% initial--final +\Initial{A} +% transitions +\EdgeL{A}{B}{a} \ArcL{B}{C}{b} \ArcL{C}{B}{b} +\LoopN{A}{a} +% +\end{VCPicture}} +\end{postscript} +\end{center} + +L'ensemble des états initiaux se résume à \(I=\{e_1\}\) et celui des états finaux à \(F=\{e_3\}\). + +La fonction de transition \(\delta\) est alors donnée par : +\begin{align*} + \delta(e_1,a)&=e_1\,|\,e_2 & \delta(e_1,b)&=\perp\\ + \delta(e_2,a)&=\perp & \delta(e_2,b)&=e_3\\ + \delta(e_3,a)&=\perp & \delta(e_3,b)&=e_2 +\end{align*} +où \(|\) signifie ``ou'' et \(\perp\) signifie ``indéfini''. + +\bigskip +Ce dernier exemple est aussi intéressant du point de vue du déterminisme du résultat. La fonction de transition \(\delta(e_1,a)\) donne deux solution possibles : \(e_1\) ou \(e_2\). Cela signifie qu'à partir de l'état \(e_1\), par la transition \(a\), un choix non déterminé peut avoir lieu. L'automate est alors dit \emph{non déterministe}. Cela peut constituer un problème et il est possible de le rendre déterministe. Mais nous n'aborderons pas ce point dans ce cours. Il convenait cependant de le mentionner. + +\medskip +Nous avons vu dans ce paragraphe une manière de \emph{reconnaître} mécaniquement un langage simplement formé de mots issus d'un alphabet à l'aide d'un automate. Nous avons vu deux manières de représenter celui-ci, par un diagramme et mathématiquement. Chacune de ces deux expressions constitue un mode de représentation qui a des avantages et des inconvénients. Les automates soulignent clairement le caractère mécanique du processus de reconnaissance. L'expression mathématique exprime quant à elle mieux les propriétés ensemblistes et relationnelles de la notion d'automate. + +\begin{ex}\label{autosimple} + Construisez un automate sur l'alphabet \(\Sigma=\{a,b\}\) qui reconnaît les mots suivants : aab, abb et bbb. + \begin{sol} + Il s'agit d'un automate simple sans grande difficultés. Il est déterministe puisque de chaque état ne part qu'une seule transition de même type. + \begin{center} + \begin{postscript} + \VCDraw{% + \begin{VCPicture}{(0,-2)(6,4)} + % states + \State[e_1]{(0,0)}{A} \State[e_2]{(2,1)}{B} \State[e_3]{(4,1)}{C} + \State[e_5]{(4,3)}{E} + \State[e_8]{(2,-1)}{G} \State[e_9]{(4,-1)}{H} + \StateLineDouble \State[e_{10}]{(6,-1)}{I} \State[e_4]{(6,1)}{D} \State[e_6]{(6,3)}{F} + % initial--final + \Initial{A} + % transitions + \EdgeL{A}{B}{a} \EdgeL{B}{C}{a} \EdgeL{C}{D}{b} + \EdgeL{B}{E}{b} \EdgeL{E}{F}{b} + \EdgeL{A}{G}{b} \EdgeL{G}{H}{b} \EdgeL{H}{I}{b} + % + \end{VCPicture}} + \end{postscript} + \end{center} + \end{sol} +\end{ex} + +\begin{ex} + Construisez un automate qui reconnaisse la succession des feux de circulation rouge-jaune-vert, allumé successivement et indéfiniment dans cet ordre. + \begin{sol} + Le problème peut sembler complexe car l'énoncé du mot allumé fait immédiatement penser à éteint. Naturellement, on pense à un état allumé et à un autre éteint. Comme il y a trois couleurs, on s'imagine alors six états et leur représentation avec des transitions du type ``s'allume'' ou ``s'éteint'' ne va pas de soit. + + Pour s'en sortir simplement, il faut au contraire penser ``quel est la couleur du feu''. Cela suppose qu'on ne considère que l'état allumé du feu. On peut alors dire rouge, puis jaune, puis vert, puis rouge, \dots On pourrait aussi dire, c'est le rouge qui s'allume, puis c'est le jaune qui s'allume, puis \dots Cela nous permet alors de représenter la succession des états allumé du feu par : + \begin{center} + \begin{postscript} + \VCDraw{% + \begin{VCPicture}{(0,-3)(11,1)} + % states + \State[r]{(0,0)}{A} \State[j]{(4,0)}{B} \State[v]{(2,-2)}{C} + \State{(7,0)}{D} \State{(11,0)}{E} \State{(9,-2)}{F} + % initial--final + \Initial{A} \Initial{D} + % transitions + \EdgeL{A}{B}{} \EdgeR{B}{C}{} \EdgeR{C}{A}{} + \EdgeL{D}{E}{r} \EdgeR{E}{F}{j} \EdgeR{F}{D}{v} + % + \end{VCPicture}} + \end{postscript} + \end{center} + En particulier, l'automate ci-dessus permet la reconnaissance de la chaîne de caractère rjvrjvrjv\dots{} qui est une représentation de la succession des couleurs (allumées) du feu de circulation. + \end{sol} +\end{ex} + +\begin{ex} + Construisez un automate qui reconnaisse les codons de l'exercice \ref{codons}. + \begin{sol} + On a vu qu'il y a 64 possibilités pour former un acide aminé constitué d'un triplet choisi parmi quatre nucléotides. Il peut sembler alors que l'automate correspondant doive être très complexe. Mais au contraire, en raison du fait qu'on peut mettre n'importe quel nucléotide à chaque place parmi les trois disponibles nous permet de constituer un automate particulièrement simple : + \begin{center} + \begin{postscript} + \VCDraw{% + \begin{VCPicture}{(0,-1)(8,1)} + % states + \State{(0,0)}{A} \State{(4,0)}{B} \State{(8,0)}{C} + \StateLineDouble \State{(12,0)}{D} + % initial--final + \Initial{A} + % transitions + \EdgeL{A}{B}{C|G|T|U} \EdgeR{B}{C}{C|G|T|U} \EdgeR{C}{D}{C|G|T|U} + % + \end{VCPicture}} + \end{postscript} + \end{center} + où la notation C|G|T|U signifie soit C, soit G, soit T, soit U. + + \smallskip + On constate ainsi la concision de la description de l'automate par rapport à l'exposé de tout les cas possibles. + \end{sol} +\end{ex} + +\subsection{Les expressions rationnelles} +Voyons maintenant une autre manière de représenter les langages reconnus par les automates : les \emph{expressions rationnelles}\footnote{On parle aussi d'expressions régulières en raison de l'anglais \emph{regular expression}.}. Elles trouvent ici leur place pour souligner qu'il existe différentes approches dans la reconnaissance des langages (dits rationnels). + +\smallskip +Premièrement, une \emph{expression rationelle} est un mot. L'alphabet permettant de construire ce mot est particulier, puisque s'y trouve obligatoirement les symboles suivants : \((\,)\,+\,.\,*\,\emptyset\) Ainsi, on peut voir l'alphabet sur lequel on construit les expressions rationelles comme composé de deux ensemble : un alphabet \(\Sigma\) choisi selon les besoins et l'ensemble des symboles donné ci-dessus. L'alphabet des expressions rationelles est donc formellement : +\[\Sigma^{\text{expressions rationnelles}}=\Sigma \cup \{(,),+,.,^*,\varepsilon\}\] +où \(\varepsilon\) est le caractère vide. + +\smallskip +Secondement, une expression rationnelle doit obéir à une \emph{syntaxe}. Une manière de construire un langage, en tant que partie de l'ensemble des mots de \(\Sigma\), est de le faire à partir d'un automate. Une autre manière est de le faire à partir d'un ensemble d'opération qui permettent de le définir complètement. On appelle ces opérations des règles de \emph{syntaxe}. En d'autre termes, ces règles autorisent ou interdisent la production de mots particuliers sans référence au sens des mots produits. Ainsi, si e et r sont des expressions rationnelle, alors il faut ajouter que les règles de syntaxe des expressions rationnelles autorisent les trois expressions (e+r), (e.r) et (e*) comme expressions rationnelles. On dit ici simplement que les trois suites de caractères précédentes sont correctement formées pour qu'il s'agisse d'expressions rationnelles. Le sens des symboles +, . et * n'est pas précisé car il ne s'agit que de règles de syntaxe. + +\medskip +En définitive donc, on peut définir l'ensemble des expressions rationnelles par une règle de base (B) et par un règle récursive (R) : +\begin{description} + \item[B.] \(\varepsilon\) et e avec \(\text{e}\in \Sigma\) sont des mots sur \(\Sigma \cup \{(,),+,.,^*,\varepsilon\}\) + \item[R.] si e et r sont des expressions rationnelles, alors (e+r), (e.r) et (e*) le sont aussi. +\end{description} + +\medskip +Voilà l'ensemble des expressions rationnelles correctement défini. Il s'agit d'une définitions complexe sur laquelle nous ne reviendrons pas dans le cadre de ce cours. Il faut cependant retenir l'idée de syntaxe comme règles de production d'expressions. Elle nous mènera plus tard à celle de grammaire. + +\medskip +Jusqu'à présent nous n'avons pas parlé de sémantique. La définition rigoureuse du sens des symboles utilisés précédemment pour les expressions rationnelles (\(+,.,*\)) passe par la construction d'une relation de l'ensemble des expressions rationnelles sur une partie de l'ensemble des mots. Ces symboles prennent alors une signification particulière : le \(+\) devient un ou (\(\cup\) ou \(|\)), le \(.\) une concaténation de deux expressions rationnelles et l'étoile devient la répétition de zéro ou plusieurs fois le caractère précédent. + +\medskip +Ainsi, en imaginant la reconnaissance du code postal d'une ville suisse constitués de quatre chiffres, on peut écrire une expression rationnelle qui exprime cette structure de la manière suivante : +\[(0|1|2|3|4|5|6|7|8|9).(0|1|2|3|4|5|6|7|8|9).(0|1|2|3|4|5|6|7|8|9).(0|1|2|3|4|5|6|7|8|9)\] +Évidemment, cette notation n'est pas très pratique et en réalité on définit d'autres symboles qui permettent d'alléger l'écriture. Par exemple, de manière évidente : +\[[0..9]\{4\}\] +On trouvera avec \cite{DB08} un ouvrage très complet sur la syntaxe et la signification des symboles les plus utilisés. + +\begin{ex}\label{autotomate} + Exprimez l'automate suivant en français et transformez-le en expression rationnelle. + \begin{center} + \begin{postscript} + \VCDraw{% + \begin{VCPicture}{(0,-1)(4,1)} + % states + \State[e_1]{(4,0)}{B} + \StateLineDouble \State[e_0]{(0,0)}{A} + % initial--final + \Initial{A} + % transitions + \ArcL{A}{B}{b} \ArcL{B}{A}{a} \LoopE{B}{b} + % + \end{VCPicture}} + \end{postscript} + \end{center} + \begin{sol} + On doit tout d'abord faire un b, puis un nombre quelconque (voir nul) de b. Comme l'état de droite n'est pas final, il faut ensuite revenir à celui de gauche par un a. Enfin, on peut faire tout cela un nombre quelconque de fois, y compris aucune. L'expression rationnelle correspondante est : +\[(bb^*a)^*\] + \end{sol} +\end{ex} + +\begin{ex} + Construisez l'expression régulière correspondant aux numéros de plaques minéralogiques suisses : deux lettres, suivies de cinq chiffres décimaux. + \begin{sol} + De manière très naturelle, on a : + \[[A..Z]\{2\}[0..9]\{5\}\] + \end{sol} +\end{ex} + +\begin{ex} + Construisez l'expression régulière correspondant à l'automate de l'exercice \ref{autosimple}, page \pageref{autosimple}. + \begin{sol} + Il faut raisonner à partir de la dernière transition avant l'état final. Il s'agit toujours d'une transition b. Donc, l'expression régulière finira par un b. Puis, on a soit deux b, soit un a suivi d'un a ou b. Cela se traduit par : + \[(bb|a(a|b))b\] + \end{sol} +\end{ex} + +\section{Langage de Lindenmayer} +Avant de nous intéresser aux langages de Lindenmayer, dit L-system, il faut pousser un peu plus avant l'étude des langages formels. + +On a vu plusieurs manières de formaliser un langage simple. La formalisation des expressions rationnelles, par exemple, a fait apparaître des règles de syntaxe qui évoquent une \emph{grammaire}. Il est intéressant ici de préciser cette notion pour pouvoir comprendre au mieux les L-system. + +\subsection{Grammaire} +Une \emph{grammaire} est un ensemble \(G\) d'objets tel que : +\begin{align*} + G=&\{T,N,R,S\}\text{ où : }\\ + &T \text{ est l'alphabet des lettres terminales,}\\ + &N\text{ est l'alphabet des lettres non terminales}\\ + &S\in N\text{ est l'axiome}\\ + &R\subseteq (T \cup N)^*\times(T \cup N)^*\text{ est un ensemble fini de règles de production.} +\end{align*} +Une convention pratique veut que les symboles non-terminaux s'écrivent en majuscule et les terminaux en minuscule. + +\medskip +Voici un exemple simple, mais assez complet. +\begin{align*} + G=&\{T,N,R,S\}\text{ où : }\\ + &T=\{a,b,m,n\}\\ + &N=\{O,P\}\\ + &S=aO\\ + &R=\{O\rightarrow aO|bP\;\text{ et }\;P\rightarrow m|n\} +\end{align*} +où le signe \(|\) signifie ou. + +\smallskip +On peut donner deux exemples de génération d'éléments à partir de cette grammaire : +\begin{enumerate} + \item \(aO\rightarrow aaO\rightarrow aaaO\rightarrow aaabP\rightarrow aaabn\) + \item \(aO\rightarrow aaO\rightarrow aabP\rightarrow aabm\) +\end{enumerate} +On comprend aisément le rôle des règles de productions et celui des lettres terminales qui vont arrêter la génération de la chaîne de caractère. + +\medskip +Un automate peut constituer un autre exemple intéressant. En effet, il est possible de traduire un automate par une grammaire appropriée. +\begin{align*} + G=&\{T,N,R,S\}\text{ où : }\\ + &T=\{\text{états terminaux}\}\\ + &N=\{\text{états non terminaux}\}\\ + &S=\text{état initial}\\ + &R=\{\text{fonction de transition}\} +\end{align*} +Pour plus de clarté, considérons l'automate de l'exercice \ref{autotomate}, page \pageref{autotomate}, et tentons de le traduire en une grammaire. +\begin{align*} + G=&\{T,N,R,S\}\text{ où : }\\ + &T=\{e_0\}\\ + &N=\{e_0,e_1\}\\ + &S=e_0\\ + &R=\{\delta(e_0,a)=\bot\;;\;\delta(e_0,b)=e_1\;;\;\delta(e_1,a)=e_0\;;\;\delta(e_1,b)=e_1\} +\end{align*} + +\medskip +Évidemment, l'existence d'une correspondance entre automate et expression régulière implique l'idée d'une formalisation des expressions régulières sous forme de grammaire. Nous n'aborderons pas ce cas ici. + +\medskip +Enfin, voici un autre exemple de grammaire où il n'y a pas de symbole terminaux. +\begin{align}\label{grammaire_algue_de_lindenmayer} + G=&\{T,N,R,S\}\text{ où : }\\\nonumber + &T=\emptyset\\\nonumber + &N=\{A,B\}\\\nonumber + &S=A\\\nonumber + &R=\{A\rightarrow AB\;\text{ et }\;B\rightarrow A\} +\end{align} +Celle-ci permet la génération de séquences infinies dont nous allons parler maintenant. + +\subsection{Génération de séquence} +La production des éléments de la grammaire donnée dans le dernier exemple ci-dessus débute par l'axiome \(A\). Ensuite, les deux règles de productions doivent être utilisées pour produire l'élément suivant : remplacer \(A\) par \(AB\) et \(B\) par \(A\). Comme il n'y a qu'un \(A\) dans l'axiome, seule la première règle s'applique tout d'abord et on obtient \(AB\). Puis, les deux règles doivent à nouveau être utilisées à partir de cet état pour produire le suivant. On remplace donc les \(A\) par \(AB\) et \(B\) par \(A\) pour chaque lettre de \(AB\). On obtient ainsi \(ABA\). Et on continue indéfiniment. On obtient alors : +\begin{align}\label{chaine_algue_de_lindenmayer} + A&\nonumber\\A&B\nonumber\\A&BA\nonumber\\A&BAAB\nonumber\\A&BAABABA\nonumber\\A&BAABABAABAAB +\end{align} +Ces chaînes n'ont pas encore de signification. Mais on peut déjà envisager différentes règles de productions permettant de générer de multiples chaînes de caractères. Beaucoup de questions peuvent alors se poser concernant les propriétés des chaînes ainsi constituées. + +\medskip +Voici enfin un autre exemple très connu de grammaire permettant de générer des chaînes auxquelles on peut attacher une signification par interprétation graphique : +\begin{align}\label{grammaire_courbe_de_koch} + G=&\{T,N,R,S\}\text{ où : }\\\nonumber + &T=\{+,-\}\\\nonumber + &N=\{F\}\\\nonumber + &S=F\\\nonumber + &R=\{F\rightarrow F+F-F-F+F\} +\end{align} +Notez la séparation entre les symboles terminaux, sans règles de production et le symbole non terminal auquel correspond une règle de production. + +La génération des chaînes est alors la suivante : +\begin{align}\label{chaine_courbe_de_koch} + F&\nonumber\\F&+F-F-F+F\nonumber\\F&+F-F-F+F+F+F-F-F+F-F+F-F-\;\hookleftarrow\nonumber\\\hookrightarrow\;&F+F-F+F-F-F+F+F+F-F-F+F +\end{align} + +\smallskip +Encore une fois, le sens de ces chaînes n'est pas encore évident. Par ailleurs, on voit aussi que la génération des caractères de ces chaînes devient très vite inaccessible au possibilités humaines en raison du nombre élevé de caractères produits. Une mécanisation de la production est donc rapidement nécessaire. + +\subsection{L-system} +Les deux derniers exemples ci-dessus ((\ref{grammaire_courbe_de_koch}) et (\ref{grammaire_algue_de_lindenmayer})) sont des systèmes de Lindenmayer ou L-system. De quoi s'agit-il ? L'ouvrage de référence en la matière, \emph{The Algorithmic Beauty of Plants} \cite[p.2-3]{LA04}, en présente le contexte en ces termes : + +\begin{quotation} + \og \selectlanguage{english}In 1968 a biologist, Aristid Lindenmayer, introduced a new type of string-rewriting mechanism, subsequently termed L-systems. The essential difference between Chomsky grammars and L-systems lies in the method of applying productions. In Chomsky grammars productions are applied sequentially, whereas in L-systems they are applied in parallel and simultaneously replace all letters in a given word. This difference reflects the biological motivation of L-systems. Productions are intended to capture cell divisions in multicellular organisms, where many divisions may occur at the same time. Parallel production application has an essential impact on the formal properties of rewriting systems.\selectlanguage{french}\fg +\end{quotation} + +L'objectif est donc de modéliser des mécanisme de division (cellulaire). Le moyen utilisé est un mécanisme de réécriture. Quoi de plus naturel donc que d'utiliser une grammaire précisément basée sur la réécriture. Les règles de production s'y prêtent bien et les deux exemples présentés ci-dessus en sont une belle démonstration. Il ne s'agit pas ici d'une division cellulaire, mais de la réplication d'un motif sur lui-même. Une grammaire en est ici l'origine et nous verrons par la suite qu'elle peut se traduire par une récursivité des algorithmes nécessaires pour la production de motifs conséquents. + +Mais, pour bien comprendre la signification des mécanismes de production des L-system dans la réalité, il faut maintenant en donner une interprétation graphique. + +\subsection{Interprétation graphique} +Il s'agit ici d'utiliser les chaînes de caractères produites par les grammaires pour en faire une représentation graphique. L'articulation d'un mécanisme de production général et d'une analyse des briques composant les structures végétales est certainement à l'origine de cette idée. Elle a trouvé son expression complète grâce aux possibilités de dessin proposés par le langage de programmation Logo et la tortue (Turtle) qui lui est attachée. + +L'interprétation consiste à utiliser les caractères de l'ensemble terminal \(N\) comme des instructions d'orientation de la tortue et les symboles non-terminaux comme des instruction de construction du chemin. Classiquement, on a : + +\begin{description} + \item[\(F\)] avancer d'un pas en avant, + \item[\(+\)] tourner à gauche d'un angle donné, + \item[\(-\)] tourner à droite d'un angle donné. +\end{description} + +Cette interprétation permet de suivre les chaînes de caractères produites par la grammaire pour en tracer une représentation graphique. + +\subsubsection{Koch} +L'exemple donné par la grammaire (\ref{grammaire_courbe_de_koch}), correspondant aux chaînes (\ref{chaine_courbe_de_koch}) peut ainsi prendre l'apparence des figures \ref{figkoch} pour un angle de 90\textdegree. + +\begin{figure*}[t] +\centering +\subfigure[F+F-F-F+F\label{figkoch1}]{\includegraphics{koch1.eps}}\qquad +\subfigure[F+F-F-F+F+F+F-F-F+F-F+F-F-F+F-F+F-F-F+F+F+F-F-F+F\label{figkoch2}]{\includegraphics{koch2.eps}}\\ +\subfigure[Plus encore ...\label{figkoch3}]{\includegraphics[width=8cm]{koch3.eps}} +\caption{La courbe de Koch\label{figkoch}} +\end{figure*} + +La figure \ref{figkoch1} permet de suivre l'interprétation graphique à la main, caractère après caractère. Mais, on hésite déjà à le faire à la figure suivante pour des raisons évidentes. + +A part l'esthétique des figures \ref{figkoch}, cette interprétation graphique semble ne pas apporter grand chose. Pourtant, les objets étudiés dans \emph{The Algorithmic Beauty +of Plants +} \cite[p.8-9]{LA04} et reproduits\footnote{Le code permettant d'obtenir ces structures sera présenté plus loin.} dans la figure \ref{figkochisland} sont déjà plus intéressants et montrent des structures qui commencent à ressembler vaguement à des cristaux par exemple. + +\begin{figure*}[ht] +\centering +\subfigure[F-F+F+FF-F-F+F (axiome F-F-F-F)\label{figkochisland1}]{\includegraphics{kochIsland1.eps}}\qquad +\subfigure[F-F+F+FF-F-F+F (axiome F-F-F-F)\label{figkochisland2}]{\includegraphics[width=5cm]{kochIsland2.eps}}\\ +\subfigure[F+FF-FF-F-F+F+FF-F-F+F+FF+FF-F (axiome F-F-F-F)\label{figkochislandquadratic2}]{\includegraphics[width=5cm]{kochIslandquadratic2.eps}} +\caption{Une autre courbe de Koch\label{figkochisland}} +\end{figure*} + +\subsubsection{Modélisation des plantes} +Un autre exemple d'utilisation d'une grammaire interprétée de manière particulière est donné par la modélisation de la croissance des plantes. Turtle est ici utilisé comme précédemment. Mais l'alphabet est étendu par deux nouveaux signes : les crochets [ et ]. Leur signification dans le cadre de Turtle est spéciale : +\begin{description} + \item[le crochet ouvrant] [ signifie qu'il faut mémoriser la position et la direction courante de la tortue et + \item[le crochet fermant] ] signifie qu'il faut faire revenir la tortue à la position et direction précédemment mémorisée. +\end{description} +Cette opération de mémorisation est nécessaire pour permettre une structure d'arbre qui n'est pas possible sans cela puisque le parcours de la tortue est alors continu. + +Voyons un exemple d'une telle grammaire. +\begin{align}\label{grammaire_plante} + G=&\{T,N,R,S\}\text{ où : }\\\nonumber + &T=\{+,-,[,]\}\\\nonumber + &N=\{F\}\\\nonumber + &S=F\\\nonumber + &R=\{F\rightarrow F[+F]F[-F]F\} +\end{align} + +Après deux fois l'application de la règle (deux générations), on obtient la chaîne suivante : +\begin{center} + F[+F]F[-F]F[+F[+F]F[-F]F]F[+F]F[-F]F[-F[+F]F[-F]F]F[+F]F[-F]F +\end{center} +Encore une fois, l'interprétation n'est pas triviale, pas plus que la programmation de la mémorisation. + +\medskip +Pourtant, la figure \ref{figplante} obtenue par l'interprétation graphique pour cinq génération et un angle de 25,7\textdegree{} est étonnante. + +\begin{figure*}[t] +\centering +\includegraphics[width=12cm]{plante1.eps} +\caption{Modélisation d'une plante\label{figplante}} +\end{figure*} + +Évidemment, on s'imagine bien que le remplacement récursif d'un même motif à partir de points clés de celui-ci mène à la construction d'arbre de ce type. Mais la figure produite est tout de même remarquable dans la structure ``botanique'' qu'elle présente, même si une tige aussi rectiligne est peu probable dans la réalité. + +\medskip +Plusieurs autres règles peuvent être utilisées. Plusieurs angles aussi. Comme présenté dans \emph{The Algorithmic Beauty of Plants} \cite[p.25]{LA04}, on obtient alors des structures dont la ressemblance avec des plantes réelles est étonnante. La figure \ref{figplantes} présente les ``plantes'' obtenues avec les règles suivantes : +\begin{align} + F &\rightarrow F[+F]F[-F]F\label{regle1}\\ + F &\rightarrow FF-[-F+F+F]+[+F-F-F]\label{regle2} +\end{align} + +\begin{figure*}[ht] +\centering +\subfigure[nbiteration = 5, angle = 20\textdegree{} (axiome F ; règle \ref{regle1})\label{figplante2}]{\includegraphics[width=10cm]{plante2.eps}}\\ +\subfigure[nbiteration = 5, angle = 22,5\textdegree{} (axiome F ; règle \ref{regle2})\label{figplante3}]{\includegraphics[width=10cm]{plante3.eps}} +\caption{Deux autres plantes\label{figplantes}} +\end{figure*} + +On peut aussi illustrer l'utilisation de plusieurs règles simultanément à travers les exemples \ref{regle3}, \ref{regle4} et \ref{regle5} où l'axiome est \(X\). Les deux règles sont appliquées successivement. +\begin{align} + X &\rightarrow F[+X]F[-X]+X\;\text{ et }\;F \rightarrow FF\label{regle3}\\ + X &\rightarrow F[+X][-X]FX\;\text{ et }\;F \rightarrow FF\label{regle4}\\ + X &\rightarrow F-[[X]+X]+F[+FX]-X\;\text{ et }\;F \rightarrow FF\label{regle5} +\end{align} + +Le résultat de l'interprétation graphique est donné à la figure \ref{figplantes2}, page \pageref{figplantes2}. + +\begin{figure*}[ht] +\centering +\subfigure[nbiteration = 7, angle = 20\textdegree{} (axiome X, règle \ref{regle3})\label{figplante4}]{\includegraphics[width=6cm]{plante4.eps}}\qquad +\subfigure[nbiteration = 7, angle = 25,7\textdegree{} (axiome X, règle \ref{regle4})\label{figplante5}]{\includegraphics[width=6cm]{plante5.eps}}\\ +\subfigure[nbiteration = 5, angle = 22,5\textdegree{} (axiome X, règle \ref{regle5})\label{figplante6}]{\includegraphics[width=8cm]{plante6.eps}} +\caption{Deux autres plantes\label{figplantes2}} +\end{figure*} + +\medskip +Bien évidemment, ces exemples ne sont pas construits au hasard. Une étude approfondie de la forme des chaînes nécessaires pour obtenir des structures réalistes de plantes doit être faite, ou tout au moins une réflection sur les motifs de base. On se reportera pour cela à l'adresse \url{http://algorithmicbotany.org/papers/} où sont présentés de nombreux exemples très crédibles. En ce qui nous concerne, l'objectif de ce petit cours n'étant pas d'étudier systématiquement ou de manière approfondie les plantes elles-mêmes, on laisse au lecteur le soin de tenter de trouver des chaînes crédibles et de les interpréter grâce à la programmation. + +Car, il est temps maintenant de considérer cet aspect moins formel, mais aussi important dans le cadre d'un cours d'informatique. Nous allons donc commencer par considérer le problème de la création de L-system de manière algorithmique. Cela nous permettra d'avoir une bonne vision des mécanismes principaux permettant la génération de ces systèmes. + +\subsection{Algorithmique\label{algo}} + +\subsubsection{Création} + +Commençons par la construction de la chaîne de Lindenmayer. Nous avons vu que tout commence avec un axiome. Celui-ci peut être constitué de deux types de symboles : non-terminaux et terminaux. La production de la chaîne consiste quant à elle à remplacer un certain nombre de fois les symboles non-terminaux par une chaîne correspondante donné par une (ou plusieurs) règle. Pour simplifier la situation, on ne considérera ici qu'une seule règle. Il est donc nécessaire de disposer du nombre d'itération choisi, de l'axiome et de la règle qu'il faut initialiser. Ensuite viennent deux boucles. L'une pour répéter les remplacements des symboles non-terminaux par ceux de la règle et l'autre pour lire l'axiome. Finalement, au bout d'un nombre de génération donné, le résultat produit est un axiome ``final'' qui sera transmis pour interprétation graphique. L'algorithme \ref{algocreation}, page \pageref{algocreation}, traduit ce raisonnement en pseudo-code. + +%\begin{comment} +%\begin{algorithm}[t] +%\caption{RRT}\label{algo_rrt} +%\begin{pseudocode}[ovalbox]{mystère}{a,b,c} +%a \PREND a+b\\ +%c \PREND a+\sqrt{b}\\ +%b \PREND a-b +%\end{pseudocode} +%\end{algorithm} +%\end{comment} + +\begin{algorithm}[t] + \SetAlgoVlined + \SetAlgoCaptionSeparator{ \textendash} + %\SetAlCapSkip{5mm} + \Donnees{nbiteration, axiome, regle} + \Res{nouvel axiome} + initialisation\; + \Tq{on est < nbiteration}{ + \Tq{on est < longueur de l'axiome}{ + on lit un caractère de l'axiome\; + \Si{on trouve un caractère non terminal}{ + on le remplace par la règle\; + } + \SinonSi{on trouve un caractère terminal}{ + on le réécrit simplement\; + } + } + } + \caption{Création de la chaîne}\label{algocreation} +\end{algorithm} + +\subsubsection{Interprétation} + +Après la création de la chaîne vient son interprétation. Pour cela, il est nécessaire de disposer de l'axiome ``final'', de la longueur choisie des segments produit par la tortue et de l'angle correspondant à chaque rotation. L'interprétation graphique se fait ensuite par la lecture de chaque caractères de la chaîne et le lancement de l'opération correspondante par la tortue. + +Si l'opération de mémorisation de la position de la tortue est simple, il faut analyser plus finement le retour à cette position. En effet, il est nécessaire de mémoriser plusieurs états différents avant d'y revenir successivement. Un tableau des états est donc nécessaire. De plus, une fois revenu sur un état mémorisé, il ne faut pas oublier de l'effacer. L'algorithme \ref{algointerpretation}, page \pageref{algointerpretation}, résume tout cela en pseudo-code. + +\begin{algorithm}[bh] + \SetAlgoVlined + \SetAlgoCaptionSeparator{ \textendash} + %\SetAlCapSkip{5mm} + \Donnees{axiome, longueur, angle\\variable : tableau des états mémorisés} + \Res{le tracé} + initialisation\; + \Tq{on est < longueur de l'axiome}{ + on lit un caractère de l'axiome\; + \Si{on trouve tel caractère}{ + on demande à la tortue de faire telle opération\; + } + \SinonSi{on trouve tel autre caractère}{ + on demande à la tortue de faire telle autre opération\; + } + \SinonSi{on trouve le caractère [}{ + on mémorise sa position et sa direction + } + \SinonSi{on trouve le caractère ]}{ + on récupère l'état, on y envoie la tortue et on efface cet état de la pile des état mémorisés + } + } + \caption{Interprétation graphique de la chaîne}\label{algointerpretation} +\end{algorithm} + +Dans le cas où plusieurs règles sont présentes, seule la procédure de création change avec deux boucles de lecture successives permettant d'effectuer le remplacement de deux caractères non-terminaux différents. Ces deux boucles ayant la même forme, on ne présentera pas ce cas. Mais le code donné en annexe \ref{codepythonfonct2} en donne une illustration compréhensible. + +\subsection{Programmation} + +\subsubsection{Approche non récursive} +Python est le langage utilisé. Mais,comme toujours dans le domaine de la programmation, il y a pour chaque problèmes beaucoup de solutions possibles. Même si l'objectif de ce cours n'est pas la programmation, deux solutions différentes vont être présentées, non récursive et récursive, répondant toutes les deux à la description algorithmique présentée au point \ref{algo}. En effet, dans le cadre d'un cours sur les langages, il est une fois de plus intéressant de montrer la diversité des approches, sans pour autant se focaliser particulièrement sur les avantages des différents modes de programmation. Par contre, on abordera pas ces problèmes en programmation orientée objet ou à travers d'autres langages tels que java par exemple. Mais il est évidemment envisageable de le faire par la suite. + +Dans la section \ref{algo}, on a vu la décomposition du problème de génération des L-system en deux procédures d'une part de production des motifs à partir de la grammaire et d'autre part d'interprétation de ceux-ci par turtle. Le code complet est donné en annexe \ref{codepythonfonct}. On ne va détailler ici que les aspects non liés au langage de programmation utilisé. Cependant, il faut signaler qu'on a utilisé un découpage du problème en \emph{fonction} pour des raisons de clarté uniquement et qu'on a aussi mélangé variables \emph{globale} et \emph{locales}, \emph{procédures} et \emph{fonctions} pour mettre en jeu ces notions dans un programme fonctionnel. Mais, on ne les définira pas ici. + +\paragraph{La création de la chaîne} consiste essentiellement en une boucle qui parcourt successivement tous les caractères. +\begin{lstlisting}[language=Python] + while j < len(axiom): + pointeur1 = axiom[j] +\end{lstlisting} +La lecture se fait en plaçant le caractère j de l'axiome dans une variable \verb|pointeur1| qui permettra de décider du caractère terminal ou non d'un caractère. En effet, par remplacement de la règle dans les symboles non terminaux de l'axiome, la boucle va créer un nouvel axiome, nommé \verb|axiome2|, qui sera utilisé comme axiome de départ à chaque itération. Ainsi, on a : +\begin{lstlisting}[language=Python] + # Symboles non terminaux + if pointeur1 == 'F': + # remplacement de la regle dans l'axiome + axiom2 = axiom2 + regle +\end{lstlisting} +On voit qu'on remplace le caractère \verb|F| par la chaîne correspondante de la règle. Par contre, la lecture d'un symbole terminal ne donne lieu à aucun remplacement. On réécrit simplement ce symbole tel quel. + +Ainsi, au fur et à mesure de la lecture des caractères,on construit par concaténation une nouvelle chaîne \verb|axiom2| qui sera à la base de la prochaine itération. + +\paragraph{L'interprétation de la chaîne} se fait à nouveau par lecture de celle-ci. Après création, elle est passée à la procédure d'interprétation via le paramètre \verb|axiom|, contenant la chaîne de caractère qui sera lue. Ensuite, pour les symboles de déplacement et d'orientation, l'interprétation est simple : +\begin{lstlisting}[language=Python] + # Interpreation + if pointeur2 == 'F': + forward(longueur) + elif pointeur2 == '+': + left(angle) + elif pointeur2 == '-': + right(angle) +\end{lstlisting} +Par contre, pour les symboles de mémorisation, la procédure est quelque peu plus complexe : +\begin{lstlisting}[language=Python] + elif pointeur2 == '[': + # memorisation de la position courante + laposition.append(position()) + langle.append(heading()) + elif pointeur2 == ']': + # retour a la derniere position et suppression de celle-ci + up() + goto(laposition[len(laposition)-1][0],\ + laposition[len(laposition)-1][1]) + del laposition[len(laposition)-1] + setheading(langle[len(langle)-1]) + del langle[len(langle)-1] + down() +\end{lstlisting} +Tout d'abord, on utilise ici un tableau, nommé \verb|laposition| pour mémoriser chaque couple abscisse et ordonnée de la tortue et un autre tableau, nommé \verb|langle| pour mémoriser sa direction. + +\smallskip +Puis, pour le retour à la position mémorisée, on réalise un \verb|goto| aux coordonnées mémorisées et un \verb|setheading| pour revenir à la direction enregistrée. Enfin, il ne faut pas oublier de supprimer des deux tableaux la position à laquelle on est revenu. Cela se fait par deux \verb|del|. Notez que pendant le retour il ne faut pas que la tortue soit en position d'écriture. On lève donc le crayon avant le retour et on le baisse après. + +\medskip +En fin de compte, grâce à l'utilisation de fonctions (et procédures), on peut écrire le programme principal avec une grande clarté : +\begin{lstlisting}[language=Python] + initialisation() + gram = grammaire() + ## gram[0] retourne l'angle + ## gram[1] retourne l'axiome + ## gram[2] retourne la regle + + axio = production(gram[1], gram[2]) + interpretation(gram[0], axio) +\end{lstlisting} +L'initialisation se fait par le placement de la tortue et la mémorisation des paramètre, axiome et règle. Puis, vient naturellement la production et l'interprétation. + +\subsubsection{Approche récursive} +Une autre approche de la génération de L-system fait usage d'une technique de programmation particulièrement intéressante : la récursivité. Il s'agit de l'appel d'une fonction à elle-même. Pour les L-system, il s'agit donc de ne plus générer la chaîne de caractère, mais de lire l'axiome en renvoyant simultanément à la fonction appelée, à chaque occurrence d'un caractère non terminal. Il n'est alors pas évident de suivre ce que fait le programme. C'est pourquoi on va maintenant reprendre une forme légèrement différente de celle donnée précédemment du flocon de Koch pour l'étudier dans le détail. Le code complet est donne en annexe \ref{codepythonfloconrecursif}. Seule la partie purement récursive est présentée ci-dessous : +\begin{lstlisting}[language=Python] + def production(nbiteration, longueur, angle, axiome, regle): + if nbiteration == 0 : + forward(longueur) + else: + i = 0 + while i < len(regle): + regle1 = regle[i] + if regle1 == 'F': + production(nbiteration-1, longueur/3,\ + angle, axiome, regle) + if regle1 == '+': + left(angle) + if regle1 == '-': + right(angle) + i = i + 1 +\end{lstlisting} +Et la figure \ref{Kochrecursif1} en donne l'interprétation graphique. + +\begin{figure*}[t] +\centering +\includegraphics[width=5cm]{kochrecursif1.eps} +\caption{Flocon de Koch récursif.\label{Kochrecursif1}} +\end{figure*} + +Le code est remarquable de concision, mais complexe à apréhender. + +\smallskip +Le c\oe ur de la récursivité se trouve dans le test de reconnaissance du caractère \verb|F|. Il s'agit du caractère non terminal et il est nécessaire de le remplacer par l'ensemble des caractères de la règle. Mais, au lieu de réaliser simplement ce remplacement, on renvoie à la fonction de production. Cela n'a pas pour effet de remplacer le caractère non terminal par les symboles de la règle, mais de renvoyer sur une procédure, en l'occurrence celle dans laquelle on se trouve, qui va le faire. Or, cette fonction va encore s'appeler elle-même à la lecture du caractère \verb|F| et ainsi de suite. + +Le même processus va ainsi se répéter un nombre de fois déterminé par la variable \verb|nbiteration|. Comme à chaque appel de la fonction de production on diminue la variable contenant le nombre d'itération, quand celle-ci devient nulle, on stoppe l'appel récursif et on trace un segment. + +Ensuite, on retourne à la dernière procédure appelée pour réaliser l'instruction qui suit l'appel récursif selon la règle donnée, par exemple une instruction terminale de rotation. Et on continue pour réaliser une suite d'instructions terminales ou si non terminales, pour réaliser un appel récursif à la fonction principale. + +Il est très difficile de suivre chaque étape de ce processus. Cependant, la figure \ref{appelrecursif} tente d'en donner une représentation graphique qui montre l'appel et le retour des processus dans l'ordre des flèches numérotées. + +\begin{figure*}[t] +\centering +\begin{psfrags} + \psfrag{production(...)}{production(...)} + \psfrag{if regle1}{if regle1 == 'F':} + \psfrag{production}{production} + \psfrag{de}{(\scriptsize nbiteration-1,longueur-1)} + \psfrag{if regle2}{if regle1 == '+'} + \psfrag{left}{left(angle)} + \psfrag{if nbiteration}{if nbiteration == 0:} + \psfrag{forward}{forward(longueur)} + \psfrag{1}{1} + \psfrag{2}{2} + \psfrag{3}{3} + \psfrag{4}{4} + \psfrag{5}{5} + \psfrag{6}{6} +\includegraphics[width=12cm]{production.eps} +\end{psfrags} +\caption{Flocon de Koch récursif.\label{appelrecursif}} +\end{figure*} + +On voit que toute une série de procédures sont progressivement lancées les unes dans les autres. Puis, arrivées à leur terme, le tracé des segment se fait et elles se terminent par un retour à la procédure qui les a lancées. Finalement, tout aussi progressivement, on termine chacune des procédures imbriquées. + +\medskip +On a déjà remarqué que le code est plus concis que celui issu d'une simple lecture-interprétation. Par contre, il est moins clair, demanderait donc pour la modification une relecture attentive qui peut prendre du temps et il faudrait se demander s'il est plus performant. + +\medskip +Voilà. Comme le but de ce cours n'est pas la programmation récursive, nous finirons là cette analyse en précisant cependant les limites de celle-ci. En effet, avec la récursivité on utilise toujours la même règle. L'axiome n'est donc pas utilisé ou considéré égal à \verb|F|. Certes, on peut envisager des moyens de contourner ce problème en réalisant l'axiome hors de la procédure récursive, en utilisant une récursivité indirecte ou en remplaçant à un moment bien précis la règle par l'axiome. Mais cela complique le code et l'avantage en concision de la récursivité s'amenuise. + +\section{Conclusion} +Nous avons parcouru avec cette introduction aux L-system beaucoup de notions d'informatiques importantes. Les langages, leur reconnaissance, les grammaires, l'algorithmique et finalement différents modes de programmation nous ont permis de comprendre le rôle essentiel joué par les structures formelles que l'informatique théorique met à notre disposition. Les limites de ce cours ne nous ont cependant pas permis d'aborder d'autres notions tout aussi importantes comme les réseaux de Pétri ou les machines de Turing. Par contre, l'étude des L-system marque clairement la puissance de la modélisation informatique des structures biologiques. Plus, elle fait entrevoir une correspondance fascinante entre le développement des organisme vivants et des constructions répétitives très structurées, finalement très bien décrites par \dots{} le langage mathématique. + +\Closesolutionfile{ans} % ferme le fichier Solutions.tex qui contient les solutions +\vfill +\pagebreak % met les solutions sur une autre page +\section{Solutions des exercices} +\Readsolutionfile{ans} % importe les données du fichier Solutions.tex + +\myclearpage + +%\appendix + +\section{Annexes} + +\subsection{Code python plante}\label{codepythonfonct} + +\begin{lstlisting}[language=Python] + # -*- coding: utf-8 -*- +from turtle import * + +# Generation de plante par grammaire L-system + +def initialisation(): + # Definition des variables independantes de la grammaire + global nbiteration, longueur + nbiteration = 5 + longueur = 3 + +def grammaire(): + # Definition de la grammaire + angle = 25.7 + axiom = 'F' + + regle = 'FF-[-F+F+F]+[+F-F-F]' + #regle = 'F[+F]F[-F][F]' + #regle = 'F[+F]F[-F]F' + #regle = 'F+FF-FF-F-F+F+FF-F-F+F+FF+FF-F' + + return angle, axiom, regle + +def production(axiom, regle): + # Generation de la chaine de caracteres + i = 0 + while i < nbiteration: + j = 0 + axiom2 = '' + while j < len(axiom): + pointeur1 = axiom[j] + + # Symboles non terminaux + if pointeur1 == 'F': + # remplacement de la regle dans l'axiome + axiom2 = axiom2 + regle + + # Symboles terminaux + if pointeur1 == '+': + axiom2 = axiom2 + '+' + if pointeur1 == '-': + axiom2 = axiom2 + '-' + if pointeur1 == '[': + axiom2 = axiom2 + '[' + if pointeur1 == ']': + axiom2 = axiom2 + ']' + + j = j + 1 + axiom = axiom2 + i = i + 1 + print axiom + return axiom + +def interpretation(angle, axiom): + # Interpretation de la chaine par turtle + k = 0 + laposition = [] + langle = [] + while k < len(axiom): + pointeur2 = axiom[k] + + # Interpreation + if pointeur2 == 'F': + forward(longueur) + elif pointeur2 == '+': + left(angle) + elif pointeur2 == '-': + right(angle) + elif pointeur2 == '[': + # memorisation de la position courante + laposition.append(position()) + langle.append(heading()) + elif pointeur2 == ']': + # retour a la derniere position et suppression de celle-ci + up() + goto(laposition[len(laposition)-1][0], laposition[len(laposition)-1][1]) + del laposition[len(laposition)-1] + setheading(langle[len(langle)-1]) + del langle[len(langle)-1] + down() + + k = k + 1 + +# Placement initial de la tortue +up() +backward(495) +down() + +# Programme principal +## gram[0] retourne l'angle +## gram[1] retourne l'axiome +## gram[2] retourne la regle + +initialisation() +gram = grammaire() + +axio = production(gram[1], gram[2]) +interpretation(gram[0], axio) + +#quit() +\end{lstlisting} + +\subsection{Code python plante à deux règles}\label{codepythonfonct2} + +\begin{lstlisting}[language=Python] +# -*- coding: utf-8 -*- +from turtle import * + +# Generation de plante par grammaire L-system + +def initialisation(): + # Definition des variables independantes de la grammaire + global nbiteration, longueur + # Pour le flocon de Koch + #nbiteration = 3 + #longueur = 2 + + # Pour le flocon de Koch Island + #nbiteration = 3 + #longueur = 2 + + # Pour la plante + #nbiteration = 5 + #longueur = 5 + + # Pour la plante 2 + #nbiteration = 5 + #longueur = 5 + + # Pour la plante 3 + #nbiteration = 4 + #longueur = 8 + + # Pour la plante 4 + #nbiteration = 7 + #longueur = 1 + + # Pour la plante 5 + #nbiteration = 7 + #longueur = 1 + + # Pour la plante 6 + nbiteration = 5 + longueur = 4 + +def grammaire(): + # Definition de la grammaire + # Pour le flocon de Koch + #angle = 90 + #axiom = 'F-F-F-F' + #regle = 'F+F-F-F+F' + + # Pour le flocon de Koch Island + #angle = 90 + #axiom = 'F-F-F-F' + #regle = 'F-F+F+FF-F-F+F' + + # Pour la plante + #angle = 25.7 + #axiom = 'F' + #regle = 'F[+F]F[-F]F' + + # Pour la plante 2 + #angle = 20 + #axiom = 'F' + #regle = 'F[+F]F[-F][F]' + + # Pour la plante 3 + #angle = 22.5 + #axiom = 'F' + #regle = 'FF-[-F+F+F]+[+F-F-F]' + + # Pour la plante 4 + #angle = 20 + #axiom = 'X' + #regle1 = 'F[+X]F[-X]+X' + #regle2 = 'FF' + + # Pour la plante 5 + angle = 25.7 + axiom = 'X' + regle1 = 'F[+X][-X]FX' + regle2 = 'FF' + + # Pour la plante 6 + angle = 22.5 + axiom = 'X' + regle1 = 'F-[[X]+X]+F[+FX]-X' + regle2 = 'FF' + + return angle, axiom, regle1, regle2 + +def production(axiom, regle1, regle2): + # Generation de la chaine de caracteres + i = 0 + while i < nbiteration: + j = 0 + axiom2 = '' + while j < len(axiom): + pointeur1 = axiom[j] + + # Symboles non terminaux + if pointeur1 == 'X': + # remplacement de la regle dans l'axiome + axiom2 = axiom2 + regle1 + + # Symboles terminaux + if pointeur1 == '+': + axiom2 = axiom2 + '+' + if pointeur1 == '-': + axiom2 = axiom2 + '-' + if pointeur1 == '[': + axiom2 = axiom2 + '[' + if pointeur1 == ']': + axiom2 = axiom2 + ']' + if pointeur1 == 'F': + axiom2 = axiom2 + 'F' + + j = j + 1 + axiom = axiom2 + print axiom + + m = 0 + axiom3 = '' + while m < len(axiom): + pointeur2 = axiom[m] + + # Symboles non terminaux + if pointeur2 == 'F': + # remplacement de la regle dans l'axiome + axiom3 = axiom3 + regle2 + + # Symboles terminaux + if pointeur2 == '+': + axiom3 = axiom3 + '+' + if pointeur2 == '-': + axiom3 = axiom3 + '-' + if pointeur2 == '[': + axiom3 = axiom3 + '[' + if pointeur2 == ']': + axiom3 = axiom3 + ']' + if pointeur2 == 'X': + axiom3 = axiom3 + 'X' + + m = m + 1 + axiom = axiom3 + i = i + 1 + + print axiom + return axiom + +def interpretation(angle, axiom): + # Interpretation de la chaine par turtle + k = 0 + laposition = [] + langle = [] + while k < len(axiom): + pointeur2 = axiom[k] + + # Interpreation + if pointeur2 == 'F': + forward(longueur) + elif pointeur2 == '+': + left(angle) + elif pointeur2 == '-': + right(angle) + elif pointeur2 == '[': + # memorisation de la position courante + laposition.append(position()) + langle.append(heading()) + elif pointeur2 == ']': + # retour a la derniere position et suppression de celle-ci + up() + goto(laposition[len(laposition)-1][0], laposition[len(laposition)-1][1]) + del laposition[len(laposition)-1] + setheading(langle[len(langle)-1]) + del langle[len(langle)-1] + down() + + k = k + 1 + +# Placement initial de la tortue +up() +backward(400) +down() + +# Programme principal +## gram[0] retourne l'angle +## gram[1] retourne l'axiome +## gram[2] retourne la regle + +initialisation() +gram = grammaire() +axio = production(gram[1], gram[2], gram[3]) +interpretation(gram[0], axio) + +#quit() + +\end{lstlisting} + +\subsection{Code python flocon de Koch récursif}\label{codepythonfloconrecursif} + +\begin{lstlisting}[language=Python] + # -*- coding: utf-8 -*- + from turtle import * + + # Generation de plante par grammaire L-system + + def initialisation(): + # Definition des variables independantes de la grammaire + # Pour le flocon de Koch + nbiteration = 3 + longueur = 200 + angle = 45 + axiome = 'F' + regle = 'F+F--F+F' + return nbiteration, longueur, angle, axiome, regle + + def production(nbiteration, longueur, angle, axiome, regle): + if nbiteration == 0 : + forward(longueur) + else: + i = 0 + while i < len(regle): + regle1 = regle[i] + if regle1 == 'F': + production(nbiteration-1, longueur/3,\ + angle, axiome, regle) + if regle1 == '+': + left(angle) + if regle1 == '-': + right(angle) + i = i + 1 + + # Placement initial de la tortue + up() + backward(200) + down() + + # Programme principal + ## init[1] : nbiteration + ## init[2] : longueur + ## init[3] : angle + ## init[4] : axiome + ## init[5] : regle + + init = initialisation() + production(init[0], init[1], init[2], init[3], init[4]) + right(120) + production(init[0], init[1], init[2], init[3], init[4]) + right(120) + production(init[0], init[1], init[2], init[3], init[4]) +\end{lstlisting} + +\myclearpage + +%\theendnotes + +\myclearpage + +%\renewcommand{\bibname}{Bibliographie} % pour mettre autre chose que bibliographie, par défaut avec babel +\nocite{CJ06,PP94,PP94,TO07,GP09,BD11} % pour mettre toutes les entrées dans la biblio, y compris celles non citées ds le texte +\bibliographystyle{apalike-fr} % le style de bibliographie apalike francisé +%\bibliographystyle{apalike} % le style de bibliographie +\bibliography{../../Bibliographies/BiblioCours} % définit le fichier des réf. biblio. nom.bib + + % le logiciel utilisé pour faire la biblio est kbibtex +%\printindex{} % attention, pour que l'index soit mis à jour il faut lancer la commande de konsole : makeindex -s ../Perso.ist nom_du_cours.idx + % le Perso.ist est un fichier disant qu'il faut mettre des lettres entre les parties de l'index + +%\renewcommand{\listfigurename}{Liste des figures} +%\listoffigures % ai tout basculé après la table des matières car je n'arrivais pas à éliminer le petit bug ci-dessous. Mais le compagnon latex fait idem. +%\listoftables % un petit problème (bug ?) d'alignement m'a fait retirer provisoirement cette liste + +\end{document} +\endinput % termine l'importation des solutions \ No newline at end of file diff --git a/Lsystem.tex.backup b/Lsystem.tex.backup new file mode 100644 index 0000000..6cc6a41 --- /dev/null +++ b/Lsystem.tex.backup @@ -0,0 +1,1285 @@ +\documentclass[11pt,a4paper,titlepage,twoside]{article} + +\input{Preambule.tex} % Toutes les instructions du préambule sont dans un fichier séparé qui est utilisé pour tous les cours. + % Il faut le mettre à l'extérieur du dossier dans lequel se trouve le cours --> ../Preambule.tex + +\usepackage{geometry} % La seule présence de ce paquet élargit les marges correctement + +\lstloadlanguages{python,TeX,XML} % chargement des languages nécessaires (ici un seul) +%\lstset{language=TeX} % language choisi par défaut + +\setlength{\headheight}{22.6pt} % pour élargir un peu l'entête +%\setcounter{tocdepth}{5} % pour définir la profondeur de la table des matières +%\renewcommand{\thefootnote}{\alph{footnote}} % pour mettre les notes de bas de page en alphabétique + +%\usepackage{etex} +%\usepackage[all,2cell]{xy} +%\UseAllTwocells +%\usepackage{xymtex} % pour la représentation de molécules chimiques +%\let\urlorig\url % pour mettre des deux points dans les url, passage en langue anglaise +%\renewcommand{\url}[1]{% +% \begin{otherlanguage}{english}\urlorig{#1}\end{otherlanguage}% +%} +%\usepackage[colorlinks=true,linkcolor=black,urlcolor=blue,pagebackref,frenchlinks]{hyperref} +\usepackage{eurosym} +\usepackage{textcomp} +\usepackage{auto-pst-pdf} +\usepackage{vaucanson-g} +%\usepackage{algorithm,algorithmique} +\usepackage[french,boxruled]{algorithm2e} + +\setlength{\headheight}{13.6pt} % spécifie la hauteur de l'entête pour qu'elle ne couvre pas le titre + +\setcounter{topnumber}{3} % nombre maximal de flottant en haut d'une page (par défaut 2) +\setcounter{bottomnumber}{2} % nombre maximal de flottant en bas d'une page (par défaut 1) +%\setcounter{totalnumber}{3} % nombre maximal de flottant sur une page (par défaut 3) +\renewcommand{\topfraction}{0.9} % proportion maximale de la page qui peut recevoir des flottants en haut de la page (par défaut 0,7) +\renewcommand{\textfraction}{0.1} % proportion minimale d'une page qui reçoit du texte (sans les pages de flottants) +\renewcommand{\floatpagefraction}{0.8} % proportion minimale de flottants sur une page pour qu'elle devienne une page uniquement de flottants +\renewcommand{\bottomfraction}{0.5} % proportion maximale de la page qui peut recevoir des flottants en bas de la page (par défaut 0,3) + +\usepackage{answers} % pour utiliser le module answers +\Newassociation{sol}{Solution}{ans} % définition de l'étiquette Solution du module answers +\newtheorem{exc}{Exercice} % définition de l'étiquette Exercice du module answers +\newenvironment{ex}{\begin{exc}\normalfont}{\end{exc}} % cela (\normalfont) annule l'italique des énoncés des exercices + +\graphicspath{{./Images/}{./Introduction/Images/}} + +%\fancyhead[RE,LO]{Lycée Blaise Cendrars\\La Chaux-de-Fonds} +\fancyhead[LE,RO]{\thepage} +\lfoot{} \cfoot{} \rfoot{} +\pagestyle{fancy} + +%\usepackage[firstpage]{draftwatermark} +%\SetWatermarkLightness{0.8} +%\SetWatermarkAngle{60} +%\SetWatermarkScale{0.8} +%\SetWatermarkFontSize{5cm} +%\SetWatermarkText{DOCUMENT À LAISSER AU LYCÉE} + +\begin{document} + +%\frontmatter +\newcommand\HRule{\noindent\rule{\linewidth}{1.5pt}} +\begin{titlepage} + \vspace*{\stretch{1}} + \HRule + \begin{flushright} + \LARGE Option complémentaire \\ + \LARGE Informatique \\ + \textsc{Langages formels et L-system} + \end{flushright} + \HRule\\ +% \textsc{Document à laisser au lycée} + \vspace*{\stretch{2}} + \begin{center} + \today + \end{center} +\end{titlepage} + +\section*{Copyright} +\addcontentsline{toc}{section}{Copyright} +Copyright (c) 2011 Guyot Vincent + +\medskip +Permission vous est donnée de copier, distribuer et/ou modifier ce document selon les termes de la Licence GNU Free Documentation License, Version 1.1 ou ultérieure publiée par la Free Software Foundation. + +\medskip +\begin{tabbing}\small +\hspace*{1cm}\=\hspace*{1cm}\=\kill +\>Avec les sections inaltérables suivantes :\\ +\>\>\textit{Pas de section inaltérable}\\ +\>avec le texte de première page de couverture suivant :\\ +\>\>\textit{Option complémentaire \textsc{Informatique}}\\ +\>avec le texte de dernière page de couverture suivant :\\ +\>\>\textit{Pas de texte de texte de dernière page de couverture}\\ +\end{tabbing} + +Une copie transparente de ce document se trouve à l'adresse suivante : +www.cvgg.org +\begin{itemize} +\item Lsystem.tex : copie transparente, +\item Lsystem.dvi : copie opaque au format dvi. +\end{itemize} + +\bigskip +Normalement une copie de la licence GFDL doit se trouver avec le présent document. Comme il est relativement court et comme on trouve facilement le texte de la GFDL sur internet, elle n'est pas reproduite ici pour ne pas utiliser inutilement du papier. +\thispagestyle{empty} + +\newpage + +\section*{Avertissement} +\addcontentsline{toc}{section}{Avertissement} + +\thispagestyle{empty} + +\tableofcontents{} +\addcontentsline{toc}{section}{Table des matières} +\thispagestyle{empty} + +\newpage + +%\mainmatter + +\Opensolutionfile{ans}[Solutions] % ouvre le fichier Solutions.tex qui va contenir les solutions + +\section{Introduction} +Ce cours s'adresse à des élèves dont les connaissances en informatiques se limitent à l'utilisation de logiciels et à des connaissances de base (boucles et fonctions) en programmation. + +\smallskip +Il ne s'adresse pas spécifiquement à des élèves non mathématiciens, car ceux-ci peuvent aussi y trouver de l'intérêt, mais surtout à des personnes qui ne se destinent ni à l'informatique, ni au mathématiques. C'est pourquoi, l'accent à été mis sur la plus grande clarté possible dans l'exposé des notions qui relèvent des mathématiques et sur les exemples. Il constitue à mon avis une approche adaptée pour des personnes intéressées par la biologie et par les langues. + +\smallskip +Les objectifs sont multiples. +\begin{description} + \item[Le premier] d'entre eux est de montrer aux élèves que l'analyse formelle est un préalable nécessaire à la compréhension des langages, en particulier informatiques. + \item[Le deuxième] est de présenter un exemple de la diversité des modes de représentation de ces langages. + \item[Le troisième] est d'illustrer le lien qui existe entre les langages et certains processus naturels, liens qui montre l'importance d'étude des formalismes présentés. + \item[Le dernier] objectif est finalement de permettre à l'élève d'utiliser des connaissances très sommaires de programmation pour analyser les processus mécaniquement dans des proportion inaccessibles à l'homme. +\end{description} + +\section{Langage formel} +Rendre compte de la notion de langage formel le plus simplement possible n'est pas chose facile. Il se trouve que c'est la rigueur du langage mathématique qui permet d'en exprimer au mieux l'idée. Nous allons donc l'utiliser le plus simplement possible étant donné les difficultés qu'il peut représenter. Il sera néanmoins bien souvent plus une aide qu'un obstacle. + +Tout langage est au départ constitué d'un alphabet. Il faut considérer le terme d'alphabet au sens large du terme comme un simple ensemble de signes ou lettres, même non alphabétiques au sens commun du terme. Ainsi, les ensembles suivants notés \(\Sigma_i\) constituent des exemples d'alphabet : +\begin{align*} + \Sigma_1 &= \{\text{\officialeuro, \maltese, \textleaf, \checkmark, 1, 4, \textvisiblespace}\}\\ + \Sigma_2 &= \{\text{\textlnot, \textdollaroldstyle, \textcircled{a}, \pounds}\}\\ + \Sigma_3 &= \{\text{a, b, \textvisiblespace, }\epsilon\} +\end{align*} +A partir de l'alphabet, on peut alors définir ce qu'est un mot. Il s'agit simplement d'une suite finie de lettres prises dans l'alphabet. Ainsi, on peut former les mots suivants sur l'alphabet \(\Sigma_1\) : \hspace{1cm}\officialeuro\officialeuro\maltese\officialeuro\officialeuro\checkmark\hspace{1cm} ou \hspace{1cm}\checkmark14\maltese\maltese\hspace{1cm} Pour l'alphabet \(\Sigma_3\), on pourrait avoir : \hspace{1cm}aababbab\hspace{1cm} ou \hspace{1cm}aabbbba \hspace{1cm}mais pas \hspace{1cm}aab\textlnot + +Si m est un mot sur un alphabet \(\Sigma\), on écrira : +\[\text{m}\in\Sigma^*\] +pour dire que le mot m appartient à l'ensemble \(\Sigma^*\) des mot possibles formé sur l'alphabet \(\Sigma\). Finalement, on définit \(\epsilon\) comme un mot vide. La raison pour laquelle on fait cela sera vue plus tard. + +\medskip +Maintenant qu'on a des mots, le problème est de pouvoir les utiliser. Si on considère les mots suivants de \(\Sigma_3\) : +\[\text{abaab\textvisiblespace\hspace{1cm}}\text{aba\textvisiblespace\hspace{1cm}}\text{aabaaba\textvisiblespace}\] +il serait possible de former queque chose qui ressemble à des phrases en plaçant ces mots les uns à côté des autres. Cette opération se nomme la concaténation et on la note par un point. Ainsi, on va formellement définir la concaténation ainsi : +\[\text{Soit m et n }\in \Sigma^*\text{\hspace{0.3cm} Alors, m.n=mn}\] +On peut ainsi tenter de reproduire l'apparence formelle d'un langage par concaténation des mots ci-dessus (en s'imaginant un espace invisible) : +\[\text{abaab\textvisiblespace}.\text{aba\textvisiblespace}.\text{aabaaba\textvisiblespace}=\text{abaab aba aabaaba}\] +Certes, le résultat ne correspond qu'en apparence à un langage. Mais, l'alphabet est très restreint. + +\medskip +De manière naturelle, on va donc définir la notion de langage comme un sous-ensemble de l'ensemble de tous les mots d'un alphabet donné. Un langage est donc un ensemble de mots. On peut donc faire l'union, l'intersection, \dots de deux langages. + +\medskip +L'exemple classique est le langage des expressions arithmétiques. Il est défini à partir de l'alphabet suivant : +\[\Sigma = \{0,1,2,3,4,5,6,7,8,9,+,-,*,/\}\] +Le langage des expressions arithmétiques s'écrit alors : +\[L(\text{expr. arithm.})=\{\dots ,342, 34+4, 45+34, 2=4, \dots\}\] +Relevez qu'il ne s'agit pas de l'ensemble de tous les mots formé sur l'alphabet, mais d'un sous ensemble qui ne contient pas l'expression 34+++43==, par exemple. Évidemment, il faut spécifier comment on réalise le choix de chaque expression. + +\begin{ex} + Sur l'alphabet des chiffres binaires (que vous noterez précisément), donnez le langage \(L\) des nombres à trois chiffres. + \begin{sol} + L'alphabet des chiffres binaires est : \(\Sigma=\{0,1\}\) et le langage des nombres binaires à trois chiffres est : + \[\Sigma^*=\{000,001,010,011,100,101,110,111\}\] + \end{sol} +\end{ex} + +\begin{ex}\label{codons} + Pour la molécule d'ADN, définissez un alphabet de ses composants et déterminez quelques éléments du langage des \emph{codons}, acide aminé constitué de triplets de nucléotides. Combien peut-il y en avoir si leur arrangement tient compte de l'ordre et qu'il peut y avoir des répétitions. + + Indication : les nucléotides sont les briques de base de l'ADN. + \begin{sol} + En considérant les quatre nucléotides composant l'ADN, la thymine (T), la cytosine (C), l'adénine (A) et la guanine (G), on peut définir l'alphabet nécessaire ainsi : + \[\Sigma=\{A,C,G,T\}\] + Les codons sont quant à eux des ensemble de trois nucléotides, pris parmi les quatre disponibles dans l'alphabet. Le langage ainsi formé est : + \[L=\{AAC,ACT,ACT,CTG,GTC, \dots\}\] + et la combinatoire donne pour un arrangement avec répétition de trois éléments pris parmi quatre : + \[\overline{A_3^4}=4^3=64\;\text{possibilités}\] + \end{sol} +\end{ex} + +\bigskip +La génération ou la vérification des expressions valables pour un langage donné peut se faire de différentes manières. Nous allons présenter dans la section suivante les moyens qui nous permettrons de décrire simplement des langages particuliers. On peut utiliser pour cela soit un automate soit une expression régulière. + +\section{Langage rationnel} + +\subsection{Les automates} + +Une manière particulièrement compréhensible de tester l'appartenance d'une expression à un langage est d'utiliser un automate. Il s'agit d'une sorte de machine symbolique capable de générer un ensemble de mots. On la suit dans la construction de ceux-ci selon des flèches qui mènent d'un état à l'autre par la concaténation successive de certaines mots de l'alphabet. Pour des raison de compréhension, on va commencer par donner un exemple d'automate à travers une représentation graphique qui fait correspondre des ronds aux états et des flèches aux éléments de l'alphabet. Un état initial est un rond sur lequel arrive une flèche vide et un état final est constitué d'un double cercle. Ainsi, on représente un automate reconnaissant le mot : +\[aba\;\text{ sur l'alphabet }\;\Sigma=\{a,b\}\] +par la figure suivante : +\begin{center} +\begin{postscript} +\VCDraw{% +\begin{VCPicture}{(0,-2)(9,0)} +% states +\State{(0,0)}{A} \State{(3,0)}{B} \State{(6,0)}{C} \FinalState{(9,0)}{D} +% initial--final +\Initial{A} +% transitions +\EdgeL{A}{B}{a} \EdgeL{B}{C}{b} \EdgeL{C}{D}{a} +% +\end{VCPicture}} +\end{postscript} +\end{center} +Relevons qu'il est possible de considérer l'unique mot construit par cet automate comme un langage qu'on peut définir mathématiquement très simplement par : +\[L=\{m\in\Sigma^*\,|\,m=aba\}\] + +où la barre verticale \(|\) signifie ``tel que''. + +\smallskip +Évidemment, l'objectif est de permettre la construction de langages plus étendus. On peut par exemple construire le langage constitué d'une séquence de ab ainsi : +\begin{center} +\begin{postscript} +\VCDraw{% +\begin{VCPicture}{(0,-1)(3,1)} +% states +\FinalState{(0,0)}{A} \State{(3,0)}{B} +% initial--final +\Initial{A} +% transitions +\ArcL{A}{B}{a} \ArcL{B}{A}{b} +% +\end{VCPicture}} +\end{postscript} +\end{center} + +Cet exemple permet d'engendrer une infinité de mots du type ab, abab, ababab, \dots + +\smallskip +Par ailleurs, la présence d'un état initial, simultanément final, implique l'existence dans le langage d'un mot vide noté \(\epsilon\). On écrira donc mathématiquement le langage ainsi : +\[L=\{m\in\Sigma^*\,|\,m=\epsilon,ab,abab,ababab,\dots\}\] + +Un dernier exemple montre qu'il est possible de générer des langages bien plus complexes. Sans dire immédiatement ce dont il s'agit, voici l'automate : + +\begin{center} +\begin{postscript} +\VCDraw{% +\begin{VCPicture}{(0,-1)(6,2)} +% states +\State{(0,0)}{A} \State{(3,0)}{B} \StateLineDouble \State{(6,0)}{C} +% initial--final +\Initial{A} +% transitions +\EdgeL{A}{B}{a} \ArcL{B}{C}{b} \ArcL{C}{B}{b} +\LoopN{A}{a} +% +\end{VCPicture}} +\end{postscript} +\end{center} + +Le langage reconnu est constitué de mots comprenant en premier lieu au moins un a, suivi par un nombre impair de b. On peut aussi écrire : +\[L=\{m\in\Sigma^*\,|\,m=a^nb^m,\,n\in\mathbb{N}^*\,\text{et}\;m=2\cdot k+1,\,k\in\mathbb{N}\}\] +où \(\mathbb{N}^*\) est l'ensemble des nombres entiers positifs sans le zéro et \(\mathbb{N}\) celui avec le zéro. + +\medskip +En conclusion, on peut définir un automate de manière générale comme un ensemble constitué +\begin{itemize} + \item d'un alphabet \(\Sigma\), + \item d'un ensemble fini d'état, généralement noté \(E\), + \item d'un ensemble d'état initiaux \(I\) tel que \(I\subseteq E\), + \item d'un ensemble d'état finaux \(F\) tel que \(F\subseteq E\) et + \item d'une fonction de transition \[\delta:E\times\Sigma\longrightarrow E\] +\end{itemize} + +Où le signe \(\subseteq\) signifie ``appartient à''. + +\smallskip +Le dernier point de cette définition mérite des explications. La représentation graphique des automates donnée ci-dessus met clairement en évidence les transitions (flèches) entre états. Celle-ci doivent aussi trouver une expression mathématique. C'est la fonction de transition. + +\smallskip +Reprenons le dernier exemple d'automate donné ci-dessus. Considérons chacun des points permettant de le définir mathématiquement. + +L'alphabet tout d'abord est clairement donné par \(\Sigma=\{a,b\}\). + +L'ensemble des états est \(E=\{e_1,e_2,e_3\}\) et on peut en donner une représentation légèrement plus complète : + +\begin{center} +\begin{postscript} +\VCDraw{% +\begin{VCPicture}{(0,-1)(6,2)} +% states +\State[e_1]{(0,0)}{A} \State[e_2]{(3,0)}{B} \StateLineDouble \State[e_3]{(6,0)}{C} +% initial--final +\Initial{A} +% transitions +\EdgeL{A}{B}{a} \ArcL{B}{C}{b} \ArcL{C}{B}{b} +\LoopN{A}{a} +% +\end{VCPicture}} +\end{postscript} +\end{center} + +L'ensemble des états initiaux se résume à \(I=\{e_1\}\) et celui des états finaux à \(F=\{e_3\}\). + +La fonction de transition \(\delta\) est alors donnée par : +\begin{align*} + \delta(e_1,a)&=e_1\,|\,e_2 & \delta(e_1,b)&=\perp\\ + \delta(e_2,a)&=\perp & \delta(e_2,b)&=e_3\\ + \delta(e_3,a)&=\perp & \delta(e_3,b)&=e_2 +\end{align*} +où \(|\) signifie ``ou'' et \(\perp\) signifie ``indéfini''. + +\bigskip +Ce dernier exemple est aussi intéressant du point de vue du déterminisme du résultat. La fonction de transition \(\delta(e_1,a)\) donne deux solution possibles : \(e_1\) ou \(e_2\). Cela signifie qu'à partir de l'état \(e_1\), par la transition \(a\), un choix non déterminé peut avoir lieu. L'automate est alors dit \emph{non déterministe}. Cela peut constituer un problème et il est possible de le rendre déterministe. Mais nous n'aborderons pas ce point dans ce cours. Il convenait cependant de le mentionner. + +\medskip +Nous avons vu dans ce paragraphe une manière de \emph{reconnaître} mécaniquement un langage simplement formé de mots issus d'un alphabet à l'aide d'un automate. Nous avons vu deux manières de représenter celui-ci, par un diagramme et mathématiquement. Chacune de ces deux expressions constitue un mode de représentation qui a des avantages et des inconvénients. Les automates soulignent clairement le caractère mécanique du processus de reconnaissance. L'expression mathématique exprime quant à elle mieux les propriétés ensemblistes et relationnelles de la notion d'automate. + +\begin{ex}\label{autosimple} + Construisez un automate sur l'alphabet \(\Sigma=\{a,b\}\) qui reconnaît les mots suivants : aba, aab, abb et bbb. + \begin{sol} + Il s'agit d'un automate simple sans grande difficultés. Il est déterministe puisque de chaque état ne part qu'une seule transition de même type. + \begin{center} + \begin{postscript} + \VCDraw{% + \begin{VCPicture}{(0,-2)(6,4)} + % states + \State[e_1]{(0,0)}{A} \State[e_2]{(2,1)}{B} \State[e_3]{(4,1)}{C} + \State[e_5]{(4,3)}{E} + \State[e_8]{(2,-1)}{G} \State[e_9]{(4,-1)}{H} + \StateLineDouble \State[e_{10}]{(6,-1)}{I} \State[e_4]{(6,1)}{D} \State[e_6]{(6,3)}{F} + % initial--final + \Initial{A} + % transitions + \EdgeL{A}{B}{a} \EdgeL{B}{C}{a} \EdgeL{C}{D}{b} + \EdgeL{B}{E}{b} \EdgeL{E}{F}{b} + \EdgeL{A}{G}{b} \EdgeL{G}{H}{b} \EdgeL{H}{I}{b} + % + \end{VCPicture}} + \end{postscript} + \end{center} + \end{sol} +\end{ex} + +\begin{ex} + Construisez un automate qui reconnaisse la succession des feux de circulation rouge-jaune-vert, allumé successivement et indéfiniment dans cet ordre. + \begin{sol} + Le problème peut sembler complexe car l'énoncé du mot allumé fait immédiatement penser à éteint. Naturellement, on pense à un état allumé et à un autre éteint. Comme il y a trois couleurs, on s'imagine alors six états et leur représentation avec des transitions du type ``s'allume'' ou ``s'éteint'' ne va pas de soit. + + Pour s'en sortir simplement, il faut au contraire penser ``quel est la couleur du feu''. Cela suppose qu'on ne considère que l'état allumé du feu. On peut alors dire rouge, puis jaune, puis vert, puis rouge, \dots On pourrait aussi dire, c'est le rouge qui s'allume, puis c'est le jaune qui s'allume, puis \dots Cela nous permet alors de représenter la succession des états allumé du feu par : + \begin{center} + \begin{postscript} + \VCDraw{% + \begin{VCPicture}{(0,-3)(11,1)} + % states + \State[r]{(0,0)}{A} \State[j]{(4,0)}{B} \State[v]{(2,-2)}{C} + \State{(7,0)}{D} \State{(11,0)}{E} \State{(9,-2)}{F} + % initial--final + \Initial{A} \Initial{D} + % transitions + \EdgeL{A}{B}{} \EdgeR{B}{C}{} \EdgeR{C}{A}{} + \EdgeL{D}{E}{r} \EdgeR{E}{F}{j} \EdgeR{F}{D}{v} + % + \end{VCPicture}} + \end{postscript} + \end{center} + En particulier, l'automate ci-dessus permet la reconnaissance de la chaîne de caractère rjvrjvrjv\dots{} qui est une représentation de la succession des couleurs (allumées) du feu de circulation. + \end{sol} +\end{ex} + +\begin{ex} + Construisez un automate qui reconnaisse les codons de l'exercice \ref{codons}. + \begin{sol} + On a vu qu'il y a 64 possibilités pour former un acide aminé constitué d'un triplet choisi parmi quatre nucléotides. Il peut sembler alors que l'automate correspondant doive être très complexe. Mais au contraire, en raison du fait qu'on peut mettre n'importe quel nucléotide à chaque place parmi les trois disponibles nous permet de constituer un automate particulièrement simple : + \begin{center} + \begin{postscript} + \VCDraw{% + \begin{VCPicture}{(0,-1)(8,1)} + % states + \State{(0,0)}{A} \State{(4,0)}{B} \State{(8,0)}{C} + \StateLineDouble \State{(12,0)}{D} + % initial--final + \Initial{A} + % transitions + \EdgeL{A}{B}{C|G|T|U} \EdgeR{B}{C}{C|G|T|U} \EdgeR{C}{D}{C|G|T|U} + % + \end{VCPicture}} + \end{postscript} + \end{center} + où la notation C|G|T|U signifie soit C, soit G, soit T, soit U. + + \smallskip + On constate ainsi la concision de la description de l'automate par rapport à l'exposé de tout les cas possibles. + \end{sol} +\end{ex} + +\subsection{Les expressions rationnelles} +Voyons maintenant une autre manière de représenter les langages reconnus par les automates : les \emph{expressions rationnelles}\footnote{On parle aussi d'expressions régulières en raison de l'anglais \emph{regular expression}.}. Elles trouvent ici leur place pour souligner qu'il existe différentes approches dans la reconnaissance des langages (dits rationnels). + +\smallskip +Premièrement, une \emph{expression rationelle} est un mot. L'alphabet permettant de construire ce mot est particulier, puisque s'y trouve obligatoirement les symboles suivants : \((\,)\,+\,.\,*\,\emptyset\) Ainsi, on peut voir l'alphabet sur lequel on construit les expressions rationelles comme composé de deux ensemble : un alphabet \(\Sigma\) choisi selon les besoins et l'ensemble des symboles donné ci-dessus. L'alphabet des expressions rationelles est donc formellement : +\[\Sigma^{\text{expressions rationnelles}}=\Sigma \cup \{(,),+,.,^*,\varepsilon\}\] +où \(\varepsilon\) est le caractère vide. + +\smallskip +Secondement, une expression rationnelle doit obéir à une \emph{syntaxe}. Une manière de construire un langage, en tant que partie de l'ensemble des mots de \(\Sigma\), est de le faire à partir d'un automate. Une autre manière est de le faire à partir d'un ensemble d'opération qui permettent de le définir complètement. On appelle ces opérations des règles de \emph{syntaxe}. En d'autre termes, ces règles autorisent ou interdisent la production de mot particuliers sans référence au sens des mots produits. Ainsi, si e et r sont des expressions rationnelle, alors il faut ajouter que les règles de syntaxe des expressions rationnelles autorisent les trois expressions (e+r), (e.r) et (e*) comme expressions rationnelles. On dit ici simplement que les trois suites de caractères précédentes sont correctement formées pour qu'il s'agisse d'expressions rationnelles. Le sens des symboles +, . et * n'est pas précisé car il ne s'agit que de règles de syntaxe. + +\medskip +En définitive donc, on peut définir l'ensemble des expressions rationnelles par une règle de base (B) et par un règle récursive (R) : +\begin{description} + \item[B.] \(\varepsilon\) et e avec \(\text{e}\in \Sigma\) sont des mots sur \(\Sigma \cup \{(,),+,.,^*,\varepsilon\}\) + \item[R.] si e et r sont des expressions rationnelles, alors (e+r), (e.r) et (e*) le sont aussi. +\end{description} + +\medskip +Voilà l'ensemble des expressions rationnelles correctement défini. Il s'agit d'une définitions complexe sur laquelle nous ne reviendrons pas dans le cadre de ce cours. Il faut cependant retenir l'idée de syntaxe comme règles de production d'expressions. Elle nous mènera plus tard à celle de grammaire. + +\medskip +Jusqu'à présent nous n'avons pas parlé de sémantique. La définition rigoureuse du sens des symboles utilisés précédemment pour les expressions rationnelles (\(+,.,*\)) passe par la construction d'une relation de l'ensemble des expressions rationnelle sur une partie de l'ensemble des mots. Ces symboles prennent alors une signification particulière : le \(+\) devient un ou (\(\cup\) ou \(|\)), le \(.\) une concaténation de deux expressions rationnelles et l'étoile devient la répétition de zéro ou plusieurs fois le caractère précédent. + +\medskip +Ainsi, en imaginant la reconnaissance du code postal d'une ville suisse constitués de quatre chiffres, on peut écrire une expression rationnelle qui exprime cette structure de la manière suivante : +\[(0|1|2|3|4|5|6|7|8|9).(0|1|2|3|4|5|6|7|8|9).(0|1|2|3|4|5|6|7|8|9).(0|1|2|3|4|5|6|7|8|9)\] +Évidemment, cette notation n'est pas très pratique et en réalité on définit d'autres symboles qui permettent d'alléger l'écriture. Par exemple, de manière évidente : +\[[0..9]\{4\}\] +On trouvera avec \cite{DB08} un ouvrage très complet sur la syntaxe et la signification des symboles les plus utilisés. + +\begin{ex}\label{autotomate} + Exprimez l'automate suivant en français et transformez-le en expression rationnelle. + \begin{center} + \begin{postscript} + \VCDraw{% + \begin{VCPicture}{(0,-1)(4,1)} + % states + \State[e_1]{(4,0)}{B} + \StateLineDouble \State[e_0]{(0,0)}{A} + % initial--final + \Initial{A} + % transitions + \ArcL{A}{B}{b} \ArcL{B}{A}{a} \LoopE{B}{b} + % + \end{VCPicture}} + \end{postscript} + \end{center} + \begin{sol} + On doit tout d'abord faire un b, puis un nombre quelconque (voir nul) de b. Comme l'état de droite n'est pas final, il faut ensuite revenir à celui de gauche par un a. Enfin, on peut faire tout cela un nombre quelconque de fois, y compris aucune. L'expression rationnelle correspondante est : +\[(bb^*a)^*\] + \end{sol} +\end{ex} + +\begin{ex} + Construisez l'expression régulière correspondant aux numéros de plaques minéralogiques suisses : deux lettres, suivies de cinq chiffres décimaux. + \begin{sol} + De manière très naturelle, on a : + \[[A..Z]\{2\}[0..9]\{5\}\] + \end{sol} +\end{ex} + +\begin{ex} + Construisez l'expression régulière correspondant à l'automate de l'exercice \ref{autosimple}. + \begin{sol} + Il faut raisonner à partir de la dernière transition avant l'état final. Il s'agit toujours d'une transition b. Donc, l'expression régulière finira par un b. Puis, on a soit deux b, soit un a suivi d'un a ou b. Cela se traduit par : + \[(bb|a(a|b))b\] + \end{sol} +\end{ex} + +\section{Langage de Lindenmayer} +Avant de nous intéresser aux langages de Lindenmayer, dit L-system, il faut pousser un peu plus avant l'étude des langages formels. + +On a vu plusieurs manières de formaliser un langage simple. La formalisation des expressions rationnelles, par exemple, a fait apparaître des règles de syntaxe qui évoquent une \emph{grammaire}. Il est intéressant ici de préciser cette notion pour pouvoir formaliser au mieux les L-system. + +\subsection{Grammaire} +Une \emph{grammaire} est un ensemble \(G\) d'objets tel que : +\begin{align*} + G=&\{T,N,R,S\}\text{ où : }\\ + &T \text{ est l'alphabet des lettres terminales,}\\ + &N\text{ est l'alphabet des lettres non terminales}\\ + &S\in N\text{ est l'axiome}\\ + &R\subseteq (T \cup N)^*\times(T \cup N)^*\text{ est un ensemble fini de règles de production.} +\end{align*} +Une convention pratique veut que les symboles non-terminaux s'écrivent en majuscule et les terminaux en minuscule. + +\medskip +Voici un exemple simple, mais assez complet. +\begin{align*} + G=&\{T,N,R,S\}\text{ où : }\\ + &T=\{a,b,m,n\}\\ + &N=\{O,P\}\\ + &S=aO\\ + &R=\{O\rightarrow aO|bP\;\text{ et }\;P\rightarrow m|n\} +\end{align*} +où le signe \(|\) signifie ou. + +\smallskip +On peut donner deux exemples de génération d'éléments à partir de cette grammaire : +\begin{enumerate} + \item \(aO\rightarrow aaO\rightarrow aaaO\rightarrow aaabP\rightarrow aaabn\) + \item \(aO\rightarrow aaO\rightarrow aabP\rightarrow aabm\) +\end{enumerate} +On comprend aisément le rôle des règles de productions et celui des lettres terminales qui vont arrêter la génération de la chaîne de caractère. + +\medskip +Un automate peut constituer un autre exemple intéressant. En effet, il est possible de traduire un automate par une grammaire appropriée. +\begin{align*} + G=&\{T,N,R,S\}\text{ où : }\\ + &T=\{\text{états terminaux}\}\\ + &N=\{\text{états non terminaux}\}\\ + &S=\text{état initial}\\ + &R=\{\text{fonction de transition}\} +\end{align*} +Pour plus de clarté, considérons l'automate de l'exercice \ref{autotomate} et tentons de le traduire en une grammaire. +\begin{align*} + G=&\{T,N,R,S\}\text{ où : }\\ + &T=\{e_0\}\\ + &N=\{e_1\}\\ + &S=e_0\\ + &R=\{\delta(e_0,a)=\bot\;;\;\delta(e_0,b)=e_1\;;\;\delta(e_1,a)=e_0\;;\;\delta(e_1,b)=e_1\} +\end{align*} + +\medskip +Évidemment, l'existence d'une correspondance entre automate et expression régulière implique l'idée d'une formalisation des expressions régulières sous forme de grammaire. Nous n'aborderons pas ce cas ici. + +\medskip +Enfin, voici un autre exemple de grammaire où il n'y a pas de symbole terminaux. +\begin{align}\label{grammaire_algue_de_lindenmayer} + G=&\{T,N,R,S\}\text{ où : }\\\nonumber + &T=\emptyset\\\nonumber + &N=\{A,B\}\\\nonumber + &S=A\\\nonumber + &R=\{A\rightarrow AB\;\text{ et }\;B\rightarrow A\} +\end{align} +Celle-ci permet la génération de séquences infinies dont nous allons parler maintenant. + +\subsection{Génération de séquence} +La production des éléments de la grammaire donnée dans le dernier exemple ci-dessus débute par l'axiome \(A\). Ensuite, les deux règles de productions doivent être utilisées pour produire l'élément suivant : remplacer \(A\) par \(AB\) et \(B\) par \(A\). Comme il n'y a qu'un \(A\) dans l'axiome, seule la première règle s'applique tout d'abord et on obtient \(AB\). Puis, les deux règles doivent à nouveau être utilisées à partir de cet état pour produire le suivant. On remplace donc les \(A\) par \(AB\) et \(B\) par \(A\) pour chaque lettre de \(AB\). On obtient ainsi \(ABA\). Et on continue indéfiniment. On obtient alors : +\begin{align}\label{chaine_algue_de_lindenmayer} + A&\nonumber\\A&B\nonumber\\A&BA\nonumber\\A&BAAB\nonumber\\A&BAABABA\nonumber\\A&BAABABAABAAB +\end{align} +Ces chaînes n'ont pas encore de signification. Mais on peut déjà envisager différentes règles de productions permettant de gérérer de multiples chaines de caractères. Beaucoup de questions peuvent alors se poser concernant les propriétés des chaines ainsi générées. + +\medskip +Voici enfin un autre exemple très connu de grammaire permettant de générer des chaînes auxquelles on peut attacher une signification par interprétation graphique : +\begin{align}\label{grammaire_courbe_de_koch} + G=&\{T,N,R,S\}\text{ où : }\\\nonumber + &T=\{+,-\}\\\nonumber + &N=\{F\}\\\nonumber + &S=F\\\nonumber + &R=\{F\rightarrow F+F-F-F+F\} +\end{align} +Notez la séparation entre les symboles terminaux, sans règles de production et le symbole non terminal auquel correspond une règle de production. + +La génération des chaines est alors la suivante : +\begin{align}\label{chaine_courbe_de_koch} + F&\nonumber\\F&+F-F-F+F\nonumber\\F&+F-F-F+F+F+F-F-F+F-F+F-F-\;\hookleftarrow\nonumber\\\hookrightarrow\;&F+F-F+F-F-F+F+F+F-F-F+F +\end{align} + +\smallskip +Encore une fois, le sens de ces chaines n'est pas encore évident. Par ailleurs, on voit aussi que la génération des caractères de ces chaines devient très vite inaccessible au possibilités humaines en raison du nombre élevé de caractères produits. Une mécanisation de la production est donc rapidement nécessaire. + +\subsection{L-system} +Les deux derniers exemples ci-dessus ((\ref{grammaire_courbe_de_koch}) et (\ref{grammaire_algue_de_lindenmayer})) sont des systèmes de Lindenmayer ou L-system. De quoi s'agit-il ? L'ouvrage de référence en la matière, \emph{The Algorithmic Beauty of Plants} \cite[p.2-3]{LA04}, en présente le contexte en ces termes : + +\begin{quotation} + \og \selectlanguage{english}In 1968 a biologist, Aristid Lindenmayer, introduced a new type of string-rewriting mechanism, subsequently termed L-systems. The essential difference between Chomsky grammars and L-systems lies in the method of applying productions. In Chomsky grammars productions are applied sequentially, whereas in L-systems they are applied in parallel and simultaneously replace all letters in a given word. This difference reflects the biological motivation of L-systems. Productions are intended to capture cell divisions in multicellular organisms, where many divisions may occur at the same time. Parallel production application has an essential impact on the formal properties of rewriting systems.\selectlanguage{french}\fg +\end{quotation} + +L'objectif est donc de modéliser des mécanisme de division (cellulaire). Le moyen utilisé est un mécanisme de réécriture. Quoi de plus naturel donc que d'utiliser une grammaire précisément basée sur la réécriture. Les règles de production s'y prètent bien et les deux exemples présentés ci-dessus en sont une belle démonstration. Il ne s'agit pas ici d'une division cellulaire, mais de la réplication d'un motif sur lui-même. Une grammaire en est ici l'origine et nous verrons par la suite qu'elle peut se traduire par une récursivité des algorithmes nécessaires pour la production de modifs conséquents. + +Mais, pour bien comprendre la signification des mécanismes de production des L-system dans la réalité, il faut maintenant en donner une interpréation graphique. + +\subsection{Interprétation graphique} +Il s'agit ici d'utiliser les chaines de caractères produites par les grammaires pour en faire une représentation graphique. L'articulation d'un mécanisme de production général et d'une analyse des briques composant les structures végétales est certainement à l'origine de cette idée. Elle a trouvé son expression complète grace aux possibilités de dessin proposés par le langage de programmation Logo et la tortue qui lui est attachée. + +L'interprétation consiste à utiliser les caractères de l'ensemble terminal \(N\) comme des instructions d'orientation de la tortue et les symboles non-terminaux comme des instruction de construction du chemin. Classiquement, on a : + +\begin{description} + \item[\(F\)] avancer d'un pas en avant, + \item[\(+\)] tourner à gauche d'un angle donné, + \item[\(-\)] tourner à droite d'un angle donné. +\end{description} + +Cette interprétation permet de suivre les chaines de caractères produites par la grammaire pour en tracer une représentation graphique. + +\subsubsection{Koch} +L'exemple donné par la grammaire (\ref{grammaire_courbe_de_koch}), correspondant aux chaines (\ref{chaine_courbe_de_koch}) peut ainsi prendre l'apparence des figures \ref{figkoch} pour un angle de 90\textdegree. + +\begin{figure*}[t] +\centering +\subfigure[F+F-F-F+F\label{figkoch1}]{\includegraphics{koch1.eps}}\qquad +\subfigure[F+F-F-F+F+F+F-F-F+F-F+F-F-F+F-F+F-F-F+F+F+F-F-F+F\label{figkoch2}]{\includegraphics{koch2.eps}}\\ +\subfigure[Plus encore ...\label{figkoch3}]{\includegraphics[width=8cm]{koch3.eps}} +\caption{La courbe de Koch\label{figkoch}} +\end{figure*} + +La figure \ref{figkoch1} permet de suivre l'interprétation graphique à la main, caractère après caractère. Mais, on hésite déjà à le faire à la figure suivante pour des raisons évidentes. + +A part l'esthétique des figures \ref{figkoch}, cette interprétation graphique semble ne pas apporter grand chose. Pourtant, les objets étudiés dans \emph{The Algorithmic Beauty +of Plants +} \cite[p.8-9]{LA04} et reproduits\footnote{Le code permettant d'obtenir ces structures sera présenté plus loin.} dans la figure \ref{figkochisland} sont déjà plus intéressants et montrent des structures qui commencent à ressembler vaguement à des cristaux par exemple. + +\begin{figure*}[ht] +\centering +\subfigure[F-F+F+FF-F-F+F (axiome F-F-F-F)\label{figkochisland1}]{\includegraphics{kochIsland1.eps}}\qquad +\subfigure[F-F+F+FF-F-F+F (axiome F-F-F-F)\label{figkochisland2}]{\includegraphics[width=5cm]{kochIsland2.eps}}\\ +\subfigure[F+FF-FF-F-F+F+FF-F-F+F+FF+FF-F (axiome F-F-F-F)\label{figkochislandquadratic2}]{\includegraphics[width=5cm]{kochIslandquadratic2.eps}} +\caption{Une autre courbe de Koch\label{figkochisland}} +\end{figure*} + +\subsubsection{Modélisation des plantes} +Un autre exemple d'utilisation d'une grammaire interprétée de manière particulière est donné par la modélisation de la croissance des plantes. Turtle est ici utilisé comme précédemment. Mais l'alphabet est étendu par deux nouveaux signes : les crochets [ et ]. Leur signification dans le cadre de Turtle est spéciale : +\begin{description} + \item[le crochet ouvrant] [ signifie qu'il faut mémoriser la position et la direction courante de la tortue et + \item[le crochet fermant] ] signifie qu'il faut faire revenir la tortue à la position et direction précédemment mémorisée. +\end{description} +Cette opération de mémorisation est nécessaire pour permettre une structure d'arbre qui n'est pas possible sans cela puisque le parcours de la tortue est alors continu. + +Voyons-en un exemple d'une telle grammaire. +\begin{align}\label{grammaire_plante} + G=&\{T,N,R,S\}\text{ où : }\\\nonumber + &T=\{+,-,[,]\}\\\nonumber + &N=\{F\}\\\nonumber + &S=F\\\nonumber + &R=\{F\rightarrow F[+F]F[-F]F\} +\end{align} + +Après deux fois l'application de la règle (deux générations), on obtient la chaîne suivante : +\begin{center} + F[+F]F[-F]F[+F[+F]F[-F]F]F[+F]F[-F]F[-F[+F]F[-F]F]F[+F]F[-F]F +\end{center} +Encore une fois, l'interprétation n'est pas triviale, pas plus que la programmation de la mémorisation. + +\medskip +Pourtant, la figure \ref{figplante} obtenue par l'interprétation graphique pour cinq génération et un angle de 25,7\textdegree{} est étonnante. + +\begin{figure*}[t] +\centering +\includegraphics[width=12cm]{plante1.eps} +\caption{Modélisation d'une plante\label{figplante}} +\end{figure*} + +Évidemment, on s'imagine bien que le remplacement récursif d'un même motif à partir de points clés de ce même motif mène à la construction d'arbre de ce type. Mais la figure produite est tout de même remarquable dans la structure ``botanique'' qu'elle présente, même si une tige aussi rectiligne est peu probable dans la réalité. + +\medskip +Plusieurs autres règles peuvent être utilisées. Plusieurs angles aussi. Comme présenté dans \emph{The Algorithmic Beauty of Plants} \cite[p.25]{LA04}, on obtient alors des structures dont la ressemblance avec des plantes réelles est étonnante. La figure \ref{figplantes} présente les ``plantes'' obtenues avec les règles suivantes : +\begin{align} + F &\rightarrow F[+F]F[-F]F\label{regle1}\\ + F &\rightarrow FF-[-F+F+F]+[+F-F-F]\label{regle2} +\end{align} + +\begin{figure*}[ht] +\centering +\subfigure[nbiteration = 5, angle = 20\textdegree{} (axiome F ; règle \ref{regle1})\label{figplante2}]{\includegraphics[width=10cm]{plante2.eps}}\\ +\subfigure[nbiteration = 5, angle = 22,5\textdegree{} (axiome F ; règle \ref{regle2})\label{figplante3}]{\includegraphics[width=10cm]{plante3.eps}} +\caption{Deux autres plantes\label{figplantes}} +\end{figure*} + +On peut aussi illustrer l'utilisation de plusieurs règles simultanément à travers les exemples \ref{regle3}, \ref{regle4} et \ref{regle5} où l'axiome est \(X\). Les deux règles sont appliquées successivement. +\begin{align} + X &\rightarrow F[+X]F[-X]+X\;\text{ et }\;F \rightarrow FF\label{regle3}\\ + X &\rightarrow F[+X][-X]FX\;\text{ et }\;F \rightarrow FF\label{regle4}\\ + X &\rightarrow F-[[X]+X]+F[+FX]-X\;\text{ et }\;F \rightarrow FF\label{regle5} +\end{align} + +Le résultat de l'interprétation graphique est donné à la figure \ref{figplantes2}, page \pageref{figplante2}. + +\begin{figure*}[ht] +\centering +\subfigure[nbiteration = 7, angle = 20\textdegree{} (axiome \ref{regle3})\label{figplante4}]{\includegraphics[width=6cm]{plante4.eps}}\qquad +\subfigure[nbiteration = 7, angle = 25,7\textdegree{} (axiome \ref{regle4})\label{figplante5}]{\includegraphics[width=6cm]{plante5.eps}}\\ +\subfigure[nbiteration = 5, angle = 22,5\textdegree{} (axiome \ref{regle5})\label{figplante6}]{\includegraphics[width=8cm]{plante6.eps}} +\caption{Deux autres plantes\label{figplantes2}} +\end{figure*} + +\medskip +Bien évidemment, ces exemples ne sont pas construits au hasard. Une étude approfondie de la forme des chaines nécessaires pour obtenir des structures réalistes de plantes doit être faite, ou tout au moins une réflection sur les motifs de base. On se reportera pour cela à l'adresse \url{http://algorithmicbotany.org/papers/} où sont présentés de nombreux exemples très crédibles. En ce qui nous concerne, l'objectif de ce petit cours n'étant pas d'étudier systèmatiquement ou de manière approfondie les plantes elles-même, on laisse au lecteur le soin de tenter de trouver des chaines crédibles et de les interpréter grâce à la programmation. + +Car, il est temps maintenant de considérer cet aspect moins formel, mais aussi important dans le cadre d'un cours d'informatique. Nous allons donc commencer par considérer le problème de la création de L-system de manière algorithmique. Cela nous permettra d'avoir une bonne vision des mécanismes principaux permettant la génération de ces systèmes. + +\subsection{Algorithmique\label{algo}} + +\subsubsection{Création} + +Commençons par la construction de la chaine de Lindenmayer. Nous avons vu que tout commence avec un axiome. Celui-ci peut être constitué de deux types de symboles : non-terminaux et terminaux. La production de la chaine consiste quant à elle à remplacer un certain nombre de fois les symboles non-terminaux par une chaine correspondante donné par une (ou plusieurs) règle. Pour simplifier la situation, on ne considérera ici qu'une seule règle. Il est donc nécessaire de disposer du nombre d'itération choisi, de l'axiome et de la règle qu'il faut initialiser. Ensuite viennent deux boucles. L'une pour répéter les remplacements des symboles non-terminaux par ceux de la règle et l'autre pour lire l'axiome. Finalement, au bout d'un nombre de génération donné, le résultat produit est un axiome ``final'' qui sera transmis pour interprétation graphique. L'algorithme \ref{algocreation}, page \pageref{algocreation}, traduit ce raisonnement en pseudo-code. + +%\begin{comment} +%\begin{algorithm}[t] +%\caption{RRT}\label{algo_rrt} +%\begin{pseudocode}[ovalbox]{mystère}{a,b,c} +%a \PREND a+b\\ +%c \PREND a+\sqrt{b}\\ +%b \PREND a-b +%\end{pseudocode} +%\end{algorithm} +%\end{comment} + +\begin{algorithm}[t] + \SetAlgoVlined + \SetAlgoCaptionSeparator{ \textendash} + %\SetAlCapSkip{5mm} + \Donnees{nbiteration, axiome, regle} + \Res{nouvel axiome} + initialisation\; + \Tq{on est < nbiteration}{ + \Tq{on est < longueur de l'axiome}{ + on lit un caractère de l'axiome\; + \Si{on trouve un caractère non terminal}{ + on le remplace par la règle\; + } + \SinonSi{on trouve un caractère terminal}{ + on le réécrit simplement\; + } + } + } + \caption{Création de la chaine}\label{algocreation} +\end{algorithm} + +\subsubsection{Interprétation} + +Après la création de la chaine vient son interprétation. Pour cela, il est nécessaire de disposer de l'axiome ``final'', de la longueur choisie des segments produit par la tortue et de l'angle correspondant à chaque rotation. L'interprétation graphique se fait ensuite par la lecture de chaque caractères de la chaine et le lancement de l'opération correspondante par la tortue. + +Si l'opération de mémorisation de la position de la tortue est simple, il faut analyser plus finement le retour à cette position. En effet, il est nécessaire de mémoriser plusieurs états différents avant d'y revenir successivement. Un tableau des états est donc nécessaire. De plus, une fois revenu sur un état mémorisé, il ne faut pas oublier de l'effacer. L'algorithme \ref{algointerpretation}, page \pageref{algointerpretation}, résume tout cela en pseudo-code. + +\begin{algorithm}[bh] + \SetAlgoVlined + \SetAlgoCaptionSeparator{ \textendash} + %\SetAlCapSkip{5mm} + \Donnees{axiome, longueur, angle\\variable : tableau des états mémorisés} + \Res{le tracé} + initialisation\; + \Tq{on est < longueur de l'axiome}{ + on lit un caractère de l'axiome\; + \Si{on trouve tel caractère}{ + on demande à la tortue de faire telle opération\; + } + \SinonSi{on trouve tel autre caractère}{ + on demande à la tortue de faire telle autre opération\; + } + \SinonSi{on trouve le caractère [}{ + on mémorise sa position et sa direction + } + \SinonSi{on trouve le caractère ]}{ + on récupère l'état, on y envoie la tortue et on efface cet état de la pile des état mémorisés + } + } + \caption{Interprétation graphique de la chaine}\label{algointerpretation} +\end{algorithm} + +Dans le cas où plusieurs règles sont présentes, seule la procédure de création change avec deux boucles de lecture successives permettant d'effectuer le remplacement de deux caractères non-terminaux différents. Ces deux boucles ayant la même forme, on ne présentera pas ce cas. Mais le code donné en annexe \ref{codepythonfonct2} en donne une illustration compréhensible. + +\subsection{Programmation} + +\subsubsection{Approche non récursive} +Comme toujours dans le domaine de la programmation, il y a pour chaque problèmes beaucoup de solutions possibles. Même si l'objectif de ce cours n'est pas la programmation, deux solutions différentes vont être présentées, non récursive et récursive, répondant toutes les deux à la description algorithmique présentée au point \ref{algo}. En effet, dans le cadre d'un cours sur les langages, il est une fois de plus intéressant de montrer la diversité des approches, sans pour autant se focaliser particulièrement sur les avantages des différents modes de programmation. Par contre, on abordera pas ces problèmes en programmation orientée objet ou à travers d'autres languages tels que java par exemple. Mais il est évidemment envisageable de le faire par la suite. + +Dans la section \ref{algo}, on a vu la décomposition du problème de génération des L-system en deux procédures d'une part de production des motifs à partir de la grammaire et d'autre part d'interprétation de ceux-ci par turtle. Le code complet est donné en annexe \ref{codepythonfonct}. On ne va détailler ici que les aspects non liés au langage de programmation utilisé. Cependant, il faut signaler qu'on a utilisé un découpage du problème en \emph{fonction} pour des raisons de clarté uniquement et qu'on a aussi mélangé variables \emph{globale} et \emph{locales}, \emph{procédures} et \emph{fonctions} pour mettre en jeu ces notions dans un programme fonctionnel. Mais, on ne les définira pas ici. + +\paragraph{La création de la chaine} consiste essentiellement en une boucle qui parcourt successivement tous les carctères. +\begin{lstlisting}[language=Python] + while j < len(axiom): + pointeur1 = axiom[j] +\end{lstlisting} +La lecture se fait en plaçant le caractère j de l'axiome dans une variable \verb|pointeur1| qui permettra de décider du caractère terminal ou non d'un caractère. En effet, par remplacement de la règle dans les symboles non terminaux de l'axiome, la boucle va créer un nouvel axiome, nommé \verb|axiome2|, qui sera utilisé comme axiome de départ à chaque itération. Ainsi, on a : +\begin{lstlisting}[language=Python] + # Symboles non terminaux + if pointeur1 == 'F': + # remplacement de la regle dans l'axiome + axiom2 = axiom2 + regle +\end{lstlisting} +On voit qu'on remplace le caractère \verb|F| par la chaine correspondante de la règle. Par contre, la lecture d'un sybole terminal ne donne lieu à aucun remplacement. On réécrit simmplement ce symbole tel quel. + +Ainsi, au fur et à mesure de la lecture des caractères,on construit par concaténation une nouvelle chaìne \verb|axiom2| qui sera à la base de la prochaine itération. + +\paragraph{L'interprétation de la chaine} se fait à nouveau par lecture de celle-ci. Après création, elle est passée à la procédure d'interprétation via le paramètre \verb|axiom|, la chaine de caractère qui sera lue. Ensuite, pour les symboles de déplacement et d'orientation, l'interprétation est simple : +\begin{lstlisting}[language=Python] + # Interpreation + if pointeur2 == 'F': + forward(longueur) + elif pointeur2 == '+': + left(angle) + elif pointeur2 == '-': + right(angle) +\end{lstlisting} +Par contre, pour les symboles de mémorisation, la procédure est quelque peu plus complexe : +\begin{lstlisting}[language=Python] + elif pointeur2 == '[': + # memorisation de la position courante + laposition.append(position()) + langle.append(heading()) + elif pointeur2 == ']': + # retour a la derniere position et suppression de celle-ci + up() + goto(laposition[len(laposition)-1][0],\ + laposition[len(laposition)-1][1]) + del laposition[len(laposition)-1] + setheading(langle[len(langle)-1]) + del langle[len(langle)-1] + down() +\end{lstlisting} +Tout d'abord, on utilise ici un tableau, nommé \verb|laposition| pour mémoriser chaque couple abscisse et ordonnée de la tortue et un autre tableau, nommé \verb|langle| pour mémoriser sa direction. + +\smallskip +Puis, pour le retour à la position mémorisée, on réalise un \verb|goto| aux coordonnées mémorisées et un \verb|setheading| pour revenir à la direction enregistrée. Enfin, il ne faut pas oublier de supprimer des deux tableaux la position à laquelle on est revenu. Cela se fait par deux \verb|del|. Notez que pendant le retour il ne faut pas que la tortue soit en position d'écriture. On lève donc le crayon avant le retour et on le baisse après. + +\medskip +En fin de compte, grâce à l'utilisation de fonctions (et procédures), on peut écrire le programme principal avec une grande clarté : +\begin{lstlisting}[language=Python] + initialisation() + gram = grammaire() + ## gram[0] retourne l'angle + ## gram[1] retourne l'axiome + ## gram[2] retourne la regle + + axio = production(gram[1], gram[2]) + interpretation(gram[0], axio) +\end{lstlisting} +L'initialisation se fait par le placement de la tortue et la mémorisation des paramètre, axiome et règle. Puis, vient naturellement la production et l'interprétation. + +\subsubsection{Approche récursive} +Une autre approche de la génération de L-system fait usage d'une technique de programmation particulièrement intéressante : la récursivité. Il s'agit de l'appel d'une fonction à elle-même. Pour les L-system, il s'agit donc de ne plus générer la chaine de caractère, mais de lire l'axiome en renvoyant simultanément à la fonction appelée, à chaque occurence d'un caractère non terminal. Il n'est alors pas évident de suivre ce que fait le programme. C'est pourquoi on va maintenant reprendre une forme légérement différente de celle donnée précédemment du flocon de Koch pour l'étudier dans le détail. Le code complet est donne en annexe \ref{codepythonfloconrecursif}. Seule la partie purement récursive est présentée ci-dessous : +\begin{lstlisting}[language=Python] + def production(nbiteration, longueur, angle, axiome, regle): + if nbiteration == 0 : + forward(longueur) + else: + i = 0 + while i < len(regle): + regle1 = regle[i] + if regle1 == 'F': + production(nbiteration-1, longueur/3,\ + angle, axiome, regle) + if regle1 == '+': + left(angle) + if regle1 == '-': + right(angle) + i = i + 1 +\end{lstlisting} +Et la figure \ref{Kochrecursif1} en donne l'interprétation graphique. + +\begin{figure*}[t] +\centering +\includegraphics[width=5cm]{kochrecursif1.eps} +\caption{Flocon de Koch récursif.\label{Kochrecursif1}} +\end{figure*} + +Le code est remarquable de concision, mais complexe à apréhender. + +\smallskip +Le c\oe ur de la récursivité se trouve dans le test de reconnaissance du caractère \verb|F|. Il s'agit du caractère non terminal et il est nécessaire de le remplacer par l'ensemble des caractères de la règle. Mais, au lieu de réaliser simplement ce remplacement, on renvoie à la fonction de production. Cela n'a pas pour effet de remplacer le caractère non terminal par les symboles de la règle, mais de renvoyer sur une fonction, en l'occurence celle dans laquelle on se trouve, qui va le faire. Or, cette fonction va encore s'appeler elle-même à la lecture du caractère \verb|F| et ainsi de suite. + +Le même processus va ainsi se répéter un nombre de fois déterminé par la variable \verb|nbiteration|. Comme à chaque appel de la fonction de production on diminue la variable contenant le nombre d'itération, quand celle-ci devient nulle, on stoppe l'apppel récursif et on trace un segment. + +Ensuite, on retourne à la dernière procédure appelée pour réaliser l'instruction qui suit l'appel récursif selon la règle donnée, par exemple une instruction terminale de rotation. Et on continue pour réaliser une suite d'instructions terminales ou si non terminales, pour réaliser un appel récursif à la fonction principale. + +Il est très difficile de réprésenter chaque étape de ce processus. Cependant, la figure \ref{appelrecursif} tente d'en donner une représentation graphique qui montre l'appel et le retour des processus dans l'ordre des flèches numérotées. + +\begin{figure*}[t] +\centering +\begin{psfrags} + \psfrag{production(...)}{production(...)} + \psfrag{if regle1}{if regle1 == 'F':} + \psfrag{production}{production} + \psfrag{de}{(\scriptsize nbiteration-1,longueur-1)} + \psfrag{if regle2}{if regle1 == '+'} + \psfrag{left}{left(angle)} + \psfrag{if nbiteration}{if nbiteration == 0:} + \psfrag{forward}{forward(longueur)} + \psfrag{1}{1} + \psfrag{2}{2} + \psfrag{3}{3} + \psfrag{4}{4} + \psfrag{5}{5} + \psfrag{6}{6} +\includegraphics[width=12cm]{production.eps} +\end{psfrags} +\caption{Flocon de Koch récursif.\label{appelrecursif}} +\end{figure*} + +On voit que toute un série de procédures sont progressivement lancées les unes dans les autres. Puis, arrivées à leur terme, le tracé des segment se fait et elles se terminent par un retour à la procédure qui les a lancées. Finalement, tout aussi progressivement, on termine chacunes des procédures imbriquées. + +\medskip +On a déjà remarqué que le code est plus concis que celui issu d'une simple lecture-interprétation. Par contre, il est moins clair, demanderait donc pour la modification une relecture attentive qui peut prendre du temps et il faudrait se demander s'il est plus performant. + +\medskip +Voilà. Comme le but de ce cours n'est pas la programmation récursive, nous finirons là cette analyse en précisant cependant les limites de celle-ci. En effet, avec la récursivité on utilise toujours la même règle. L'axiome n'est donc pas utilisé ou considéré égal à \verb|F|. Certes, on peut envisager des moyens de contourner ce problème en réalisant l'axiome hors de la procédure récursive, en utilisant une récursivité indirecte ou en remplaçant à un moment bien précis la règle par l'axiome. Mais cela complique le code et l'avantage en concision de la récursivité s'amenuise. + +\section{Conclusion} +Nous avons parcouru avec cette introduction aux L-system beaucoup de notions d'informatiques importantes. Les langages, leur reconnaissance, les grammaires, l'algorithmique et finalement différents modes de programmation nous ont permis de comprendre le rôle essentiel joué par les structures formelles que l'informatique théorique met à notre disposition. Les limites de ce cours ne nous ont cependant pas permis d'aborder d'autres notions tout aussi importantes comme les résaux de Pétri ou les machines de Turing. Par contre, l'étude des L-system marque clairement la puissance de la modélisation informatique des structures biologiques. Plus, elle fait entrevoir une correspondance fascinante entre le développenent des organisme vivants et des constructions répétitives très structurées, finalement très bien décrites par le langage mathématique. + +\Closesolutionfile{ans} % ferme le fichier Solutions.tex qui contient les solutions +\vfill +\pagebreak % met les solutions sur une autre page +\section{Solutions des exercices} +\Readsolutionfile{ans} % importe les données du fichier Solutions.tex + +\myclearpage + +%\appendix + +\section{Annexes} + +\subsection{Code python plante}\label{codepythonfonct} + +\begin{lstlisting}[language=Python] + # -*- coding: utf-8 -*- +from turtle import * + +# Generation de plante par grammaire L-system + +def initialisation(): + # Definition des variables independantes de la grammaire + global nbiteration, longueur + nbiteration = 5 + longueur = 3 + +def grammaire(): + # Definition de la grammaire + angle = 25.7 + axiom = 'F' + + regle = 'FF-[-F+F+F]+[+F-F-F]' + #regle = 'F[+F]F[-F][F]' + #regle = 'F[+F]F[-F]F' + #regle = 'F+FF-FF-F-F+F+FF-F-F+F+FF+FF-F' + + return angle, axiom, regle + +def production(axiom, regle): + # Generation de la chaine de caracteres + i = 0 + while i < nbiteration: + j = 0 + axiom2 = '' + while j < len(axiom): + pointeur1 = axiom[j] + + # Symboles non terminaux + if pointeur1 == 'F': + # remplacement de la regle dans l'axiome + axiom2 = axiom2 + regle + + # Symboles terminaux + if pointeur1 == '+': + axiom2 = axiom2 + '+' + if pointeur1 == '-': + axiom2 = axiom2 + '-' + if pointeur1 == '[': + axiom2 = axiom2 + '[' + if pointeur1 == ']': + axiom2 = axiom2 + ']' + + j = j + 1 + axiom = axiom2 + i = i + 1 + print axiom + return axiom + +def interpretation(angle, axiom): + # Interpretation de la chaine par turtle + k = 0 + laposition = [] + langle = [] + while k < len(axiom): + pointeur2 = axiom[k] + + # Interpreation + if pointeur2 == 'F': + forward(longueur) + elif pointeur2 == '+': + left(angle) + elif pointeur2 == '-': + right(angle) + elif pointeur2 == '[': + # memorisation de la position courante + laposition.append(position()) + langle.append(heading()) + elif pointeur2 == ']': + # retour a la derniere position et suppression de celle-ci + up() + goto(laposition[len(laposition)-1][0], laposition[len(laposition)-1][1]) + del laposition[len(laposition)-1] + setheading(langle[len(langle)-1]) + del langle[len(langle)-1] + down() + + k = k + 1 + +# Placement initial de la tortue +up() +backward(495) +down() + +# Programme principal +## gram[0] retourne l'angle +## gram[1] retourne l'axiome +## gram[2] retourne la regle + +initialisation() +gram = grammaire() + +axio = production(gram[1], gram[2]) +interpretation(gram[0], axio) + +#quit() +\end{lstlisting} + +\subsection{Code python plante à deux règles}\label{codepythonfonct2} + +\begin{lstlisting}[language=Python] +# -*- coding: utf-8 -*- +from turtle import * + +# Generation de plante par grammaire L-system + +def initialisation(): + # Definition des variables independantes de la grammaire + global nbiteration, longueur + # Pour le flocon de Koch + #nbiteration = 3 + #longueur = 2 + + # Pour le flocon de Koch Island + #nbiteration = 3 + #longueur = 2 + + # Pour la plante + #nbiteration = 5 + #longueur = 5 + + # Pour la plante 2 + #nbiteration = 5 + #longueur = 5 + + # Pour la plante 3 + #nbiteration = 4 + #longueur = 8 + + # Pour la plante 4 + #nbiteration = 7 + #longueur = 1 + + # Pour la plante 5 + #nbiteration = 7 + #longueur = 1 + + # Pour la plante 6 + nbiteration = 5 + longueur = 4 + +def grammaire(): + # Definition de la grammaire + # Pour le flocon de Koch + #angle = 90 + #axiom = 'F-F-F-F' + #regle = 'F+F-F-F+F' + + # Pour le flocon de Koch Island + #angle = 90 + #axiom = 'F-F-F-F' + #regle = 'F-F+F+FF-F-F+F' + + # Pour la plante + #angle = 25.7 + #axiom = 'F' + #regle = 'F[+F]F[-F]F' + + # Pour la plante 2 + #angle = 20 + #axiom = 'F' + #regle = 'F[+F]F[-F][F]' + + # Pour la plante 3 + #angle = 22.5 + #axiom = 'F' + #regle = 'FF-[-F+F+F]+[+F-F-F]' + + # Pour la plante 4 + #angle = 20 + #axiom = 'X' + #regle1 = 'F[+X]F[-X]+X' + #regle2 = 'FF' + + # Pour la plante 5 + angle = 25.7 + axiom = 'X' + regle1 = 'F[+X][-X]FX' + regle2 = 'FF' + + # Pour la plante 6 + angle = 22.5 + axiom = 'X' + regle1 = 'F-[[X]+X]+F[+FX]-X' + regle2 = 'FF' + + return angle, axiom, regle1, regle2 + +def production(axiom, regle1, regle2): + # Generation de la chaine de caracteres + i = 0 + while i < nbiteration: + j = 0 + axiom2 = '' + while j < len(axiom): + pointeur1 = axiom[j] + + # Symboles non terminaux + if pointeur1 == 'X': + # remplacement de la regle dans l'axiome + axiom2 = axiom2 + regle1 + + # Symboles terminaux + if pointeur1 == '+': + axiom2 = axiom2 + '+' + if pointeur1 == '-': + axiom2 = axiom2 + '-' + if pointeur1 == '[': + axiom2 = axiom2 + '[' + if pointeur1 == ']': + axiom2 = axiom2 + ']' + if pointeur1 == 'F': + axiom2 = axiom2 + 'F' + + j = j + 1 + axiom = axiom2 + print axiom + + m = 0 + axiom3 = '' + while m < len(axiom): + pointeur2 = axiom[m] + + # Symboles non terminaux + if pointeur2 == 'F': + # remplacement de la regle dans l'axiome + axiom3 = axiom3 + regle2 + + # Symboles terminaux + if pointeur2 == '+': + axiom3 = axiom3 + '+' + if pointeur2 == '-': + axiom3 = axiom3 + '-' + if pointeur2 == '[': + axiom3 = axiom3 + '[' + if pointeur2 == ']': + axiom3 = axiom3 + ']' + if pointeur2 == 'X': + axiom3 = axiom3 + 'X' + + m = m + 1 + axiom = axiom3 + i = i + 1 + + print axiom + return axiom + +def interpretation(angle, axiom): + # Interpretation de la chaine par turtle + k = 0 + laposition = [] + langle = [] + while k < len(axiom): + pointeur2 = axiom[k] + + # Interpreation + if pointeur2 == 'F': + forward(longueur) + elif pointeur2 == '+': + left(angle) + elif pointeur2 == '-': + right(angle) + elif pointeur2 == '[': + # memorisation de la position courante + laposition.append(position()) + langle.append(heading()) + elif pointeur2 == ']': + # retour a la derniere position et suppression de celle-ci + up() + goto(laposition[len(laposition)-1][0], laposition[len(laposition)-1][1]) + del laposition[len(laposition)-1] + setheading(langle[len(langle)-1]) + del langle[len(langle)-1] + down() + + k = k + 1 + +# Placement initial de la tortue +up() +backward(400) +down() + +# Programme principal +## gram[0] retourne l'angle +## gram[1] retourne l'axiome +## gram[2] retourne la regle + +initialisation() +gram = grammaire() +axio = production(gram[1], gram[2], gram[3]) +interpretation(gram[0], axio) + +#quit() + +\end{lstlisting} + +\subsection{Code python flocon de Koch récursif}\label{codepythonfloconrecursif} + +\begin{lstlisting}[language=Python] + # -*- coding: utf-8 -*- + from turtle import * + + # Generation de plante par grammaire L-system + + def initialisation(): + # Definition des variables independantes de la grammaire + # Pour le flocon de Koch + nbiteration = 3 + longueur = 200 + angle = 45 + axiome = 'F' + regle = 'F+F--F+F' + return nbiteration, longueur, angle, axiome, regle + + def production(nbiteration, longueur, angle, axiome, regle): + if nbiteration == 0 : + forward(longueur) + else: + i = 0 + while i < len(regle): + regle1 = regle[i] + if regle1 == 'F': + production(nbiteration-1, longueur/3,\ + angle, axiome, regle) + if regle1 == '+': + left(angle) + if regle1 == '-': + right(angle) + i = i + 1 + + # Placement initial de la tortue + up() + backward(200) + down() + + # Programme principal + ## init[1] : nbiteration + ## init[2] : longueur + ## init[3] : angle + ## init[4] : axiome + ## init[5] : regle + + init = initialisation() + production(init[0], init[1], init[2], init[3], init[4]) + right(120) + production(init[0], init[1], init[2], init[3], init[4]) + right(120) + production(init[0], init[1], init[2], init[3], init[4]) +\end{lstlisting} + +\myclearpage + +%\theendnotes + +\myclearpage + +%\renewcommand{\bibname}{Bibliographie} % pour mettre autre chose que bibliographie, par défaut avec babel +\nocite{CJ06,PP94,PP94,TO07,GP09,BD11} % pour mettre toutes les entrées dans la biblio, y compris celles non citées ds le texte +\bibliographystyle{apalike-fr} % le style de bibliographie apalike francisé +%\bibliographystyle{apalike} % le style de bibliographie +\bibliography{../../Bibliographies/BiblioCours} % définit le fichier des réf. biblio. nom.bib + + % le logiciel utilisé pour faire la biblio est kbibtex +%\printindex{} % attention, pour que l'index soit mis à jour il faut lancer la commande de konsole : makeindex -s ../Perso.ist nom_du_cours.idx + % le Perso.ist est un fichier disant qu'il faut mettre des lettres entre les parties de l'index + +%\renewcommand{\listfigurename}{Liste des figures} +%\listoffigures % ai tout basculé après la table des matières car je n'arrivais pas à éliminer le petit bug ci-dessous. Mais le compagnon latex fait idem. +%\listoftables % un petit problème (bug ?) d'alignement m'a fait retirer provisoirement cette liste + +\end{document} +\endinput % termine l'importation des solutions \ No newline at end of file diff --git a/Lsystem.tex~ b/Lsystem.tex~ new file mode 100644 index 0000000..389096e --- /dev/null +++ b/Lsystem.tex~ @@ -0,0 +1,1285 @@ +\documentclass[11pt,a4paper,titlepage,twoside]{article} + +\input{Preambule.tex} % Toutes les instructions du préambule sont dans un fichier séparé qui est utilisé pour tous les cours. + % Il faut le mettre à l'extérieur du dossier dans lequel se trouve le cours --> ../Preambule.tex + +\usepackage{geometry} % La seule présence de ce paquet élargit les marges correctement + +\lstloadlanguages{python,TeX,XML} % chargement des languages nécessaires (ici un seul) +%\lstset{language=TeX} % language choisi par défaut + +\setlength{\headheight}{22.6pt} % pour élargir un peu l'entête +%\setcounter{tocdepth}{5} % pour définir la profondeur de la table des matières +%\renewcommand{\thefootnote}{\alph{footnote}} % pour mettre les notes de bas de page en alphabétique + +%\usepackage{etex} +%\usepackage[all,2cell]{xy} +%\UseAllTwocells +%\usepackage{xymtex} % pour la représentation de molécules chimiques +%\let\urlorig\url % pour mettre des deux points dans les url, passage en langue anglaise +%\renewcommand{\url}[1]{% +% \begin{otherlanguage}{english}\urlorig{#1}\end{otherlanguage}% +%} +%\usepackage[colorlinks=true,linkcolor=black,urlcolor=blue,pagebackref,frenchlinks]{hyperref} +\usepackage{eurosym} +\usepackage{textcomp} +\usepackage{auto-pst-pdf} +\usepackage{vaucanson-g} +%\usepackage{algorithm,algorithmique} +\usepackage[french,boxruled]{algorithm2e} + +\setlength{\headheight}{13.6pt} % spécifie la hauteur de l'entête pour qu'elle ne couvre pas le titre + +\setcounter{topnumber}{3} % nombre maximal de flottant en haut d'une page (par défaut 2) +\setcounter{bottomnumber}{2} % nombre maximal de flottant en bas d'une page (par défaut 1) +%\setcounter{totalnumber}{3} % nombre maximal de flottant sur une page (par défaut 3) +\renewcommand{\topfraction}{0.9} % proportion maximale de la page qui peut recevoir des flottants en haut de la page (par défaut 0,7) +\renewcommand{\textfraction}{0.1} % proportion minimale d'une page qui reçoit du texte (sans les pages de flottants) +\renewcommand{\floatpagefraction}{0.8} % proportion minimale de flottants sur une page pour qu'elle devienne une page uniquement de flottants +\renewcommand{\bottomfraction}{0.5} % proportion maximale de la page qui peut recevoir des flottants en bas de la page (par défaut 0,3) + +\usepackage{answers} % pour utiliser le module answers +\Newassociation{sol}{Solution}{ans} % définition de l'étiquette Solution du module answers +\newtheorem{exc}{Exercice} % définition de l'étiquette Exercice du module answers +\newenvironment{ex}{\begin{exc}\normalfont}{\end{exc}} % cela (\normalfont) annule l'italique des énoncés des exercices + +\graphicspath{{./Images/}{./Introduction/Images/}} + +%\fancyhead[RE,LO]{Lycée Blaise Cendrars\\La Chaux-de-Fonds} +\fancyhead[LE,RO]{\thepage} +\lfoot{} \cfoot{} \rfoot{} +\pagestyle{fancy} + +%\usepackage[firstpage]{draftwatermark} +%\SetWatermarkLightness{0.8} +%\SetWatermarkAngle{60} +%\SetWatermarkScale{0.8} +%\SetWatermarkFontSize{5cm} +%\SetWatermarkText{DOCUMENT À LAISSER AU LYCÉE} + +\begin{document} + +%\frontmatter +\newcommand\HRule{\noindent\rule{\linewidth}{1.5pt}} +\begin{titlepage} + \vspace*{\stretch{1}} + \HRule + \begin{flushright} + \LARGE Option complémentaire \\ + \LARGE Informatique \\ + \textsc{Langages formels et L-system} + \end{flushright} + \HRule\\ +% \textsc{Document à laisser au lycée} + \vspace*{\stretch{2}} + \begin{center} + \today + \end{center} +\end{titlepage} + +\section*{Copyright} +\addcontentsline{toc}{section}{Copyright} +Copyright (c) 2011 Guyot Vincent + +\medskip +Permission vous est donnée de copier, distribuer et/ou modifier ce document selon les termes de la Licence GNU Free Documentation License, Version 1.1 ou ultérieure publiée par la Free Software Foundation. + +\medskip +\begin{tabbing}\small +\hspace*{1cm}\=\hspace*{1cm}\=\kill +\>Avec les sections inaltérables suivantes :\\ +\>\>\textit{Pas de section inaltérable}\\ +\>avec le texte de première page de couverture suivant :\\ +\>\>\textit{Option complémentaire \textsc{Informatique}}\\ +\>avec le texte de dernière page de couverture suivant :\\ +\>\>\textit{Pas de texte de texte de dernière page de couverture}\\ +\end{tabbing} + +Une copie transparente de ce document se trouve à l'adresse suivante : +www.cvgg.org +\begin{itemize} +\item Lsystem.tex : copie transparente, +\item Lsystem.dvi : copie opaque au format dvi. +\end{itemize} + +\bigskip +Normalement une copie de la licence GFDL doit se trouver avec le présent document. Comme il est relativement court et comme on trouve facilement le texte de la GFDL sur internet, elle n'est pas reproduite ici pour ne pas utiliser inutilement du papier. +\thispagestyle{empty} + +\newpage + +\section*{Avertissement} +\addcontentsline{toc}{section}{Avertissement} + +\thispagestyle{empty} + +\tableofcontents{} +\addcontentsline{toc}{section}{Table des matières} +\thispagestyle{empty} + +\newpage + +%\mainmatter + +\Opensolutionfile{ans}[Solutions] % ouvre le fichier Solutions.tex qui va contenir les solutions + +\section{Introduction} +Ce cours s'adresse à des élèves dont les connaissances en informatiques se limitent à l'utilisation de logiciels et à des connaissances de base (boucles et fonctions) en programmation. + +\smallskip +Il ne s'adresse pas spécifiquement à des élèves non mathématiciens, car ceux-ci peuvent aussi y trouver de l'intérêt, mais surtout à des personnes qui ne se destinent ni à l'informatique, ni au mathématiques. C'est pourquoi, l'accent à été mis sur la plus grande clarté possible dans l'exposé des notions qui relèvent des mathématiques et sur les exemples. Il constitue à mon avis une approche adaptée pour des personnes intéressées par la biologie et par les langues. + +\smallskip +Les objectifs sont multiples. +\begin{description} + \item[Le premier] d'entre eux est de montrer aux élèves que l'analyse formelle est un préalable nécessaire à la compréhension des langages, en particulier informatiques. + \item[Le deuxième] est de présenter un exemple de la diversité des modes de représentation de ces langages. + \item[Le troisième] est d'illustrer le lien qui existe entre les langages et certains processus naturels, lien qui montre l'importance de l'étude des formalismes présentés. + \item[Le dernier] objectif est finalement de permettre à l'élève d'utiliser des connaissances très sommaires de programmation pour analyser les processus mécaniquement dans des proportion inaccessibles à l'homme. +\end{description} + +\section{Langage formel} +Rendre compte de la notion de langage formel le plus simplement possible n'est pas chose facile. Il se trouve que c'est la rigueur du langage mathématique qui permet d'en exprimer au mieux l'idée. Nous allons donc l'utiliser le plus simplement possible étant donné les difficultés qu'il peut représenter. Il sera néanmoins bien souvent plus une aide qu'un obstacle. + +Tout langage est au départ constitué d'un alphabet. Il faut considérer le terme d'alphabet au sens large du terme comme un simple ensemble de signes ou lettres, même non alphabétiques au sens commun du terme. Ainsi, les ensembles suivants notés \(\Sigma_i\) constituent des exemples d'alphabet : +\begin{align*} + \Sigma_1 &= \{\text{\officialeuro, \maltese, \textleaf, \checkmark, 1, 4, \textvisiblespace}\}\\ + \Sigma_2 &= \{\text{\textlnot, \textdollaroldstyle, \textcircled{a}, \pounds}\}\\ + \Sigma_3 &= \{\text{a, b, \textvisiblespace, }\epsilon\} +\end{align*} +A partir de l'alphabet, on peut alors définir ce qu'est un mot. Il s'agit simplement d'une suite finie de lettres prises dans l'alphabet. Ainsi, on peut former les mots suivants sur l'alphabet \(\Sigma_1\) : \hspace{1cm}\officialeuro\officialeuro\maltese\officialeuro\officialeuro\checkmark\hspace{1cm} ou \hspace{1cm}\checkmark14\maltese\maltese\hspace{1cm} Pour l'alphabet \(\Sigma_3\), on pourrait avoir : \hspace{1cm}aababbab\hspace{1cm} ou \hspace{1cm}aabbbba \hspace{1cm}mais pas \hspace{1cm}aab\textlnot + +Si m est un mot sur un alphabet \(\Sigma\), on écrira : +\[\text{m}\in\Sigma^*\] +pour dire que le mot m appartient à l'ensemble \(\Sigma^*\) des mot possibles formé sur l'alphabet \(\Sigma\). Finalement, on définit \(\epsilon\) comme un mot vide. La raison pour laquelle on fait cela sera vue plus tard. + +\medskip +Maintenant qu'on a des mots, le problème est de pouvoir les utiliser. Si on considère les mots suivants de \(\Sigma_3\) : +\[\text{abaab\textvisiblespace\hspace{1cm}}\text{aba\textvisiblespace\hspace{1cm}}\text{aabaaba\textvisiblespace}\] +il serait possible de former queque chose qui ressemble à des phrases en plaçant ces mots les uns à côté des autres. Cette opération se nomme la concaténation et on la note par un point. Ainsi, on va formellement définir la concaténation ainsi : +\[\text{Soit m et n }\in \Sigma^*\text{\hspace{0.3cm} Alors, m.n=mn}\] +On peut ainsi tenter de reproduire l'apparence formelle d'un langage par concaténation des mots ci-dessus (en s'imaginant un espace invisible) : +\[\text{abaab\textvisiblespace}.\text{aba\textvisiblespace}.\text{aabaaba\textvisiblespace}=\text{abaab aba aabaaba}\] +Certes, le résultat ne correspond qu'en apparence à un langage. Mais, l'alphabet est très restreint. + +\medskip +De manière naturelle, on va donc définir la notion de langage comme un sous-ensemble de l'ensemble de tous les mots d'un alphabet donné. Un langage est donc un ensemble de mots. On peut donc faire l'union, l'intersection, \dots de deux langages. + +\medskip +L'exemple classique est le langage des expressions arithmétiques. Il est défini à partir de l'alphabet suivant : +\[\Sigma = \{0,1,2,3,4,5,6,7,8,9,+,-,*,/\}\] +Le langage des expressions arithmétiques s'écrit alors : +\[L(\text{expr. arithm.})=\{\dots ,342, 34+4, 45+34, 2=4, \dots\}\] +Relevez qu'il ne s'agit pas de l'ensemble de tous les mots formé sur l'alphabet, mais d'un sous ensemble qui ne contient pas l'expression 34+++43==, par exemple. Évidemment, il faut spécifier comment on réalise le choix de chaque expression. + +\begin{ex} + Sur l'alphabet des chiffres binaires (que vous noterez précisément), donnez le langage \(L\) des nombres à trois chiffres. + \begin{sol} + L'alphabet des chiffres binaires est : \(\Sigma=\{0,1\}\) et le langage des nombres binaires à trois chiffres est : + \[L=\{000,001,010,011,100,101,110,111\}\] + \end{sol} +\end{ex} + +\begin{ex}\label{codons} + Pour la molécule d'ADN, définissez un alphabet de ses composants et déterminez quelques éléments du langage des \emph{codons}, acide aminé constitué de triplets de nucléotides. Combien peut-il y en avoir si leur arrangement tient compte de l'ordre et qu'il peut y avoir des répétitions. + + Indication : les nucléotides sont les briques de base de l'ADN. + \begin{sol} + En considérant les quatre nucléotides composant l'ADN, la thymine (T), la cytosine (C), l'adénine (A) et la guanine (G), on peut définir l'alphabet nécessaire ainsi : + \[\Sigma=\{A,C,G,T\}\] + Les codons sont quant à eux des ensemble de trois nucléotides, pris parmi les quatre disponibles dans l'alphabet. Le langage ainsi formé est : + \[L=\{AAC,ACT,ACT,CTG,GTC, \dots\}\] + et la combinatoire donne pour un arrangement avec répétition de trois éléments pris parmi quatre : + \[\overline{A_3^4}=4^3=64\;\text{possibilités}\] + \end{sol} +\end{ex} + +\bigskip +La génération ou la vérification des expressions valables pour un langage donné peut se faire de différentes manières. Nous allons présenter dans la section suivante les moyens qui nous permettrons de décrire simplement des langages particuliers. On peut utiliser pour cela soit un automate soit une expression régulière. + +\section{Langage rationnel} + +\subsection{Les automates} + +Une manière particulièrement compréhensible de tester l'appartenance d'une expression à un langage est d'utiliser un automate. Il s'agit d'une sorte de machine symbolique capable de générer un ensemble de mots. On la suit dans la construction de ceux-ci selon des flèches qui mènent d'un état à l'autre par la concaténation successive de certaines mots de l'alphabet. Pour des raison de compréhension, on va commencer par donner un exemple d'automate à travers une représentation graphique qui fait correspondre des ronds aux états et des flèches aux éléments de l'alphabet. Un état initial est un rond sur lequel arrive une flèche vide et un état final est constitué d'un double cercle. Ainsi, on représente un automate reconnaissant le mot : +\[aba\;\text{ sur l'alphabet }\;\Sigma=\{a,b\}\] +par la figure suivante : +\begin{center} +\begin{postscript} +\VCDraw{% +\begin{VCPicture}{(0,-2)(9,0)} +% states +\State{(0,0)}{A} \State{(3,0)}{B} \State{(6,0)}{C} \FinalState{(9,0)}{D} +% initial--final +\Initial{A} +% transitions +\EdgeL{A}{B}{a} \EdgeL{B}{C}{b} \EdgeL{C}{D}{a} +% +\end{VCPicture}} +\end{postscript} +\end{center} +Relevons qu'il est possible de considérer l'unique mot construit par cet automate comme un langage qu'on peut définir mathématiquement très simplement par : +\[L=\{m\in\Sigma^*\,|\,m=aba\}\] + +où la barre verticale \(|\) signifie ``tel que''. + +\smallskip +Évidemment, l'objectif est de permettre la construction de langages plus étendus. On peut par exemple construire le langage constitué d'une séquence de ab ainsi : +\begin{center} +\begin{postscript} +\VCDraw{% +\begin{VCPicture}{(0,-1)(3,1)} +% states +\FinalState{(0,0)}{A} \State{(3,0)}{B} +% initial--final +\Initial{A} +% transitions +\ArcL{A}{B}{a} \ArcL{B}{A}{b} +% +\end{VCPicture}} +\end{postscript} +\end{center} + +Cet exemple permet d'engendrer une infinité de mots du type ab, abab, ababab, \dots + +\smallskip +Par ailleurs, la présence d'un état initial, simultanément final, implique l'existence dans le langage d'un mot vide noté \(\epsilon\). On écrira donc mathématiquement le langage ainsi : +\[L=\{m\in\Sigma^*\,|\,m=\epsilon,ab,abab,ababab,\dots\}\] + +Un dernier exemple montre qu'il est possible de générer des langages bien plus complexes. Sans dire immédiatement ce dont il s'agit, voici l'automate : + +\begin{center} +\begin{postscript} +\VCDraw{% +\begin{VCPicture}{(0,-1)(6,2)} +% states +\State{(0,0)}{A} \State{(3,0)}{B} \StateLineDouble \State{(6,0)}{C} +% initial--final +\Initial{A} +% transitions +\EdgeL{A}{B}{a} \ArcL{B}{C}{b} \ArcL{C}{B}{b} +\LoopN{A}{a} +% +\end{VCPicture}} +\end{postscript} +\end{center} + +Le langage reconnu est constitué de mots comprenant en premier lieu au moins un a, suivi par un nombre impair de b. On peut aussi écrire : +\[L=\{m\in\Sigma^*\,|\,m=a^nb^m,\,n\in\mathbb{N}^*\,\text{et}\;m=2\cdot k+1,\,k\in\mathbb{N}\}\] +où \(\mathbb{N}^*\) est l'ensemble des nombres naturels (entiers positifs) sans le zéro et \(\mathbb{N}\) celui avec le zéro. + +\medskip +En conclusion, on peut définir un automate de manière générale comme un ensemble constitué +\begin{itemize} + \item d'un alphabet \(\Sigma\), + \item d'un ensemble fini d'état, généralement noté \(E\), + \item d'un ensemble d'état initiaux \(I\) tel que \(I\subseteq E\), + \item d'un ensemble d'état finaux \(F\) tel que \(F\subseteq E\) et + \item d'une fonction de transition \[\delta:E\times\Sigma\longrightarrow E\] +\end{itemize} + +Où le signe \(\subseteq\) signifie ``appartient à''. + +\smallskip +Le dernier point de cette définition mérite des explications. La représentation graphique des automates donnée ci-dessus met clairement en évidence les transitions (flèches) entre états. Celles-ci doivent aussi trouver une expression mathématique. C'est la fonction de transition. + +\smallskip +Reprenons le dernier exemple d'automate donné ci-dessus. Considérons chacun des points permettant de le définir mathématiquement. + +L'alphabet tout d'abord est clairement donné par \(\Sigma=\{a,b\}\). + +L'ensemble des états est \(E=\{e_1,e_2,e_3\}\) et on peut en donner une représentation légèrement plus complète ainsi : + +\begin{center} +\begin{postscript} +\VCDraw{% +\begin{VCPicture}{(0,-1)(6,2)} +% states +\State[e_1]{(0,0)}{A} \State[e_2]{(3,0)}{B} \StateLineDouble \State[e_3]{(6,0)}{C} +% initial--final +\Initial{A} +% transitions +\EdgeL{A}{B}{a} \ArcL{B}{C}{b} \ArcL{C}{B}{b} +\LoopN{A}{a} +% +\end{VCPicture}} +\end{postscript} +\end{center} + +L'ensemble des états initiaux se résume à \(I=\{e_1\}\) et celui des états finaux à \(F=\{e_3\}\). + +La fonction de transition \(\delta\) est alors donnée par : +\begin{align*} + \delta(e_1,a)&=e_1\,|\,e_2 & \delta(e_1,b)&=\perp\\ + \delta(e_2,a)&=\perp & \delta(e_2,b)&=e_3\\ + \delta(e_3,a)&=\perp & \delta(e_3,b)&=e_2 +\end{align*} +où \(|\) signifie ``ou'' et \(\perp\) signifie ``indéfini''. + +\bigskip +Ce dernier exemple est aussi intéressant du point de vue du déterminisme du résultat. La fonction de transition \(\delta(e_1,a)\) donne deux solution possibles : \(e_1\) ou \(e_2\). Cela signifie qu'à partir de l'état \(e_1\), par la transition \(a\), un choix non déterminé peut avoir lieu. L'automate est alors dit \emph{non déterministe}. Cela peut constituer un problème et il est possible de le rendre déterministe. Mais nous n'aborderons pas ce point dans ce cours. Il convenait cependant de le mentionner. + +\medskip +Nous avons vu dans ce paragraphe une manière de \emph{reconnaître} mécaniquement un langage simplement formé de mots issus d'un alphabet à l'aide d'un automate. Nous avons vu deux manières de représenter celui-ci, par un diagramme et mathématiquement. Chacune de ces deux expressions constitue un mode de représentation qui a des avantages et des inconvénients. Les automates soulignent clairement le caractère mécanique du processus de reconnaissance. L'expression mathématique exprime quant à elle mieux les propriétés ensemblistes et relationnelles de la notion d'automate. + +\begin{ex}\label{autosimple} + Construisez un automate sur l'alphabet \(\Sigma=\{a,b\}\) qui reconnaît les mots suivants : aab, abb et bbb. + \begin{sol} + Il s'agit d'un automate simple sans grande difficultés. Il est déterministe puisque de chaque état ne part qu'une seule transition de même type. + \begin{center} + \begin{postscript} + \VCDraw{% + \begin{VCPicture}{(0,-2)(6,4)} + % states + \State[e_1]{(0,0)}{A} \State[e_2]{(2,1)}{B} \State[e_3]{(4,1)}{C} + \State[e_5]{(4,3)}{E} + \State[e_8]{(2,-1)}{G} \State[e_9]{(4,-1)}{H} + \StateLineDouble \State[e_{10}]{(6,-1)}{I} \State[e_4]{(6,1)}{D} \State[e_6]{(6,3)}{F} + % initial--final + \Initial{A} + % transitions + \EdgeL{A}{B}{a} \EdgeL{B}{C}{a} \EdgeL{C}{D}{b} + \EdgeL{B}{E}{b} \EdgeL{E}{F}{b} + \EdgeL{A}{G}{b} \EdgeL{G}{H}{b} \EdgeL{H}{I}{b} + % + \end{VCPicture}} + \end{postscript} + \end{center} + \end{sol} +\end{ex} + +\begin{ex} + Construisez un automate qui reconnaisse la succession des feux de circulation rouge-jaune-vert, allumé successivement et indéfiniment dans cet ordre. + \begin{sol} + Le problème peut sembler complexe car l'énoncé du mot allumé fait immédiatement penser à éteint. Naturellement, on pense à un état allumé et à un autre éteint. Comme il y a trois couleurs, on s'imagine alors six états et leur représentation avec des transitions du type ``s'allume'' ou ``s'éteint'' ne va pas de soit. + + Pour s'en sortir simplement, il faut au contraire penser ``quel est la couleur du feu''. Cela suppose qu'on ne considère que l'état allumé du feu. On peut alors dire rouge, puis jaune, puis vert, puis rouge, \dots On pourrait aussi dire, c'est le rouge qui s'allume, puis c'est le jaune qui s'allume, puis \dots Cela nous permet alors de représenter la succession des états allumé du feu par : + \begin{center} + \begin{postscript} + \VCDraw{% + \begin{VCPicture}{(0,-3)(11,1)} + % states + \State[r]{(0,0)}{A} \State[j]{(4,0)}{B} \State[v]{(2,-2)}{C} + \State{(7,0)}{D} \State{(11,0)}{E} \State{(9,-2)}{F} + % initial--final + \Initial{A} \Initial{D} + % transitions + \EdgeL{A}{B}{} \EdgeR{B}{C}{} \EdgeR{C}{A}{} + \EdgeL{D}{E}{r} \EdgeR{E}{F}{j} \EdgeR{F}{D}{v} + % + \end{VCPicture}} + \end{postscript} + \end{center} + En particulier, l'automate ci-dessus permet la reconnaissance de la chaîne de caractère rjvrjvrjv\dots{} qui est une représentation de la succession des couleurs (allumées) du feu de circulation. + \end{sol} +\end{ex} + +\begin{ex} + Construisez un automate qui reconnaisse les codons de l'exercice \ref{codons}. + \begin{sol} + On a vu qu'il y a 64 possibilités pour former un acide aminé constitué d'un triplet choisi parmi quatre nucléotides. Il peut sembler alors que l'automate correspondant doive être très complexe. Mais au contraire, en raison du fait qu'on peut mettre n'importe quel nucléotide à chaque place parmi les trois disponibles nous permet de constituer un automate particulièrement simple : + \begin{center} + \begin{postscript} + \VCDraw{% + \begin{VCPicture}{(0,-1)(8,1)} + % states + \State{(0,0)}{A} \State{(4,0)}{B} \State{(8,0)}{C} + \StateLineDouble \State{(12,0)}{D} + % initial--final + \Initial{A} + % transitions + \EdgeL{A}{B}{C|G|T|U} \EdgeR{B}{C}{C|G|T|U} \EdgeR{C}{D}{C|G|T|U} + % + \end{VCPicture}} + \end{postscript} + \end{center} + où la notation C|G|T|U signifie soit C, soit G, soit T, soit U. + + \smallskip + On constate ainsi la concision de la description de l'automate par rapport à l'exposé de tout les cas possibles. + \end{sol} +\end{ex} + +\subsection{Les expressions rationnelles} +Voyons maintenant une autre manière de représenter les langages reconnus par les automates : les \emph{expressions rationnelles}\footnote{On parle aussi d'expressions régulières en raison de l'anglais \emph{regular expression}.}. Elles trouvent ici leur place pour souligner qu'il existe différentes approches dans la reconnaissance des langages (dits rationnels). + +\smallskip +Premièrement, une \emph{expression rationelle} est un mot. L'alphabet permettant de construire ce mot est particulier, puisque s'y trouve obligatoirement les symboles suivants : \((\,)\,+\,.\,*\,\emptyset\) Ainsi, on peut voir l'alphabet sur lequel on construit les expressions rationelles comme composé de deux ensemble : un alphabet \(\Sigma\) choisi selon les besoins et l'ensemble des symboles donné ci-dessus. L'alphabet des expressions rationelles est donc formellement : +\[\Sigma^{\text{expressions rationnelles}}=\Sigma \cup \{(,),+,.,^*,\varepsilon\}\] +où \(\varepsilon\) est le caractère vide. + +\smallskip +Secondement, une expression rationnelle doit obéir à une \emph{syntaxe}. Une manière de construire un langage, en tant que partie de l'ensemble des mots de \(\Sigma\), est de le faire à partir d'un automate. Une autre manière est de le faire à partir d'un ensemble d'opération qui permettent de le définir complètement. On appelle ces opérations des règles de \emph{syntaxe}. En d'autre termes, ces règles autorisent ou interdisent la production de mots particuliers sans référence au sens des mots produits. Ainsi, si e et r sont des expressions rationnelle, alors il faut ajouter que les règles de syntaxe des expressions rationnelles autorisent les trois expressions (e+r), (e.r) et (e*) comme expressions rationnelles. On dit ici simplement que les trois suites de caractères précédentes sont correctement formées pour qu'il s'agisse d'expressions rationnelles. Le sens des symboles +, . et * n'est pas précisé car il ne s'agit que de règles de syntaxe. + +\medskip +En définitive donc, on peut définir l'ensemble des expressions rationnelles par une règle de base (B) et par un règle récursive (R) : +\begin{description} + \item[B.] \(\varepsilon\) et e avec \(\text{e}\in \Sigma\) sont des mots sur \(\Sigma \cup \{(,),+,.,^*,\varepsilon\}\) + \item[R.] si e et r sont des expressions rationnelles, alors (e+r), (e.r) et (e*) le sont aussi. +\end{description} + +\medskip +Voilà l'ensemble des expressions rationnelles correctement défini. Il s'agit d'une définitions complexe sur laquelle nous ne reviendrons pas dans le cadre de ce cours. Il faut cependant retenir l'idée de syntaxe comme règles de production d'expressions. Elle nous mènera plus tard à celle de grammaire. + +\medskip +Jusqu'à présent nous n'avons pas parlé de sémantique. La définition rigoureuse du sens des symboles utilisés précédemment pour les expressions rationnelles (\(+,.,*\)) passe par la construction d'une relation de l'ensemble des expressions rationnelles sur une partie de l'ensemble des mots. Ces symboles prennent alors une signification particulière : le \(+\) devient un ou (\(\cup\) ou \(|\)), le \(.\) une concaténation de deux expressions rationnelles et l'étoile devient la répétition de zéro ou plusieurs fois le caractère précédent. + +\medskip +Ainsi, en imaginant la reconnaissance du code postal d'une ville suisse constitués de quatre chiffres, on peut écrire une expression rationnelle qui exprime cette structure de la manière suivante : +\[(0|1|2|3|4|5|6|7|8|9).(0|1|2|3|4|5|6|7|8|9).(0|1|2|3|4|5|6|7|8|9).(0|1|2|3|4|5|6|7|8|9)\] +Évidemment, cette notation n'est pas très pratique et en réalité on définit d'autres symboles qui permettent d'alléger l'écriture. Par exemple, de manière évidente : +\[[0..9]\{4\}\] +On trouvera avec \cite{DB08} un ouvrage très complet sur la syntaxe et la signification des symboles les plus utilisés. + +\begin{ex}\label{autotomate} + Exprimez l'automate suivant en français et transformez-le en expression rationnelle. + \begin{center} + \begin{postscript} + \VCDraw{% + \begin{VCPicture}{(0,-1)(4,1)} + % states + \State[e_1]{(4,0)}{B} + \StateLineDouble \State[e_0]{(0,0)}{A} + % initial--final + \Initial{A} + % transitions + \ArcL{A}{B}{b} \ArcL{B}{A}{a} \LoopE{B}{b} + % + \end{VCPicture}} + \end{postscript} + \end{center} + \begin{sol} + On doit tout d'abord faire un b, puis un nombre quelconque (voir nul) de b. Comme l'état de droite n'est pas final, il faut ensuite revenir à celui de gauche par un a. Enfin, on peut faire tout cela un nombre quelconque de fois, y compris aucune. L'expression rationnelle correspondante est : +\[(bb^*a)^*\] + \end{sol} +\end{ex} + +\begin{ex} + Construisez l'expression régulière correspondant aux numéros de plaques minéralogiques suisses : deux lettres, suivies de cinq chiffres décimaux. + \begin{sol} + De manière très naturelle, on a : + \[[A..Z]\{2\}[0..9]\{5\}\] + \end{sol} +\end{ex} + +\begin{ex} + Construisez l'expression régulière correspondant à l'automate de l'exercice \ref{autosimple}, page \pageref{autosimple}. + \begin{sol} + Il faut raisonner à partir de la dernière transition avant l'état final. Il s'agit toujours d'une transition b. Donc, l'expression régulière finira par un b. Puis, on a soit deux b, soit un a suivi d'un a ou b. Cela se traduit par : + \[(bb|a(a|b))b\] + \end{sol} +\end{ex} + +\section{Langage de Lindenmayer} +Avant de nous intéresser aux langages de Lindenmayer, dit L-system, il faut pousser un peu plus avant l'étude des langages formels. + +On a vu plusieurs manières de formaliser un langage simple. La formalisation des expressions rationnelles, par exemple, a fait apparaître des règles de syntaxe qui évoquent une \emph{grammaire}. Il est intéressant ici de préciser cette notion pour pouvoir comprendre au mieux les L-system. + +\subsection{Grammaire} +Une \emph{grammaire} est un ensemble \(G\) d'objets tel que : +\begin{align*} + G=&\{T,N,R,S\}\text{ où : }\\ + &T \text{ est l'alphabet des lettres terminales,}\\ + &N\text{ est l'alphabet des lettres non terminales}\\ + &S\in N\text{ est l'axiome}\\ + &R\subseteq (T \cup N)^*\times(T \cup N)^*\text{ est un ensemble fini de règles de production.} +\end{align*} +Une convention pratique veut que les symboles non-terminaux s'écrivent en majuscule et les terminaux en minuscule. + +\medskip +Voici un exemple simple, mais assez complet. +\begin{align*} + G=&\{T,N,R,S\}\text{ où : }\\ + &T=\{a,b,m,n\}\\ + &N=\{O,P\}\\ + &S=aO\\ + &R=\{O\rightarrow aO|bP\;\text{ et }\;P\rightarrow m|n\} +\end{align*} +où le signe \(|\) signifie ou. + +\smallskip +On peut donner deux exemples de génération d'éléments à partir de cette grammaire : +\begin{enumerate} + \item \(aO\rightarrow aaO\rightarrow aaaO\rightarrow aaabP\rightarrow aaabn\) + \item \(aO\rightarrow aaO\rightarrow aabP\rightarrow aabm\) +\end{enumerate} +On comprend aisément le rôle des règles de productions et celui des lettres terminales qui vont arrêter la génération de la chaîne de caractère. + +\medskip +Un automate peut constituer un autre exemple intéressant. En effet, il est possible de traduire un automate par une grammaire appropriée. +\begin{align*} + G=&\{T,N,R,S\}\text{ où : }\\ + &T=\{\text{états terminaux}\}\\ + &N=\{\text{états non terminaux}\}\\ + &S=\text{état initial}\\ + &R=\{\text{fonction de transition}\} +\end{align*} +Pour plus de clarté, considérons l'automate de l'exercice \ref{autotomate}, page \pageref{autotomate}, et tentons de le traduire en une grammaire. +\begin{align*} + G=&\{T,N,R,S\}\text{ où : }\\ + &T=\{e_0\}\\ + &N=\{e_0,e_1\}\\ + &S=e_0\\ + &R=\{\delta(e_0,a)=\bot\;;\;\delta(e_0,b)=e_1\;;\;\delta(e_1,a)=e_0\;;\;\delta(e_1,b)=e_1\} +\end{align*} + +\medskip +Évidemment, l'existence d'une correspondance entre automate et expression régulière implique l'idée d'une formalisation des expressions régulières sous forme de grammaire. Nous n'aborderons pas ce cas ici. + +\medskip +Enfin, voici un autre exemple de grammaire où il n'y a pas de symbole terminaux. +\begin{align}\label{grammaire_algue_de_lindenmayer} + G=&\{T,N,R,S\}\text{ où : }\\\nonumber + &T=\emptyset\\\nonumber + &N=\{A,B\}\\\nonumber + &S=A\\\nonumber + &R=\{A\rightarrow AB\;\text{ et }\;B\rightarrow A\} +\end{align} +Celle-ci permet la génération de séquences infinies dont nous allons parler maintenant. + +\subsection{Génération de séquence} +La production des éléments de la grammaire donnée dans le dernier exemple ci-dessus débute par l'axiome \(A\). Ensuite, les deux règles de productions doivent être utilisées pour produire l'élément suivant : remplacer \(A\) par \(AB\) et \(B\) par \(A\). Comme il n'y a qu'un \(A\) dans l'axiome, seule la première règle s'applique tout d'abord et on obtient \(AB\). Puis, les deux règles doivent à nouveau être utilisées à partir de cet état pour produire le suivant. On remplace donc les \(A\) par \(AB\) et \(B\) par \(A\) pour chaque lettre de \(AB\). On obtient ainsi \(ABA\). Et on continue indéfiniment. On obtient alors : +\begin{align}\label{chaine_algue_de_lindenmayer} + A&\nonumber\\A&B\nonumber\\A&BA\nonumber\\A&BAAB\nonumber\\A&BAABABA\nonumber\\A&BAABABAABAAB +\end{align} +Ces chaînes n'ont pas encore de signification. Mais on peut déjà envisager différentes règles de productions permettant de générer de multiples chaînes de caractères. Beaucoup de questions peuvent alors se poser concernant les propriétés des chaînes ainsi constituées. + +\medskip +Voici enfin un autre exemple très connu de grammaire permettant de générer des chaînes auxquelles on peut attacher une signification par interprétation graphique : +\begin{align}\label{grammaire_courbe_de_koch} + G=&\{T,N,R,S\}\text{ où : }\\\nonumber + &T=\{+,-\}\\\nonumber + &N=\{F\}\\\nonumber + &S=F\\\nonumber + &R=\{F\rightarrow F+F-F-F+F\} +\end{align} +Notez la séparation entre les symboles terminaux, sans règles de production et le symbole non terminal auquel correspond une règle de production. + +La génération des chaînes est alors la suivante : +\begin{align}\label{chaine_courbe_de_koch} + F&\nonumber\\F&+F-F-F+F\nonumber\\F&+F-F-F+F+F+F-F-F+F-F+F-F-\;\hookleftarrow\nonumber\\\hookrightarrow\;&F+F-F+F-F-F+F+F+F-F-F+F +\end{align} + +\smallskip +Encore une fois, le sens de ces chaînes n'est pas encore évident. Par ailleurs, on voit aussi que la génération des caractères de ces chaînes devient très vite inaccessible au possibilités humaines en raison du nombre élevé de caractères produits. Une mécanisation de la production est donc rapidement nécessaire. + +\subsection{L-system} +Les deux derniers exemples ci-dessus ((\ref{grammaire_courbe_de_koch}) et (\ref{grammaire_algue_de_lindenmayer})) sont des systèmes de Lindenmayer ou L-system. De quoi s'agit-il ? L'ouvrage de référence en la matière, \emph{The Algorithmic Beauty of Plants} \cite[p.2-3]{LA04}, en présente le contexte en ces termes : + +\begin{quotation} + \og \selectlanguage{english}In 1968 a biologist, Aristid Lindenmayer, introduced a new type of string-rewriting mechanism, subsequently termed L-systems. The essential difference between Chomsky grammars and L-systems lies in the method of applying productions. In Chomsky grammars productions are applied sequentially, whereas in L-systems they are applied in parallel and simultaneously replace all letters in a given word. This difference reflects the biological motivation of L-systems. Productions are intended to capture cell divisions in multicellular organisms, where many divisions may occur at the same time. Parallel production application has an essential impact on the formal properties of rewriting systems.\selectlanguage{french}\fg +\end{quotation} + +L'objectif est donc de modéliser des mécanisme de division (cellulaire). Le moyen utilisé est un mécanisme de réécriture. Quoi de plus naturel donc que d'utiliser une grammaire précisément basée sur la réécriture. Les règles de production s'y prêtent bien et les deux exemples présentés ci-dessus en sont une belle démonstration. Il ne s'agit pas ici d'une division cellulaire, mais de la réplication d'un motif sur lui-même. Une grammaire en est ici l'origine et nous verrons par la suite qu'elle peut se traduire par une récursivité des algorithmes nécessaires pour la production de motifs conséquents. + +Mais, pour bien comprendre la signification des mécanismes de production des L-system dans la réalité, il faut maintenant en donner une interprétation graphique. + +\subsection{Interprétation graphique} +Il s'agit ici d'utiliser les chaînes de caractères produites par les grammaires pour en faire une représentation graphique. L'articulation d'un mécanisme de production général et d'une analyse des briques composant les structures végétales est certainement à l'origine de cette idée. Elle a trouvé son expression complète grâce aux possibilités de dessin proposés par le langage de programmation Logo et la tortue (Turtle) qui lui est attachée. + +L'interprétation consiste à utiliser les caractères de l'ensemble terminal \(N\) comme des instructions d'orientation de la tortue et les symboles non-terminaux comme des instruction de construction du chemin. Classiquement, on a : + +\begin{description} + \item[\(F\)] avancer d'un pas en avant, + \item[\(+\)] tourner à gauche d'un angle donné, + \item[\(-\)] tourner à droite d'un angle donné. +\end{description} + +Cette interprétation permet de suivre les chaînes de caractères produites par la grammaire pour en tracer une représentation graphique. + +\subsubsection{Koch} +L'exemple donné par la grammaire (\ref{grammaire_courbe_de_koch}), correspondant aux chaînes (\ref{chaine_courbe_de_koch}) peut ainsi prendre l'apparence des figures \ref{figkoch} pour un angle de 90\textdegree. + +\begin{figure*}[t] +\centering +\subfigure[F+F-F-F+F\label{figkoch1}]{\includegraphics{koch1.eps}}\qquad +\subfigure[F+F-F-F+F+F+F-F-F+F-F+F-F-F+F-F+F-F-F+F+F+F-F-F+F\label{figkoch2}]{\includegraphics{koch2.eps}}\\ +\subfigure[Plus encore ...\label{figkoch3}]{\includegraphics[width=8cm]{koch3.eps}} +\caption{La courbe de Koch\label{figkoch}} +\end{figure*} + +La figure \ref{figkoch1} permet de suivre l'interprétation graphique à la main, caractère après caractère. Mais, on hésite déjà à le faire à la figure suivante pour des raisons évidentes. + +A part l'esthétique des figures \ref{figkoch}, cette interprétation graphique semble ne pas apporter grand chose. Pourtant, les objets étudiés dans \emph{The Algorithmic Beauty +of Plants +} \cite[p.8-9]{LA04} et reproduits\footnote{Le code permettant d'obtenir ces structures sera présenté plus loin.} dans la figure \ref{figkochisland} sont déjà plus intéressants et montrent des structures qui commencent à ressembler vaguement à des cristaux par exemple. + +\begin{figure*}[ht] +\centering +\subfigure[F-F+F+FF-F-F+F (axiome F-F-F-F)\label{figkochisland1}]{\includegraphics{kochIsland1.eps}}\qquad +\subfigure[F-F+F+FF-F-F+F (axiome F-F-F-F)\label{figkochisland2}]{\includegraphics[width=5cm]{kochIsland2.eps}}\\ +\subfigure[F+FF-FF-F-F+F+FF-F-F+F+FF+FF-F (axiome F-F-F-F)\label{figkochislandquadratic2}]{\includegraphics[width=5cm]{kochIslandquadratic2.eps}} +\caption{Une autre courbe de Koch\label{figkochisland}} +\end{figure*} + +\subsubsection{Modélisation des plantes} +Un autre exemple d'utilisation d'une grammaire interprétée de manière particulière est donné par la modélisation de la croissance des plantes. Turtle est ici utilisé comme précédemment. Mais l'alphabet est étendu par deux nouveaux signes : les crochets [ et ]. Leur signification dans le cadre de Turtle est spéciale : +\begin{description} + \item[le crochet ouvrant] [ signifie qu'il faut mémoriser la position et la direction courante de la tortue et + \item[le crochet fermant] ] signifie qu'il faut faire revenir la tortue à la position et direction précédemment mémorisée. +\end{description} +Cette opération de mémorisation est nécessaire pour permettre une structure d'arbre qui n'est pas possible sans cela puisque le parcours de la tortue est alors continu. + +Voyons un exemple d'une telle grammaire. +\begin{align}\label{grammaire_plante} + G=&\{T,N,R,S\}\text{ où : }\\\nonumber + &T=\{+,-,[,]\}\\\nonumber + &N=\{F\}\\\nonumber + &S=F\\\nonumber + &R=\{F\rightarrow F[+F]F[-F]F\} +\end{align} + +Après deux fois l'application de la règle (deux générations), on obtient la chaîne suivante : +\begin{center} + F[+F]F[-F]F[+F[+F]F[-F]F]F[+F]F[-F]F[-F[+F]F[-F]F]F[+F]F[-F]F +\end{center} +Encore une fois, l'interprétation n'est pas triviale, pas plus que la programmation de la mémorisation. + +\medskip +Pourtant, la figure \ref{figplante} obtenue par l'interprétation graphique pour cinq génération et un angle de 25,7\textdegree{} est étonnante. + +\begin{figure*}[t] +\centering +\includegraphics[width=12cm]{plante1.eps} +\caption{Modélisation d'une plante\label{figplante}} +\end{figure*} + +Évidemment, on s'imagine bien que le remplacement récursif d'un même motif à partir de points clés de celui-ci mène à la construction d'arbre de ce type. Mais la figure produite est tout de même remarquable dans la structure ``botanique'' qu'elle présente, même si une tige aussi rectiligne est peu probable dans la réalité. + +\medskip +Plusieurs autres règles peuvent être utilisées. Plusieurs angles aussi. Comme présenté dans \emph{The Algorithmic Beauty of Plants} \cite[p.25]{LA04}, on obtient alors des structures dont la ressemblance avec des plantes réelles est étonnante. La figure \ref{figplantes} présente les ``plantes'' obtenues avec les règles suivantes : +\begin{align} + F &\rightarrow F[+F]F[-F]F\label{regle1}\\ + F &\rightarrow FF-[-F+F+F]+[+F-F-F]\label{regle2} +\end{align} + +\begin{figure*}[ht] +\centering +\subfigure[nbiteration = 5, angle = 20\textdegree{} (axiome F ; règle \ref{regle1})\label{figplante2}]{\includegraphics[width=10cm]{plante2.eps}}\\ +\subfigure[nbiteration = 5, angle = 22,5\textdegree{} (axiome F ; règle \ref{regle2})\label{figplante3}]{\includegraphics[width=10cm]{plante3.eps}} +\caption{Deux autres plantes\label{figplantes}} +\end{figure*} + +On peut aussi illustrer l'utilisation de plusieurs règles simultanément à travers les exemples \ref{regle3}, \ref{regle4} et \ref{regle5} où l'axiome est \(X\). Les deux règles sont appliquées successivement. +\begin{align} + X &\rightarrow F[+X]F[-X]+X\;\text{ et }\;F \rightarrow FF\label{regle3}\\ + X &\rightarrow F[+X][-X]FX\;\text{ et }\;F \rightarrow FF\label{regle4}\\ + X &\rightarrow F-[[X]+X]+F[+FX]-X\;\text{ et }\;F \rightarrow FF\label{regle5} +\end{align} + +Le résultat de l'interprétation graphique est donné à la figure \ref{figplantes2}, page \pageref{figplantes2}. + +\begin{figure*}[ht] +\centering +\subfigure[nbiteration = 7, angle = 20\textdegree{} (axiome X, règle \ref{regle3})\label{figplante4}]{\includegraphics[width=6cm]{plante4.eps}}\qquad +\subfigure[nbiteration = 7, angle = 25,7\textdegree{} (axiome X, règle \ref{regle4})\label{figplante5}]{\includegraphics[width=6cm]{plante5.eps}}\\ +\subfigure[nbiteration = 5, angle = 22,5\textdegree{} (axiome X, règle \ref{regle5})\label{figplante6}]{\includegraphics[width=8cm]{plante6.eps}} +\caption{Deux autres plantes\label{figplantes2}} +\end{figure*} + +\medskip +Bien évidemment, ces exemples ne sont pas construits au hasard. Une étude approfondie de la forme des chaînes nécessaires pour obtenir des structures réalistes de plantes doit être faite, ou tout au moins une réflection sur les motifs de base. On se reportera pour cela à l'adresse \url{http://algorithmicbotany.org/papers/} où sont présentés de nombreux exemples très crédibles. En ce qui nous concerne, l'objectif de ce petit cours n'étant pas d'étudier systématiquement ou de manière approfondie les plantes elles-mêmes, on laisse au lecteur le soin de tenter de trouver des chaînes crédibles et de les interpréter grâce à la programmation. + +Car, il est temps maintenant de considérer cet aspect moins formel, mais aussi important dans le cadre d'un cours d'informatique. Nous allons donc commencer par considérer le problème de la création de L-system de manière algorithmique. Cela nous permettra d'avoir une bonne vision des mécanismes principaux permettant la génération de ces systèmes. + +\subsection{Algorithmique\label{algo}} + +\subsubsection{Création} + +Commençons par la construction de la chaîne de Lindenmayer. Nous avons vu que tout commence avec un axiome. Celui-ci peut être constitué de deux types de symboles : non-terminaux et terminaux. La production de la chaîne consiste quant à elle à remplacer un certain nombre de fois les symboles non-terminaux par une chaîne correspondante donné par une (ou plusieurs) règle. Pour simplifier la situation, on ne considérera ici qu'une seule règle. Il est donc nécessaire de disposer du nombre d'itération choisi, de l'axiome et de la règle qu'il faut initialiser. Ensuite viennent deux boucles. L'une pour répéter les remplacements des symboles non-terminaux par ceux de la règle et l'autre pour lire l'axiome. Finalement, au bout d'un nombre de génération donné, le résultat produit est un axiome ``final'' qui sera transmis pour interprétation graphique. L'algorithme \ref{algocreation}, page \pageref{algocreation}, traduit ce raisonnement en pseudo-code. + +%\begin{comment} +%\begin{algorithm}[t] +%\caption{RRT}\label{algo_rrt} +%\begin{pseudocode}[ovalbox]{mystère}{a,b,c} +%a \PREND a+b\\ +%c \PREND a+\sqrt{b}\\ +%b \PREND a-b +%\end{pseudocode} +%\end{algorithm} +%\end{comment} + +\begin{algorithm}[t] + \SetAlgoVlined + \SetAlgoCaptionSeparator{ \textendash} + %\SetAlCapSkip{5mm} + \Donnees{nbiteration, axiome, regle} + \Res{nouvel axiome} + initialisation\; + \Tq{on est < nbiteration}{ + \Tq{on est < longueur de l'axiome}{ + on lit un caractère de l'axiome\; + \Si{on trouve un caractère non terminal}{ + on le remplace par la règle\; + } + \SinonSi{on trouve un caractère terminal}{ + on le réécrit simplement\; + } + } + } + \caption{Création de la chaîne}\label{algocreation} +\end{algorithm} + +\subsubsection{Interprétation} + +Après la création de la chaîne vient son interprétation. Pour cela, il est nécessaire de disposer de l'axiome ``final'', de la longueur choisie des segments produit par la tortue et de l'angle correspondant à chaque rotation. L'interprétation graphique se fait ensuite par la lecture de chaque caractères de la chaîne et le lancement de l'opération correspondante par la tortue. + +Si l'opération de mémorisation de la position de la tortue est simple, il faut analyser plus finement le retour à cette position. En effet, il est nécessaire de mémoriser plusieurs états différents avant d'y revenir successivement. Un tableau des états est donc nécessaire. De plus, une fois revenu sur un état mémorisé, il ne faut pas oublier de l'effacer. L'algorithme \ref{algointerpretation}, page \pageref{algointerpretation}, résume tout cela en pseudo-code. + +\begin{algorithm}[bh] + \SetAlgoVlined + \SetAlgoCaptionSeparator{ \textendash} + %\SetAlCapSkip{5mm} + \Donnees{axiome, longueur, angle\\variable : tableau des états mémorisés} + \Res{le tracé} + initialisation\; + \Tq{on est < longueur de l'axiome}{ + on lit un caractère de l'axiome\; + \Si{on trouve tel caractère}{ + on demande à la tortue de faire telle opération\; + } + \SinonSi{on trouve tel autre caractère}{ + on demande à la tortue de faire telle autre opération\; + } + \SinonSi{on trouve le caractère [}{ + on mémorise sa position et sa direction + } + \SinonSi{on trouve le caractère ]}{ + on récupère l'état, on y envoie la tortue et on efface cet état de la pile des état mémorisés + } + } + \caption{Interprétation graphique de la chaîne}\label{algointerpretation} +\end{algorithm} + +Dans le cas où plusieurs règles sont présentes, seule la procédure de création change avec deux boucles de lecture successives permettant d'effectuer le remplacement de deux caractères non-terminaux différents. Ces deux boucles ayant la même forme, on ne présentera pas ce cas. Mais le code donné en annexe \ref{codepythonfonct2} en donne une illustration compréhensible. + +\subsection{Programmation} + +\subsubsection{Approche non récursive} +Python est le langage utilisé. Mais,comme toujours dans le domaine de la programmation, il y a pour chaque problèmes beaucoup de solutions possibles. Même si l'objectif de ce cours n'est pas la programmation, deux solutions différentes vont être présentées, non récursive et récursive, répondant toutes les deux à la description algorithmique présentée au point \ref{algo}. En effet, dans le cadre d'un cours sur les langages, il est une fois de plus intéressant de montrer la diversité des approches, sans pour autant se focaliser particulièrement sur les avantages des différents modes de programmation. Par contre, on abordera pas ces problèmes en programmation orientée objet ou à travers d'autres langages tels que java par exemple. Mais il est évidemment envisageable de le faire par la suite. + +Dans la section \ref{algo}, on a vu la décomposition du problème de génération des L-system en deux procédures d'une part de production des motifs à partir de la grammaire et d'autre part d'interprétation de ceux-ci par turtle. Le code complet est donné en annexe \ref{codepythonfonct}. On ne va détailler ici que les aspects non liés au langage de programmation utilisé. Cependant, il faut signaler qu'on a utilisé un découpage du problème en \emph{fonction} pour des raisons de clarté uniquement et qu'on a aussi mélangé variables \emph{globale} et \emph{locales}, \emph{procédures} et \emph{fonctions} pour mettre en jeu ces notions dans un programme fonctionnel. Mais, on ne les définira pas ici. + +\paragraph{La création de la chaîne} consiste essentiellement en une boucle qui parcourt successivement tous les caractères. +\begin{lstlisting}[language=Python] + while j < len(axiom): + pointeur1 = axiom[j] +\end{lstlisting} +La lecture se fait en plaçant le caractère j de l'axiome dans une variable \verb|pointeur1| qui permettra de décider du caractère terminal ou non d'un caractère. En effet, par remplacement de la règle dans les symboles non terminaux de l'axiome, la boucle va créer un nouvel axiome, nommé \verb|axiome2|, qui sera utilisé comme axiome de départ à chaque itération. Ainsi, on a : +\begin{lstlisting}[language=Python] + # Symboles non terminaux + if pointeur1 == 'F': + # remplacement de la regle dans l'axiome + axiom2 = axiom2 + regle +\end{lstlisting} +On voit qu'on remplace le caractère \verb|F| par la chaîne correspondante de la règle. Par contre, la lecture d'un symbole terminal ne donne lieu à aucun remplacement. On réécrit simplement ce symbole tel quel. + +Ainsi, au fur et à mesure de la lecture des caractères,on construit par concaténation une nouvelle chaîne \verb|axiom2| qui sera à la base de la prochaine itération. + +\paragraph{L'interprétation de la chaîne} se fait à nouveau par lecture de celle-ci. Après création, elle est passée à la procédure d'interprétation via le paramètre \verb|axiom|, contenant la chaîne de caractère qui sera lue. Ensuite, pour les symboles de déplacement et d'orientation, l'interprétation est simple : +\begin{lstlisting}[language=Python] + # Interpreation + if pointeur2 == 'F': + forward(longueur) + elif pointeur2 == '+': + left(angle) + elif pointeur2 == '-': + right(angle) +\end{lstlisting} +Par contre, pour les symboles de mémorisation, la procédure est quelque peu plus complexe : +\begin{lstlisting}[language=Python] + elif pointeur2 == '[': + # memorisation de la position courante + laposition.append(position()) + langle.append(heading()) + elif pointeur2 == ']': + # retour a la derniere position et suppression de celle-ci + up() + goto(laposition[len(laposition)-1][0],\ + laposition[len(laposition)-1][1]) + del laposition[len(laposition)-1] + setheading(langle[len(langle)-1]) + del langle[len(langle)-1] + down() +\end{lstlisting} +Tout d'abord, on utilise ici un tableau, nommé \verb|laposition| pour mémoriser chaque couple abscisse et ordonnée de la tortue et un autre tableau, nommé \verb|langle| pour mémoriser sa direction. + +\smallskip +Puis, pour le retour à la position mémorisée, on réalise un \verb|goto| aux coordonnées mémorisées et un \verb|setheading| pour revenir à la direction enregistrée. Enfin, il ne faut pas oublier de supprimer des deux tableaux la position à laquelle on est revenu. Cela se fait par deux \verb|del|. Notez que pendant le retour il ne faut pas que la tortue soit en position d'écriture. On lève donc le crayon avant le retour et on le baisse après. + +\medskip +En fin de compte, grâce à l'utilisation de fonctions (et procédures), on peut écrire le programme principal avec une grande clarté : +\begin{lstlisting}[language=Python] + initialisation() + gram = grammaire() + ## gram[0] retourne l'angle + ## gram[1] retourne l'axiome + ## gram[2] retourne la regle + + axio = production(gram[1], gram[2]) + interpretation(gram[0], axio) +\end{lstlisting} +L'initialisation se fait par le placement de la tortue et la mémorisation des paramètre, axiome et règle. Puis, vient naturellement la production et l'interprétation. + +\subsubsection{Approche récursive} +Une autre approche de la génération de L-system fait usage d'une technique de programmation particulièrement intéressante : la récursivité. Il s'agit de l'appel d'une fonction à elle-même. Pour les L-system, il s'agit donc de ne plus générer la chaîne de caractère, mais de lire l'axiome en renvoyant simultanément à la fonction appelée, à chaque occurrence d'un caractère non terminal. Il n'est alors pas évident de suivre ce que fait le programme. C'est pourquoi on va maintenant reprendre une forme légèrement différente de celle donnée précédemment du flocon de Koch pour l'étudier dans le détail. Le code complet est donne en annexe \ref{codepythonfloconrecursif}. Seule la partie purement récursive est présentée ci-dessous : +\begin{lstlisting}[language=Python] + def production(nbiteration, longueur, angle, axiome, regle): + if nbiteration == 0 : + forward(longueur) + else: + i = 0 + while i < len(regle): + regle1 = regle[i] + if regle1 == 'F': + production(nbiteration-1, longueur/3,\ + angle, axiome, regle) + if regle1 == '+': + left(angle) + if regle1 == '-': + right(angle) + i = i + 1 +\end{lstlisting} +Et la figure \ref{Kochrecursif1} en donne l'interprétation graphique. + +\begin{figure*}[t] +\centering +\includegraphics[width=5cm]{kochrecursif1.eps} +\caption{Flocon de Koch récursif.\label{Kochrecursif1}} +\end{figure*} + +Le code est remarquable de concision, mais complexe à apréhender. + +\smallskip +Le c\oe ur de la récursivité se trouve dans le test de reconnaissance du caractère \verb|F|. Il s'agit du caractère non terminal et il est nécessaire de le remplacer par l'ensemble des caractères de la règle. Mais, au lieu de réaliser simplement ce remplacement, on renvoie à la fonction de production. Cela n'a pas pour effet de remplacer le caractère non terminal par les symboles de la règle, mais de renvoyer sur une procédure, en l'occurrence celle dans laquelle on se trouve, qui va le faire. Or, cette fonction va encore s'appeler elle-même à la lecture du caractère \verb|F| et ainsi de suite. + +Le même processus va ainsi se répéter un nombre de fois déterminé par la variable \verb|nbiteration|. Comme à chaque appel de la fonction de production on diminue la variable contenant le nombre d'itération, quand celle-ci devient nulle, on stoppe l'appel récursif et on trace un segment. + +Ensuite, on retourne à la dernière procédure appelée pour réaliser l'instruction qui suit l'appel récursif selon la règle donnée, par exemple une instruction terminale de rotation. Et on continue pour réaliser une suite d'instructions terminales ou si non terminales, pour réaliser un appel récursif à la fonction principale. + +Il est très difficile de suivre chaque étape de ce processus. Cependant, la figure \ref{appelrecursif} tente d'en donner une représentation graphique qui montre l'appel et le retour des processus dans l'ordre des flèches numérotées. + +\begin{figure*}[t] +\centering +\begin{psfrags} + \psfrag{production(...)}{production(...)} + \psfrag{if regle1}{if regle1 == 'F':} + \psfrag{production}{production} + \psfrag{de}{(\scriptsize nbiteration-1,longueur-1)} + \psfrag{if regle2}{if regle1 == '+'} + \psfrag{left}{left(angle)} + \psfrag{if nbiteration}{if nbiteration == 0:} + \psfrag{forward}{forward(longueur)} + \psfrag{1}{1} + \psfrag{2}{2} + \psfrag{3}{3} + \psfrag{4}{4} + \psfrag{5}{5} + \psfrag{6}{6} +\includegraphics[width=12cm]{production.eps} +\end{psfrags} +\caption{Flocon de Koch récursif.\label{appelrecursif}} +\end{figure*} + +On voit que toute une série de procédures sont progressivement lancées les unes dans les autres. Puis, arrivées à leur terme, le tracé des segment se fait et elles se terminent par un retour à la procédure qui les a lancées. Finalement, tout aussi progressivement, on termine chacune des procédures imbriquées. + +\medskip +On a déjà remarqué que le code est plus concis que celui issu d'une simple lecture-interprétation. Par contre, il est moins clair, demanderait donc pour la modification une relecture attentive qui peut prendre du temps et il faudrait se demander s'il est plus performant. + +\medskip +Voilà. Comme le but de ce cours n'est pas la programmation récursive, nous finirons là cette analyse en précisant cependant les limites de celle-ci. En effet, avec la récursivité on utilise toujours la même règle. L'axiome n'est donc pas utilisé ou considéré égal à \verb|F|. Certes, on peut envisager des moyens de contourner ce problème en réalisant l'axiome hors de la procédure récursive, en utilisant une récursivité indirecte ou en remplaçant à un moment bien précis la règle par l'axiome. Mais cela complique le code et l'avantage en concision de la récursivité s'amenuise. + +\section{Conclusion} +Nous avons parcouru avec cette introduction aux L-system beaucoup de notions d'informatiques importantes. Les langages, leur reconnaissance, les grammaires, l'algorithmique et finalement différents modes de programmation nous ont permis de comprendre le rôle essentiel joué par les structures formelles que l'informatique théorique met à notre disposition. Les limites de ce cours ne nous ont cependant pas permis d'aborder d'autres notions tout aussi importantes comme les réseaux de Pétri ou les machines de Turing. Par contre, l'étude des L-system marque clairement la puissance de la modélisation informatique des structures biologiques. Plus, elle fait entrevoir une correspondance fascinante entre le développement des organisme vivants et des constructions répétitives très structurées, finalement très bien décrites par \dots{} le langage mathématique. + +\Closesolutionfile{ans} % ferme le fichier Solutions.tex qui contient les solutions +\vfill +\pagebreak % met les solutions sur une autre page +\section{Solutions des exercices} +\Readsolutionfile{ans} % importe les données du fichier Solutions.tex + +\myclearpage + +%\appendix + +\section{Annexes} + +\subsection{Code python plante}\label{codepythonfonct} + +\begin{lstlisting}[language=Python] + # -*- coding: utf-8 -*- +from turtle import * + +# Generation de plante par grammaire L-system + +def initialisation(): + # Definition des variables independantes de la grammaire + global nbiteration, longueur + nbiteration = 5 + longueur = 3 + +def grammaire(): + # Definition de la grammaire + angle = 25.7 + axiom = 'F' + + regle = 'FF-[-F+F+F]+[+F-F-F]' + #regle = 'F[+F]F[-F][F]' + #regle = 'F[+F]F[-F]F' + #regle = 'F+FF-FF-F-F+F+FF-F-F+F+FF+FF-F' + + return angle, axiom, regle + +def production(axiom, regle): + # Generation de la chaine de caracteres + i = 0 + while i < nbiteration: + j = 0 + axiom2 = '' + while j < len(axiom): + pointeur1 = axiom[j] + + # Symboles non terminaux + if pointeur1 == 'F': + # remplacement de la regle dans l'axiome + axiom2 = axiom2 + regle + + # Symboles terminaux + if pointeur1 == '+': + axiom2 = axiom2 + '+' + if pointeur1 == '-': + axiom2 = axiom2 + '-' + if pointeur1 == '[': + axiom2 = axiom2 + '[' + if pointeur1 == ']': + axiom2 = axiom2 + ']' + + j = j + 1 + axiom = axiom2 + i = i + 1 + print axiom + return axiom + +def interpretation(angle, axiom): + # Interpretation de la chaine par turtle + k = 0 + laposition = [] + langle = [] + while k < len(axiom): + pointeur2 = axiom[k] + + # Interpreation + if pointeur2 == 'F': + forward(longueur) + elif pointeur2 == '+': + left(angle) + elif pointeur2 == '-': + right(angle) + elif pointeur2 == '[': + # memorisation de la position courante + laposition.append(position()) + langle.append(heading()) + elif pointeur2 == ']': + # retour a la derniere position et suppression de celle-ci + up() + goto(laposition[len(laposition)-1][0], laposition[len(laposition)-1][1]) + del laposition[len(laposition)-1] + setheading(langle[len(langle)-1]) + del langle[len(langle)-1] + down() + + k = k + 1 + +# Placement initial de la tortue +up() +backward(495) +down() + +# Programme principal +## gram[0] retourne l'angle +## gram[1] retourne l'axiome +## gram[2] retourne la regle + +initialisation() +gram = grammaire() + +axio = production(gram[1], gram[2]) +interpretation(gram[0], axio) + +#quit() +\end{lstlisting} + +\subsection{Code python plante à deux règles}\label{codepythonfonct2} + +\begin{lstlisting}[language=Python] +# -*- coding: utf-8 -*- +from turtle import * + +# Generation de plante par grammaire L-system + +def initialisation(): + # Definition des variables independantes de la grammaire + global nbiteration, longueur + # Pour le flocon de Koch + #nbiteration = 3 + #longueur = 2 + + # Pour le flocon de Koch Island + #nbiteration = 3 + #longueur = 2 + + # Pour la plante + #nbiteration = 5 + #longueur = 5 + + # Pour la plante 2 + #nbiteration = 5 + #longueur = 5 + + # Pour la plante 3 + #nbiteration = 4 + #longueur = 8 + + # Pour la plante 4 + #nbiteration = 7 + #longueur = 1 + + # Pour la plante 5 + #nbiteration = 7 + #longueur = 1 + + # Pour la plante 6 + nbiteration = 5 + longueur = 4 + +def grammaire(): + # Definition de la grammaire + # Pour le flocon de Koch + #angle = 90 + #axiom = 'F-F-F-F' + #regle = 'F+F-F-F+F' + + # Pour le flocon de Koch Island + #angle = 90 + #axiom = 'F-F-F-F' + #regle = 'F-F+F+FF-F-F+F' + + # Pour la plante + #angle = 25.7 + #axiom = 'F' + #regle = 'F[+F]F[-F]F' + + # Pour la plante 2 + #angle = 20 + #axiom = 'F' + #regle = 'F[+F]F[-F][F]' + + # Pour la plante 3 + #angle = 22.5 + #axiom = 'F' + #regle = 'FF-[-F+F+F]+[+F-F-F]' + + # Pour la plante 4 + #angle = 20 + #axiom = 'X' + #regle1 = 'F[+X]F[-X]+X' + #regle2 = 'FF' + + # Pour la plante 5 + angle = 25.7 + axiom = 'X' + regle1 = 'F[+X][-X]FX' + regle2 = 'FF' + + # Pour la plante 6 + angle = 22.5 + axiom = 'X' + regle1 = 'F-[[X]+X]+F[+FX]-X' + regle2 = 'FF' + + return angle, axiom, regle1, regle2 + +def production(axiom, regle1, regle2): + # Generation de la chaine de caracteres + i = 0 + while i < nbiteration: + j = 0 + axiom2 = '' + while j < len(axiom): + pointeur1 = axiom[j] + + # Symboles non terminaux + if pointeur1 == 'X': + # remplacement de la regle dans l'axiome + axiom2 = axiom2 + regle1 + + # Symboles terminaux + if pointeur1 == '+': + axiom2 = axiom2 + '+' + if pointeur1 == '-': + axiom2 = axiom2 + '-' + if pointeur1 == '[': + axiom2 = axiom2 + '[' + if pointeur1 == ']': + axiom2 = axiom2 + ']' + if pointeur1 == 'F': + axiom2 = axiom2 + 'F' + + j = j + 1 + axiom = axiom2 + print axiom + + m = 0 + axiom3 = '' + while m < len(axiom): + pointeur2 = axiom[m] + + # Symboles non terminaux + if pointeur2 == 'F': + # remplacement de la regle dans l'axiome + axiom3 = axiom3 + regle2 + + # Symboles terminaux + if pointeur2 == '+': + axiom3 = axiom3 + '+' + if pointeur2 == '-': + axiom3 = axiom3 + '-' + if pointeur2 == '[': + axiom3 = axiom3 + '[' + if pointeur2 == ']': + axiom3 = axiom3 + ']' + if pointeur2 == 'X': + axiom3 = axiom3 + 'X' + + m = m + 1 + axiom = axiom3 + i = i + 1 + + print axiom + return axiom + +def interpretation(angle, axiom): + # Interpretation de la chaine par turtle + k = 0 + laposition = [] + langle = [] + while k < len(axiom): + pointeur2 = axiom[k] + + # Interpreation + if pointeur2 == 'F': + forward(longueur) + elif pointeur2 == '+': + left(angle) + elif pointeur2 == '-': + right(angle) + elif pointeur2 == '[': + # memorisation de la position courante + laposition.append(position()) + langle.append(heading()) + elif pointeur2 == ']': + # retour a la derniere position et suppression de celle-ci + up() + goto(laposition[len(laposition)-1][0], laposition[len(laposition)-1][1]) + del laposition[len(laposition)-1] + setheading(langle[len(langle)-1]) + del langle[len(langle)-1] + down() + + k = k + 1 + +# Placement initial de la tortue +up() +backward(400) +down() + +# Programme principal +## gram[0] retourne l'angle +## gram[1] retourne l'axiome +## gram[2] retourne la regle + +initialisation() +gram = grammaire() +axio = production(gram[1], gram[2], gram[3]) +interpretation(gram[0], axio) + +#quit() + +\end{lstlisting} + +\subsection{Code python flocon de Koch récursif}\label{codepythonfloconrecursif} + +\begin{lstlisting}[language=Python] + # -*- coding: utf-8 -*- + from turtle import * + + # Generation de plante par grammaire L-system + + def initialisation(): + # Definition des variables independantes de la grammaire + # Pour le flocon de Koch + nbiteration = 3 + longueur = 200 + angle = 45 + axiome = 'F' + regle = 'F+F--F+F' + return nbiteration, longueur, angle, axiome, regle + + def production(nbiteration, longueur, angle, axiome, regle): + if nbiteration == 0 : + forward(longueur) + else: + i = 0 + while i < len(regle): + regle1 = regle[i] + if regle1 == 'F': + production(nbiteration-1, longueur/3,\ + angle, axiome, regle) + if regle1 == '+': + left(angle) + if regle1 == '-': + right(angle) + i = i + 1 + + # Placement initial de la tortue + up() + backward(200) + down() + + # Programme principal + ## init[1] : nbiteration + ## init[2] : longueur + ## init[3] : angle + ## init[4] : axiome + ## init[5] : regle + + init = initialisation() + production(init[0], init[1], init[2], init[3], init[4]) + right(120) + production(init[0], init[1], init[2], init[3], init[4]) + right(120) + production(init[0], init[1], init[2], init[3], init[4]) +\end{lstlisting} + +\myclearpage + +%\theendnotes + +\myclearpage + +%\renewcommand{\bibname}{Bibliographie} % pour mettre autre chose que bibliographie, par défaut avec babel +\nocite{CJ06,PP94,PP94,TO07,GP09,BD11} % pour mettre toutes les entrées dans la biblio, y compris celles non citées ds le texte +\bibliographystyle{apalike-fr} % le style de bibliographie apalike francisé +%\bibliographystyle{apalike} % le style de bibliographie +\bibliography{../../Bibliographies/BiblioCours} % définit le fichier des réf. biblio. nom.bib + + % le logiciel utilisé pour faire la biblio est kbibtex +%\printindex{} % attention, pour que l'index soit mis à jour il faut lancer la commande de konsole : makeindex -s ../Perso.ist nom_du_cours.idx + % le Perso.ist est un fichier disant qu'il faut mettre des lettres entre les parties de l'index + +%\renewcommand{\listfigurename}{Liste des figures} +%\listoffigures % ai tout basculé après la table des matières car je n'arrivais pas à éliminer le petit bug ci-dessous. Mais le compagnon latex fait idem. +%\listoftables % un petit problème (bug ?) d'alignement m'a fait retirer provisoirement cette liste + +\end{document} +\endinput % termine l'importation des solutions \ No newline at end of file diff --git a/Lsystem.toc b/Lsystem.toc new file mode 100644 index 0000000..b00b9cf --- /dev/null +++ b/Lsystem.toc @@ -0,0 +1,34 @@ +\babel@toc {french}{} +\contentsline {section}{Copyright}{2}% +\contentsline {section}{Avertissement}{3}% +\contentsline {section}{Table des mati\IeC {\`e}res}{3}% +\contentsline {section}{\numberline {1}Introduction}{4}% +\contentsline {section}{\numberline {2}Langage formel}{4}% +\contentsline {section}{\numberline {3}Langage rationnel}{6}% +\contentsline {subsection}{\numberline {3.1}Les automates}{6}% +\contentsline {subsection}{\numberline {3.2}Les expressions rationnelles}{8}% +\contentsline {section}{\numberline {4}Langage de Lindenmayer}{10}% +\contentsline {subsection}{\numberline {4.1}Grammaire}{10}% +\contentsline {subsection}{\numberline {4.2}G\IeC {\'e}n\IeC {\'e}ration de s\IeC {\'e}quence}{11}% +\contentsline {subsection}{\numberline {4.3}L-system}{12}% +\babel@toc {english}{} +\babel@toc {french}{} +\babel@toc {english}{} +\babel@toc {french}{} +\contentsline {subsection}{\numberline {4.4}Interpr\IeC {\'e}tation graphique}{13}% +\contentsline {subsubsection}{\numberline {4.4.1}Koch}{13}% +\contentsline {subsubsection}{\numberline {4.4.2}Mod\IeC {\'e}lisation des plantes}{14}% +\contentsline {subsection}{\numberline {4.5}Algorithmique}{18}% +\contentsline {subsubsection}{\numberline {4.5.1}Cr\IeC {\'e}ation}{18}% +\contentsline {subsubsection}{\numberline {4.5.2}Interpr\IeC {\'e}tation}{19}% +\contentsline {subsection}{\numberline {4.6}Programmation}{19}% +\contentsline {subsubsection}{\numberline {4.6.1}Approche non r\IeC {\'e}cursive}{19}% +\contentsline {paragraph}{La cr\IeC {\'e}ation de la cha\IeC {\^\i }ne}{20}% +\contentsline {paragraph}{L'interpr\IeC {\'e}tation de la cha\IeC {\^\i }ne}{21}% +\contentsline {subsubsection}{\numberline {4.6.2}Approche r\IeC {\'e}cursive}{22}% +\contentsline {section}{\numberline {5}Conclusion}{24}% +\contentsline {section}{\numberline {6}Solutions des exercices}{25}% +\contentsline {section}{\numberline {7}Annexes}{27}% +\contentsline {subsection}{\numberline {7.1}Code python plante}{27}% +\contentsline {subsection}{\numberline {7.2}Code python plante \IeC {\`a} deux r\IeC {\`e}gles}{29}% +\contentsline {subsection}{\numberline {7.3}Code python flocon de Koch r\IeC {\'e}cursif}{34}% diff --git a/Preambule.tex b/Preambule.tex new file mode 100644 index 0000000..90a8902 --- /dev/null +++ b/Preambule.tex @@ -0,0 +1,91 @@ +% Ce fichier constitue le préambule des différents cours de physique. Les définitions de modules propres à chaque cours se trouvent dans le préambule de ces cours. On trouve ici les définitions des modules communs. +% Ce fichier est appelé par chaque cours à travers l'instruction \input{../Preambule.tex} +% Attention, les modules précédés de trois commentaires : %%% sont en conflit avec le module vaucanson-g. C'est pourquoi il ont été retirés. + +\usepackage[T1]{fontenc} % écrire avec les accents +\usepackage[utf8]{inputenc} % écrire avec les accents +%\usepackage{babel} +\usepackage[english,french]{babel} % pour le support des documents multilingues. Il vaut mieux mettre french là plutôt que dans le documentclass + % pour pouvoir gérer l'anglais ou autres langues. +%\usepackage{lmodern} +%\usepackage{geometry} % pour gérer les marges de la page +%\geometry{verbose,a4paper,lmargin=2cm,rmargin=2cm} % règle les marges de la page + +\setcounter{tocdepth}{5} % pour définir la profondeur de la table des matières +\usepackage{makeidx} % pour permettre de construire un index +\makeindex % pour construire effectivement l'index + +\makeatletter % pour faire de @ une lettre simple (et non un caractère associé à une macro interne +\usepackage{path} % pour mettre du code non interprété +\usepackage{fancybox} % pour faire des boites entourée de différents types de cadres +\usepackage{framed} % pour mettre des sortes de minipages encadrées sur plusieures pages, sur fond coloré +\usepackage[dvips]{color} % pour mettre du gris ou de la couleur dans les boites +%%%\usepackage{pst-grad} % pour faire des dégradés en pstricks (! il faut le mettre juste ici pour que cela fonctionne !) +\definecolor{shadecolor}{rgb}{.9,.9,.9} % pour définir la couleur de fond du package framed +\usepackage{boxedminipage} % pour mettre des bordures aux minipages +\usepackage{relsize,fancyvrb} % pour mettre des textes non interprétés encadrés et changer la taille des caractères à l'intérieur +\usepackage{array} % pour étendre les possiblités de tabular par défaut pour les tableaux +\usepackage{multirow} % pour faire une cellule de tableau sur plusieurs lignes +\usepackage{colortbl} % pour mettre de la couleur dans un tableau +\usepackage{hhline} % pour faire des filets spéciaux dans les tableaux +\usepackage{supertabular} % pour faire des tableaux sur plusieurs pages +\usepackage{float} % permet un placement obligatoire +\usepackage{url} % permet une bonne gestion de la césure des url +\usepackage{graphicx} % pour charger le module étendu graphicx du module graphics. Pour différents effets graphiques dont l'orientation et le redimentionnement des eps +%\usepackage{graphics} % idem que graphicx, mais sans orientation et redimentionnement +\usepackage{wrapfig} % pour mettre des figures dans le flot du texte +\usepackage{psfrag} % pour mettre des équations latex dans des figures (ne fonctionne qu'avec ps) +%%%\usepackage{pstricks} % pour étendre les possiblités de graphisme des figures de latex construite avec jpicedt +%\usepackage{pstricks-add} % plein de bug semble-t-il, mais aurait été utile +%\usepackage{pstcol} +\usepackage{multido} % pour répéter des figures en pstricks +%\usepackage[figuresright ou figuresleft]{rotating} % permet de faire tourner des figures \sidewaysfigure ou des textes \rotatebox dans un sens ou un autre +\usepackage{rotating} % permet de faire tourner des figures \sidewaysfigure ou des textes \rotatebox +\usepackage{subfigure} % pour faire des figures comprenant plusieures figures +\usepackage{fancyhdr} % pour des entêtes et pieds de page entièrement personnalisés +\usepackage{fancyvrb} % pour mettre des \verb dans les minipages +\usepackage[Lenny]{fncychap} % pour des titres de chapitres différents ... +\usepackage{multicol} % pour mettre un \begin{multicols}...\end{multicols} et faire du multicolonnage localement dans une page +\usepackage{amsmath} % pour avoir beaucoup de symboles mathématiques et des alignements complexes +\usepackage{amssymb} % nécessaire pour avoir certains symboles mathématiques +\usepackage{amsfonts} % nécessaire pour avoir certains symboles mathématiques +%\usepackage{chappg} % numérotation par chapitre +\usepackage{epsfig} % pour inclure des fichiers .eps. Mais il semble que les modules graphics ou graphicx devraient le remplacer +\makeatother % pour refaire de @ une lettre différente des autres (exploitée dans des macros) +\usepackage{caption} % pour gérer encore mieux les légendes +\renewcommand{\captionfont}{\it \small} % pour avoir de l'italique pour le texte de légende +\renewcommand{\captionlabelfont}{\it \bf \small} +\DeclareCaptionLabelSeparator{endash}{ -- } % pour définir un tiret après le numéro des légentes +\captionsetup{labelsep=endash,justification=centering,belowskip=10pt} % pour avoir un petit trait après le no de légende +\usepackage{verbatim} % pour faire des commentaires longs \begin{comment} +\usepackage{endnotes} % pour faire des notes de fin +%\renewcommand{\theendnote}{\Roman{endnote}} % pour mettre les notes de fin en romain +\renewcommand{\thefootnote}{\alph{footnote}} % pour mettre les notes de bas de page en alphabétique +%\usepackage{apalike} % nécessaire pour que le style de bibliographie apalike soit correctement appliqué + % attention, le module apalike ne doit être utilisé qu'en anglais. Le titre bibliography est codé en dur à l'intérieur du module +\usepackage{natbib} % à utiliser impérativement avec apalike-fr pour parfaire les entrées +%\usepackage{textcomp} % pour mettre de beaux degrés +\usepackage{textcomp} % pour le symbole pourmille : \textperthousand (! pas en mode math) et pour le symbole de copyleft \textcopyleft +%%%\usepackage{boites,boites_exemples} % pour faire des paragraphes encadrés sur plusieurs colonnes +\usepackage{textpos} % pour positionner du texte ou des images à une place très précise de la page. +%%\usepackage{floatflt} % pour placer des tableaux entourés de texte +\usepackage{nextpage} % pour retirer les entêtes de la dernière page vide d'un chapitre ; +\newcommand\myclearpage{\cleartooddpage[\thispagestyle{plain}]} % il faut ensuite placer \myclearpage à la fin de chaque chapitre. +\def\siecle#1{\textsc{\romannumeral #1}\textsuperscript{e}~siècle} % macro pour écrire les siècles : \siecle{19} par exemple + +%\usepackage{showidx} % pour voir les entrées d'index dans la marge. ATTENTION : commenter avant la compilation. + +%\usepackage{aeguill} % fontes utilisées par pdflatex (pour un pdf avec liens cliquables ; pas encore testé) +%\usepackage{hyperref} % pour faire des hyperréférences (pas encore testé) +%\hypersetup{colorlinks,citecolor=black,filecolor=black,linkcolor=black,urlcolor=black,pdftex} % pour définir la couleur des liens en noir, pour une meilleure impression (pas encore testé) +%\usepackage{unitsdef} % pour l'écriture des unités +%%\usepackage[derivedinbase,squaren,cdot,Gray]{SIunits} % pour utiliser des commandes relatives au unités du système internationnal ; il y a aussi les préfixes m M G ... + % le squaren est là pour éviter une incompatibilité avec le symbole ams de la racine. + % le derivedinbase est là pour fournir une traduction des unités SI dérivées en unités SI de base +%\usepackage{hepunits} % unités telles que Hz, eV, ... +%\usepackage{siunitx} % meilleure gestion des unités que siunits, mais incompatible avec celui-ci ! +%\usepackage[fixlanguage]{babelbib} % pour avoir une bibliographie en français ou différentes entrées de langues différentes (pour cela il faut renseigner un champ de langue) +%\selectbiblanguage{french} % sélection de la langue (attention, pour apalike, il faut sélectionner apalike-fr) +\usepackage{lettrine} % pour mettre des lettrines +\usepackage{listings} % pour mettre du code informatique non interprété +\usepackage{xspace} % pour gérer les expaces nott. entre X et e pour les siècles \ No newline at end of file diff --git a/Preambule.tex.backup b/Preambule.tex.backup new file mode 100644 index 0000000..eff5a4d --- /dev/null +++ b/Preambule.tex.backup @@ -0,0 +1,90 @@ +% Ce fichier constitue le préambule des différents cours de physique. Les définitions de modules propres à chaque cours se trouvent dans le préambule de ces cours. On trouve ici les définitions des modules communs. +% Ce fichier est appelé par chaque cours à travers l'instruction \input{../Preambule.tex} + +\usepackage[T1]{fontenc} % écrire avec les accents +\usepackage[utf8]{inputenc} % écrire avec les accents +%\usepackage{babel} +\usepackage[english,french]{babel} % pour le support des documents multilingues. Il vaut mieux mettre french là plutôt que dans le documentclass + % pour pouvoir gérer l'anglais ou autres langues. +%\usepackage{lmodern} +%\usepackage{geometry} % pour gérer les marges de la page +%\geometry{verbose,a4paper,lmargin=2cm,rmargin=2cm} % règle les marges de la page + +\setcounter{tocdepth}{5} % pour définir la profondeur de la table des matières +\usepackage{makeidx} % pour permettre de construire un index +\makeindex % pour construire effectivement l'index + +\makeatletter % pour faire de @ une lettre simple (et non un caractère associé à une macro interne +\usepackage{path} % pour mettre du code non interprété +\usepackage{fancybox} % pour faire des boites entourée de différents types de cadres +\usepackage{framed} % pour mettre des sortes de minipages encadrées sur plusieures pages, sur fond coloré +\usepackage[dvips]{color} % pour mettre du gris ou de la couleur dans les boites +%\usepackage{pst-grad} % pour faire des dégradés en pstricks (! il faut le mettre juste ici pour que cela fonctionne !) +\definecolor{shadecolor}{rgb}{.9,.9,.9} % pour définir la couleur de fond du package framed +\usepackage{boxedminipage} % pour mettre des bordures aux minipages +\usepackage{relsize,fancyvrb} % pour mettre des textes non interprétés encadrés et changer la taille des caractères à l'intérieur +\usepackage{array} % pour étendre les possiblités de tabular par défaut pour les tableaux +\usepackage{multirow} % pour faire une cellule de tableau sur plusieurs lignes +\usepackage{colortbl} % pour mettre de la couleur dans un tableau +\usepackage{hhline} % pour faire des filets spéciaux dans les tableaux +\usepackage{supertabular} % pour faire des tableaux sur plusieurs pages +\usepackage{float} % permet un placement obligatoire +\usepackage{url} % permet une bonne gestion de la césure des url +\usepackage{graphicx} % pour charger le module étendu graphicx du module graphics. Pour différents effets graphiques dont l'orientation et le redimentionnement des eps +%\usepackage{graphics} % idem que graphicx, mais sans orientation et redimentionnement +\usepackage{wrapfig} % pour mettre des figures dans le flot du texte +\usepackage{psfrag} % pour mettre des équations latex dans des figures (ne fonctionne qu'avec ps) +\usepackage{pstricks} % pour étendre les possiblités de graphisme des figures de latex construite avec jpicedt +%\usepackage{pstricks-add} % plein de bug semble-t-il, mais aurait été utile +%\usepackage{pstcol} +\usepackage{multido} % pour répéter des figures en pstricks +%\usepackage[figuresright ou figuresleft]{rotating} % permet de faire tourner des figures \sidewaysfigure ou des textes \rotatebox dans un sens ou un autre +\usepackage{rotating} % permet de faire tourner des figures \sidewaysfigure ou des textes \rotatebox +\usepackage{subfigure} % pour faire des figures comprenant plusieures figures +\usepackage{fancyhdr} % pour des entêtes et pieds de page entièrement personnalisés +\usepackage{fancyvrb} % pour mettre des \verb dans les minipages +\usepackage[Lenny]{fncychap} % pour des titres de chapitres différents ... +\usepackage{multicol} % pour mettre un \begin{multicols}...\end{multicols} et faire du multicolonnage localement dans une page +\usepackage{amsmath} % pour avoir beaucoup de symboles mathématiques et des alignements complexes +\usepackage{amssymb} % nécessaire pour avoir certains symboles mathématiques +\usepackage{amsfonts} % nécessaire pour avoir certains symboles mathématiques +%\usepackage{chappg} % numérotation par chapitre +\usepackage{epsfig} % pour inclure des fichiers .eps. Mais il semble que les modules graphics ou graphicx devraient le remplacer +\makeatother % pour refaire de @ une lettre différente des autres (exploitée dans des macros) +\usepackage{caption} % pour gérer encore mieux les légendes +\renewcommand{\captionfont}{\it \small} % pour avoir de l'italique pour le texte de légende +\renewcommand{\captionlabelfont}{\it \bf \small} +\DeclareCaptionLabelSeparator{endash}{ -- } % pour définir un tiret après le numéro des légentes +\captionsetup{labelsep=endash,justification=centering,belowskip=10pt} % pour avoir un petit trait après le no de légende +\usepackage{verbatim} % pour faire des commentaires longs \begin{comment} +\usepackage{endnotes} % pour faire des notes de fin +%\renewcommand{\theendnote}{\Roman{endnote}} % pour mettre les notes de fin en romain +\renewcommand{\thefootnote}{\alph{footnote}} % pour mettre les notes de bas de page en alphabétique +%\usepackage{apalike} % nécessaire pour que le style de bibliographie apalike soit correctement appliqué + % attention, le module apalike ne doit être utilisé qu'en anglais. Le titre bibliography est codé en dur à l'intérieur du module +\usepackage{natbib} % à utiliser impérativement avec apalike-fr pour parfaire les entrées +%\usepackage{textcomp} % pour mettre de beaux degrés +\usepackage{textcomp} % pour le symbole pourmille : \textperthousand (! pas en mode math) et pour le symbole de copyleft \textcopyleft +\usepackage{boites,boites_exemples} % pour faire des paragraphes encadrés sur plusieurs colonnes +\usepackage{textpos} % pour positionner du texte ou des images à une place très précise de la page. +\usepackage{floatflt} % pour placer des tableaux entourés de texte +\usepackage{nextpage} % pour retirer les entêtes de la dernière page vide d'un chapitre ; +\newcommand\myclearpage{\cleartooddpage[\thispagestyle{plain}]} % il faut ensuite placer \myclearpage à la fin de chaque chapitre. +\def\siecle#1{\textsc{\romannumeral #1}\textsuperscript{e}~siècle} % macro pour écrire les siècles : \siecle{19} par exemple + +%\usepackage{showidx} % pour voir les entrées d'index dans la marge. ATTENTION : commenter avant la compilation. + +%\usepackage{aeguill} % fontes utilisées par pdflatex (pour un pdf avec liens cliquables ; pas encore testé) +%\usepackage{hyperref} % pour faire des hyperréférences (pas encore testé) +%\hypersetup{colorlinks,citecolor=black,filecolor=black,linkcolor=black,urlcolor=black,pdftex} % pour définir la couleur des liens en noir, pour une meilleure impression (pas encore testé) +%\usepackage{unitsdef} % pour l'écriture des unités +\usepackage[derivedinbase,squaren,cdot,Gray]{SIunits} % pour utiliser des commandes relatives au unités du système internationnal ; il y a aussi les préfixes m M G ... + % le squaren est là pour éviter une incompatibilité avec le symbole ams de la racine. + % le derivedinbase est là pour fournir une traduction des unités SI dérivées en unités SI de base +%\usepackage{hepunits} % unités telles que Hz, eV, ... +%\usepackage{siunitx} % meilleure gestion des unités que siunits, mais incompatible avec celui-ci ! +%\usepackage[fixlanguage]{babelbib} % pour avoir une bibliographie en français ou différentes entrées de langues différentes (pour cela il faut renseigner un champ de langue) +%\selectbiblanguage{french} % sélection de la langue (attention, pour apalike, il faut sélectionner apalike-fr) +\usepackage{lettrine} % pour mettre des lettrines +\usepackage{listings} % pour mettre du code informatique non interprété +\usepackage{xspace} % pour gérer les expaces nott. entre X et e pour les siècles \ No newline at end of file diff --git a/Preambule.tex~ b/Preambule.tex~ new file mode 100644 index 0000000..56f032a --- /dev/null +++ b/Preambule.tex~ @@ -0,0 +1,91 @@ +% Ce fichier constitue le préambule des différents cours de physique. Les définitions de modules propres à chaque cours se trouvent dans le préambule de ces cours. On trouve ici les définitions des modules communs. +% Ce fichier est appelé par chaque cours à travers l'instruction \input{../Preambule.tex} +% Attention, les modules précédés de trois commentaires : %%% sont en conflit avec le module vaucanson-g. C'est pourquoi il ont été retirés. + +\usepackage[T1]{fontenc} % écrire avec les accents +\usepackage[utf8]{inputenc} % écrire avec les accents +%\usepackage{babel} +\usepackage[english,french]{babel} % pour le support des documents multilingues. Il vaut mieux mettre french là plutôt que dans le documentclass + % pour pouvoir gérer l'anglais ou autres langues. +%\usepackage{lmodern} +%\usepackage{geometry} % pour gérer les marges de la page +%\geometry{verbose,a4paper,lmargin=2cm,rmargin=2cm} % règle les marges de la page + +\setcounter{tocdepth}{5} % pour définir la profondeur de la table des matières +\usepackage{makeidx} % pour permettre de construire un index +\makeindex % pour construire effectivement l'index + +\makeatletter % pour faire de @ une lettre simple (et non un caractère associé à une macro interne +\usepackage{path} % pour mettre du code non interprété +\usepackage{fancybox} % pour faire des boites entourée de différents types de cadres +\usepackage{framed} % pour mettre des sortes de minipages encadrées sur plusieures pages, sur fond coloré +\usepackage[dvips]{color} % pour mettre du gris ou de la couleur dans les boites +%%%\usepackage{pst-grad} % pour faire des dégradés en pstricks (! il faut le mettre juste ici pour que cela fonctionne !) +\definecolor{shadecolor}{rgb}{.9,.9,.9} % pour définir la couleur de fond du package framed +\usepackage{boxedminipage} % pour mettre des bordures aux minipages +\usepackage{relsize,fancyvrb} % pour mettre des textes non interprétés encadrés et changer la taille des caractères à l'intérieur +\usepackage{array} % pour étendre les possiblités de tabular par défaut pour les tableaux +\usepackage{multirow} % pour faire une cellule de tableau sur plusieurs lignes +\usepackage{colortbl} % pour mettre de la couleur dans un tableau +\usepackage{hhline} % pour faire des filets spéciaux dans les tableaux +\usepackage{supertabular} % pour faire des tableaux sur plusieurs pages +\usepackage{float} % permet un placement obligatoire +\usepackage{url} % permet une bonne gestion de la césure des url +\usepackage{graphicx} % pour charger le module étendu graphicx du module graphics. Pour différents effets graphiques dont l'orientation et le redimentionnement des eps +%\usepackage{graphics} % idem que graphicx, mais sans orientation et redimentionnement +\usepackage{wrapfig} % pour mettre des figures dans le flot du texte +\usepackage{psfrag} % pour mettre des équations latex dans des figures (ne fonctionne qu'avec ps) +%%%\usepackage{pstricks} % pour étendre les possiblités de graphisme des figures de latex construite avec jpicedt +%\usepackage{pstricks-add} % plein de bug semble-t-il, mais aurait été utile +%\usepackage{pstcol} +\usepackage{multido} % pour répéter des figures en pstricks +%\usepackage[figuresright ou figuresleft]{rotating} % permet de faire tourner des figures \sidewaysfigure ou des textes \rotatebox dans un sens ou un autre +\usepackage{rotating} % permet de faire tourner des figures \sidewaysfigure ou des textes \rotatebox +\usepackage{subfigure} % pour faire des figures comprenant plusieures figures +\usepackage{fancyhdr} % pour des entêtes et pieds de page entièrement personnalisés +\usepackage{fancyvrb} % pour mettre des \verb dans les minipages +\usepackage[Lenny]{fncychap} % pour des titres de chapitres différents ... +\usepackage{multicol} % pour mettre un \begin{multicols}...\end{multicols} et faire du multicolonnage localement dans une page +\usepackage{amsmath} % pour avoir beaucoup de symboles mathématiques et des alignements complexes +\usepackage{amssymb} % nécessaire pour avoir certains symboles mathématiques +\usepackage{amsfonts} % nécessaire pour avoir certains symboles mathématiques +%\usepackage{chappg} % numérotation par chapitre +\usepackage{epsfig} % pour inclure des fichiers .eps. Mais il semble que les modules graphics ou graphicx devraient le remplacer +\makeatother % pour refaire de @ une lettre différente des autres (exploitée dans des macros) +\usepackage{caption} % pour gérer encore mieux les légendes +\renewcommand{\captionfont}{\it \small} % pour avoir de l'italique pour le texte de légende +\renewcommand{\captionlabelfont}{\it \bf \small} +\DeclareCaptionLabelSeparator{endash}{ -- } % pour définir un tiret après le numéro des légentes +\captionsetup{labelsep=endash,justification=centering,belowskip=10pt} % pour avoir un petit trait après le no de légende +\usepackage{verbatim} % pour faire des commentaires longs \begin{comment} +\usepackage{endnotes} % pour faire des notes de fin +%\renewcommand{\theendnote}{\Roman{endnote}} % pour mettre les notes de fin en romain +\renewcommand{\thefootnote}{\alph{footnote}} % pour mettre les notes de bas de page en alphabétique +%\usepackage{apalike} % nécessaire pour que le style de bibliographie apalike soit correctement appliqué + % attention, le module apalike ne doit être utilisé qu'en anglais. Le titre bibliography est codé en dur à l'intérieur du module +\usepackage{natbib} % à utiliser impérativement avec apalike-fr pour parfaire les entrées +%\usepackage{textcomp} % pour mettre de beaux degrés +\usepackage{textcomp} % pour le symbole pourmille : \textperthousand (! pas en mode math) et pour le symbole de copyleft \textcopyleft +%%%\usepackage{boites,boites_exemples} % pour faire des paragraphes encadrés sur plusieurs colonnes +\usepackage{textpos} % pour positionner du texte ou des images à une place très précise de la page. +%%\usepackage{floatflt} % pour placer des tableaux entourés de texte +\usepackage{nextpage} % pour retirer les entêtes de la dernière page vide d'un chapitre ; +\newcommand\myclearpage{\cleartooddpage[\thispagestyle{plain}]} % il faut ensuite placer \myclearpage à la fin de chaque chapitre. +\def\siecle#1{\textsc{\romannumeral #1}\textsuperscript{e}~siècle} % macro pour écrire les siècles : \siecle{19} par exemple + +%\usepackage{showidx} % pour voir les entrées d'index dans la marge. ATTENTION : commenter avant la compilation. + +%\usepackage{aeguill} % fontes utilisées par pdflatex (pour un pdf avec liens cliquables ; pas encore testé) +%\usepackage{hyperref} % pour faire des hyperréférences (pas encore testé) +%\hypersetup{colorlinks,citecolor=black,filecolor=black,linkcolor=black,urlcolor=black,pdftex} % pour définir la couleur des liens en noir, pour une meilleure impression (pas encore testé) +%\usepackage{unitsdef} % pour l'écriture des unités +\usepackage[derivedinbase,squaren,cdot,Gray]{SIunits} % pour utiliser des commandes relatives au unités du système internationnal ; il y a aussi les préfixes m M G ... + % le squaren est là pour éviter une incompatibilité avec le symbole ams de la racine. + % le derivedinbase est là pour fournir une traduction des unités SI dérivées en unités SI de base +%\usepackage{hepunits} % unités telles que Hz, eV, ... +%\usepackage{siunitx} % meilleure gestion des unités que siunits, mais incompatible avec celui-ci ! +%\usepackage[fixlanguage]{babelbib} % pour avoir une bibliographie en français ou différentes entrées de langues différentes (pour cela il faut renseigner un champ de langue) +%\selectbiblanguage{french} % sélection de la langue (attention, pour apalike, il faut sélectionner apalike-fr) +\usepackage{lettrine} % pour mettre des lettrines +\usepackage{listings} % pour mettre du code informatique non interprété +\usepackage{xspace} % pour gérer les expaces nott. entre X et e pour les siècles \ No newline at end of file diff --git a/Python/Arbre1.py b/Python/Arbre1.py new file mode 100644 index 0000000..f81c45d --- /dev/null +++ b/Python/Arbre1.py @@ -0,0 +1,65 @@ +# -*- coding: utf-8 -*- +from turtle import * + +# Génération de plante + +def regle(): + nbiteration = 5 + longueur = 3 + angle = 25.7 + regle = 'F[+F]F[-F]F' + #regle = 'F+FF-FF-F-F+F+FF-F-F+F+FF+FF-F' + axiom = 'F' + i = 0 + while i < nbiteration: + j = 0 + axiom2 = '' + while j < len(axiom): + pointeur1 = axiom[j] + if pointeur1 == 'F': + axiom2 = axiom2 + regle + if pointeur1 == '+': + axiom2 = axiom2 + '+' + if pointeur1 == '-': + axiom2 = axiom2 + '-' + if pointeur1 == '[': + axiom2 = axiom2 + '[' + if pointeur1 == ']': + axiom2 = axiom2 + ']' + j = j + 1 + axiom = axiom2 + i = i + 1 + print axiom + k = 0 + laposition = [] + langle = [] + while k < len(axiom): + pointeur2 = axiom[k] + if pointeur2 == 'F': + forward(longueur) + elif pointeur2 == '+': + left(angle) + elif pointeur2 == '-': + right(angle) + elif pointeur2 == '[': + laposition.append(position()) + langle.append(heading()) + elif pointeur2 == ']': + goto(laposition[len(laposition)-1][0], laposition[len(laposition)-1][1]) + del laposition[len(laposition)-1] + setheading(langle[len(langle)-1]) + del langle[len(langle)-1] + k = k + 1 + +up() +#left(180) +#forward(400) +#left(90) +#forward(200) +#left(90) +#down() +backward(495) +down() +regle() + +#quit() diff --git a/Python/Arbre2.py b/Python/Arbre2.py new file mode 100644 index 0000000..3439800 --- /dev/null +++ b/Python/Arbre2.py @@ -0,0 +1,65 @@ +# -*- coding: utf-8 -*- +from turtle import * + +# La courbe de KochIsland + +def regle(): + nbiteration = 5 + longueur = 6 + angle = 20 + regle = 'F[+F]F[-F][F]' + #regle = 'F+FF-FF-F-F+F+FF-F-F+F+FF+FF-F' + axiom = 'F' + i = 0 + while i < nbiteration: + j = 0 + axiom2 = '' + while j < len(axiom): + pointeur1 = axiom[j] + if pointeur1 == 'F': + axiom2 = axiom2 + regle + if pointeur1 == '+': + axiom2 = axiom2 + '+' + if pointeur1 == '-': + axiom2 = axiom2 + '-' + if pointeur1 == '[': + axiom2 = axiom2 + '[' + if pointeur1 == ']': + axiom2 = axiom2 + ']' + j = j + 1 + axiom = axiom2 + i = i + 1 + print axiom + k = 0 + laposition = [] + langle = [] + while k < len(axiom): + pointeur2 = axiom[k] + if pointeur2 == 'F': + forward(longueur) + elif pointeur2 == '+': + left(angle) + elif pointeur2 == '-': + right(angle) + elif pointeur2 == '[': + laposition.append(position()) + langle.append(heading()) + elif pointeur2 == ']': + goto(laposition[len(laposition)-1][0], laposition[len(laposition)-1][1]) + del laposition[len(laposition)-1] + setheading(langle[len(langle)-1]) + del langle[len(langle)-1] + k = k + 1 + +up() +#left(180) +#forward(400) +#left(90) +#forward(200) +#left(90) +#down() +backward(495) +down() +regle() + +#quit() diff --git a/Python/ArbreFonctionnel.py b/Python/ArbreFonctionnel.py new file mode 100644 index 0000000..941b213 --- /dev/null +++ b/Python/ArbreFonctionnel.py @@ -0,0 +1,93 @@ +# -*- coding: utf-8 -*- +from turtle import * + +# Génération de plante par grammaire L-system + +def initialisation(): + # Définition des variables indépendantes de la grammaire + global nbiteration, longueur + nbiteration = 5 + longueur = 3 + +def grammaire(): + # Définition de la grammaire + angle = 25.7 + axiom = 'F' + regle = 'F[+F]F[-F]F' + #regle = 'F+FF-FF-F-F+F+FF-F-F+F+FF+FF-F' + return angle, axiom, regle + +def production(axiom, regle): + # Génération de la chaine de caractères + i = 0 + while i < nbiteration: + j = 0 + axiom2 = '' + while j < len(axiom): + pointeur1 = axiom[j] + + # Symboles non terminaux + if pointeur1 == 'F': + # remplacement de la règle dans l'axiome + axiom2 = axiom2 + regle + + # Symboles terminaux + if pointeur1 == '+': + axiom2 = axiom2 + '+' + if pointeur1 == '-': + axiom2 = axiom2 + '-' + if pointeur1 == '[': + axiom2 = axiom2 + '[' + if pointeur1 == ']': + axiom2 = axiom2 + ']' + + j = j + 1 + axiom = axiom2 + i = i + 1 + print axiom + return axiom + +def interpretation(angle, axiom): + # Interprétation de la chaine par turtle + k = 0 + laposition = [] + langle = [] + while k < len(axiom): + pointeur2 = axiom[k] + + # Interpréation + if pointeur2 == 'F': + forward(longueur) + elif pointeur2 == '+': + left(angle) + elif pointeur2 == '-': + right(angle) + elif pointeur2 == '[': + # mémorisation de la position courante + laposition.append(position()) + langle.append(heading()) + elif pointeur2 == ']': + # retour à la dernière position et suppression de celle-ci + goto(laposition[len(laposition)-1][0], laposition[len(laposition)-1][1]) + del laposition[len(laposition)-1] + setheading(langle[len(langle)-1]) + del langle[len(langle)-1] + + k = k + 1 + +# Placement initial de la tortue +up() +backward(400) +down() + +# Programme principal +## gram[0] retourne l'angle +## gram[1] retourne l'axiome +## gram[2] retourne la règle + +initialisation() +gram = grammaire() +axio = production(gram[1], gram[2]) +interpretation(gram[0], axio) + +#quit() diff --git a/Python/ArbreFonctionnel2.py b/Python/ArbreFonctionnel2.py new file mode 100644 index 0000000..f50d7b9 --- /dev/null +++ b/Python/ArbreFonctionnel2.py @@ -0,0 +1,132 @@ +# -*- coding: utf-8 -*- +from turtle import * + +# Génération de plante par grammaire L-system + +def initialisation(): + # Définition des variables indépendantes de la grammaire + global nbiteration, longueur + # Pour le flocon de Koch + nbiteration = 3 + longueur = 2 + + # Pour le flocon de Koch Island + #nbiteration = 3 + #longueur = 2 + + # Pour la plante + #nbiteration = 5 + #longueur = 5 + + # Pour la plante 2 + #nbiteration = 5 + #longueur = 5 + + # Pour la plante 3 + #nbiteration = 5 + #longueur = 3 + +def grammaire(): + # Définition de la grammaire + # Pour le flocon de Koch + angle = 90 + axiom = 'F-F-F-F' + regle = 'F+F-F-F+F' + + # Pour le flocon de Koch Island + #angle = 90 + #axiom = 'F-F-F-F' + #regle = 'F-F+F+FF-F-F+F' + + # Pour la plante + #angle = 25.7 + #axiom = 'F' + #regle = 'F[+F]F[-F]F' + + # Pour la plante 2 + #angle = 20 + #axiom = 'F' + #regle = 'F[+F]F[-F][F]' + + # Pour la plante 3 + #angle = 22.5 + #axiom = 'F' + #regle = 'FF-[-F+F+F]+[+F-F-F]' + return angle, axiom, regle + +def production(axiom, regle): + # Génération de la chaine de caractères + i = 0 + while i < nbiteration: + j = 0 + axiom2 = '' + while j < len(axiom): + pointeur1 = axiom[j] + + # Symboles non terminaux + if pointeur1 == 'F': + # remplacement de la règle dans l'axiome + axiom2 = axiom2 + regle + + # Symboles terminaux + if pointeur1 == '+': + axiom2 = axiom2 + '+' + if pointeur1 == '-': + axiom2 = axiom2 + '-' + if pointeur1 == '[': + axiom2 = axiom2 + '[' + if pointeur1 == ']': + axiom2 = axiom2 + ']' + + j = j + 1 + axiom = axiom2 + i = i + 1 + print axiom + return axiom + +def interpretation(angle, axiom): + # Interprétation de la chaine par turtle + k = 0 + laposition = [] + langle = [] + while k < len(axiom): + pointeur2 = axiom[k] + + # Interpréation + if pointeur2 == 'F': + forward(longueur) + elif pointeur2 == '+': + left(angle) + elif pointeur2 == '-': + right(angle) + elif pointeur2 == '[': + # mémorisation de la position courante + laposition.append(position()) + langle.append(heading()) + elif pointeur2 == ']': + # retour à la dernière position et suppression de celle-ci + up() + goto(laposition[len(laposition)-1][0], laposition[len(laposition)-1][1]) + del laposition[len(laposition)-1] + setheading(langle[len(langle)-1]) + del langle[len(langle)-1] + down() + + k = k + 1 + +# Placement initial de la tortue +up() +backward(400) +down() + +# Programme principal +## gram[0] retourne l'angle +## gram[1] retourne l'axiome +## gram[2] retourne la règle + +initialisation() +gram = grammaire() +axio = production(gram[1], gram[2]) +interpretation(gram[0], axio) + +#quit() diff --git a/Python/ArbreFonctionnel3.py b/Python/ArbreFonctionnel3.py new file mode 100644 index 0000000..82dce97 --- /dev/null +++ b/Python/ArbreFonctionnel3.py @@ -0,0 +1,192 @@ +# -*- coding: utf-8 -*- +from turtle import * + +# Génération de plante par grammaire L-system + +def initialisation(): + # Définition des variables indépendantes de la grammaire + global nbiteration, longueur + # Pour le flocon de Koch + #nbiteration = 3 + #longueur = 2 + + # Pour le flocon de Koch Island + #nbiteration = 3 + #longueur = 2 + + # Pour la plante + #nbiteration = 5 + #longueur = 5 + + # Pour la plante 2 + #nbiteration = 5 + #longueur = 5 + + # Pour la plante 3 + #nbiteration = 4 + #longueur = 8 + + # Pour la plante 4 + #nbiteration = 7 + #longueur = 1 + + # Pour la plante 5 + #nbiteration = 7 + #longueur = 1 + + # Pour la plante 6 + nbiteration = 5 + longueur = 4 + +def grammaire(): + # Définition de la grammaire + # Pour le flocon de Koch + #angle = 90 + #axiom = 'F-F-F-F' + #regle = 'F+F-F-F+F' + + # Pour le flocon de Koch Island + #angle = 90 + #axiom = 'F-F-F-F' + #regle = 'F-F+F+FF-F-F+F' + + # Pour la plante + #angle = 25.7 + #axiom = 'F' + #regle = 'F[+F]F[-F]F' + + # Pour la plante 2 + #angle = 20 + #axiom = 'F' + #regle = 'F[+F]F[-F][F]' + + # Pour la plante 3 + #angle = 22.5 + #axiom = 'F' + #regle = 'FF-[-F+F+F]+[+F-F-F]' + + # Pour la plante 4 + #angle = 20 + #axiom = 'X' + #regle1 = 'F[+X]F[-X]+X' + #regle2 = 'FF' + + # Pour la plante 5 + angle = 25.7 + axiom = 'X' + regle1 = 'F[+X][-X]FX' + regle2 = 'FF' + + # Pour la plante 6 + angle = 22.5 + axiom = 'X' + regle1 = 'F-[[X]+X]+F[+FX]-X' + regle2 = 'FF' + + return angle, axiom, regle1, regle2 + +def production(axiom, regle1, regle2): + # Génération de la chaine de caractères + i = 0 + while i < nbiteration: + j = 0 + axiom2 = '' + while j < len(axiom): + pointeur1 = axiom[j] + + # Symboles non terminaux + if pointeur1 == 'X': + # remplacement de la règle dans l'axiome + axiom2 = axiom2 + regle1 + + # Symboles terminaux + if pointeur1 == '+': + axiom2 = axiom2 + '+' + if pointeur1 == '-': + axiom2 = axiom2 + '-' + if pointeur1 == '[': + axiom2 = axiom2 + '[' + if pointeur1 == ']': + axiom2 = axiom2 + ']' + if pointeur1 == 'F': + axiom2 = axiom2 + 'F' + + j = j + 1 + axiom = axiom2 + print axiom + + m = 0 + axiom3 = '' + while m < len(axiom): + pointeur2 = axiom[m] + + # Symboles non terminaux + if pointeur2 == 'F': + # remplacement de la règle dans l'axiome + axiom3 = axiom3 + regle2 + + # Symboles terminaux + if pointeur2 == '+': + axiom3 = axiom3 + '+' + if pointeur2 == '-': + axiom3 = axiom3 + '-' + if pointeur2 == '[': + axiom3 = axiom3 + '[' + if pointeur2 == ']': + axiom3 = axiom3 + ']' + if pointeur2 == 'X': + axiom3 = axiom3 + 'X' + + m = m + 1 + axiom = axiom3 + i = i + 1 + + print axiom + return axiom + +def interpretation(angle, axiom): + # Interprétation de la chaine par turtle + k = 0 + laposition = [] + langle = [] + while k < len(axiom): + pointeur2 = axiom[k] + + # Interpréation + if pointeur2 == 'F': + forward(longueur) + elif pointeur2 == '+': + left(angle) + elif pointeur2 == '-': + right(angle) + elif pointeur2 == '[': + # mémorisation de la position courante + laposition.append(position()) + langle.append(heading()) + elif pointeur2 == ']': + # retour à la dernière position et suppression de celle-ci + up() + goto(laposition[len(laposition)-1][0], laposition[len(laposition)-1][1]) + del laposition[len(laposition)-1] + setheading(langle[len(langle)-1]) + del langle[len(langle)-1] + down() + + k = k + 1 + +# Placement initial de la tortue +up() +backward(400) +down() + +# Programme principal +## gram[0] retourne l'angle +## gram[1] retourne l'axiome +## gram[2] retourne la règle + +initialisation() +gram = grammaire() +axio = production(gram[1], gram[2], gram[3]) +interpretation(gram[0], axio) + +#quit() diff --git a/Python/ArbreRecursif1.py b/Python/ArbreRecursif1.py new file mode 100644 index 0000000..f9621cc --- /dev/null +++ b/Python/ArbreRecursif1.py @@ -0,0 +1,49 @@ +# -*- coding: utf-8 -*- +from turtle import * + +# Génération de plante par grammaire L-system + +def initialisation(): + # Définition des variables indépendantes de la grammaire + # Pour le flocon de Koch + nbiteration = 3 + longueur = 200 + angle = 45 + axiome = 'F' + regle = 'F+F--F+F' + return nbiteration, longueur, angle, axiome, regle + +def production(nbiteration, longueur, angle, axiome, regle): + if nbiteration == 0 : + forward(longueur) + else: + i = 0 + while i < len(regle): + regle1 = regle[i] + if regle1 == 'F': + production(nbiteration-1, longueur/3, angle, axiome, regle) + if regle1 == '+': + left(angle) + if regle1 == '-': + right(angle) + i = i + 1 + +# Placement initial de la tortue +up() +backward(200) +down() + +# Programme principal +## init[1] : nbiteration +## init[2] : longueur +## init[3] : angle +## init[4] : axiome +## init[5] : regle +init = initialisation() +production(init[0], init[1], init[2], init[3], init[4]) +right(120) +production(init[0], init[1], init[2], init[3], init[4]) +right(120) +production(init[0], init[1], init[2], init[3], init[4]) + +#quit() diff --git a/Python/ArbreRecursif2.py b/Python/ArbreRecursif2.py new file mode 100644 index 0000000..a2b5f84 --- /dev/null +++ b/Python/ArbreRecursif2.py @@ -0,0 +1,45 @@ +# -*- coding: utf-8 -*- +from turtle import * + +# Génération de plante par grammaire L-system + +def initialisation(): + # Définition des variables indépendantes de la grammaire + # Pour le flocon de Koch + nbiteration = 3 + longueur = 200 + angle = 90 + axiome = 'F' + regle = 'F+F-F-F+F' + return nbiteration, longueur, angle, axiome, regle + +def production(nbiteration, longueur, angle, axiome, regle): + if nbiteration == 0 : + forward(longueur) + else: + i = 0 + while i < len(regle): + regle1 = regle[i] + if regle1 == 'F': + production(nbiteration-1, longueur/3, angle, axiome, regle) + if regle1 == '+': + left(angle) + if regle1 == '-': + right(angle) + i = i + 1 + +# Placement initial de la tortue +up() +backward(200) +down() + +# Programme principal +## init[1] : nbiteration +## init[2] : longueur +## init[3] : angle +## init[4] : axiome +## init[5] : regle +init = initialisation() +production(init[0], init[1], init[2], init[3], init[4]) + +#quit() diff --git a/Python/ArbreRecursif3.py b/Python/ArbreRecursif3.py new file mode 100644 index 0000000..ac10743 --- /dev/null +++ b/Python/ArbreRecursif3.py @@ -0,0 +1,45 @@ +# -*- coding: utf-8 -*- +from turtle import * + +# Génération de plante par grammaire L-system + +def initialisation(): + # Définition des variables indépendantes de la grammaire + # Pour le flocon de Koch + nbiteration = 3 + longueur = 200 + angle = 90 + axiome = 'F-F-F-F' + regle = 'FF-F+F-F-FF' + return nbiteration, longueur, angle, axiome, regle + +def production(nbiteration, longueur, angle, axiome, regle): + if nbiteration == 0 : + forward(longueur) + else: + i = 0 + while i < len(regle): + regle1 = regle[i] + if regle1 == 'F': + production(nbiteration-1, longueur/3, angle, axiome, regle) + if regle1 == '+': + left(angle) + if regle1 == '-': + right(angle) + i = i + 1 + +# Placement initial de la tortue +up() +backward(200) +down() + +# Programme principal +## init[1] : nbiteration +## init[2] : longueur +## init[3] : angle +## init[4] : axiome +## init[5] : regle +init = initialisation() +production(init[0], init[1], init[2], init[3], init[4]) + +#quit() diff --git a/Solutions.tex b/Solutions.tex new file mode 100644 index 0000000..bc22ad4 --- /dev/null +++ b/Solutions.tex @@ -0,0 +1,98 @@ +\begin{Solution}{1} + L'alphabet des chiffres binaires est : \(\Sigma=\{0,1\}\) et le langage des nombres binaires à trois chiffres est : + \[L=\{000,001,010,011,100,101,110,111\}\] + +\end{Solution} +\begin{Solution}{2} + En considérant les quatre nucléotides composant l'ADN, la thymine (T), la cytosine (C), l'adénine (A) et la guanine (G), on peut définir l'alphabet nécessaire ainsi : + \[\Sigma=\{A,C,G,T\}\] + Les codons sont quant à eux des ensemble de trois nucléotides, pris parmi les quatre disponibles dans l'alphabet. Le langage ainsi formé est : + \[L=\{AAC,ACT,ACT,CTG,GTC, \dots\}\] + et la combinatoire donne pour un arrangement avec répétition de trois éléments pris parmi quatre : + \[\overline{A_3^4}=4^3=64\;\text{possibilités}\] + +\end{Solution} +\begin{Solution}{3} + Il s'agit d'un automate simple sans grande difficultés. Il est déterministe puisque de chaque état ne part qu'une seule transition de même type. + \begin{center} + \begin{postscript} + \VCDraw{% + \begin{VCPicture}{(0,-2)(6,4)} + % states + \State[e_1]{(0,0)}{A} \State[e_2]{(2,1)}{B} \State[e_3]{(4,1)}{C} + \State[e_5]{(4,3)}{E} + \State[e_8]{(2,-1)}{G} \State[e_9]{(4,-1)}{H} + \StateLineDouble \State[e_{10}]{(6,-1)}{I} \State[e_4]{(6,1)}{D} \State[e_6]{(6,3)}{F} + % initial--final + \Initial{A} + % transitions + \EdgeL{A}{B}{a} \EdgeL{B}{C}{a} \EdgeL{C}{D}{b} + \EdgeL{B}{E}{b} \EdgeL{E}{F}{b} + \EdgeL{A}{G}{b} \EdgeL{G}{H}{b} \EdgeL{H}{I}{b} + % + \end{VCPicture}} + \end{postscript} + \end{center} + +\end{Solution} +\begin{Solution}{4} + Le problème peut sembler complexe car l'énoncé du mot allumé fait immédiatement penser à éteint. Naturellement, on pense à un état allumé et à un autre éteint. Comme il y a trois couleurs, on s'imagine alors six états et leur représentation avec des transitions du type ``s'allume'' ou ``s'éteint'' ne va pas de soit. + + Pour s'en sortir simplement, il faut au contraire penser ``quel est la couleur du feu''. Cela suppose qu'on ne considère que l'état allumé du feu. On peut alors dire rouge, puis jaune, puis vert, puis rouge, \dots On pourrait aussi dire, c'est le rouge qui s'allume, puis c'est le jaune qui s'allume, puis \dots Cela nous permet alors de représenter la succession des états allumé du feu par : + \begin{center} + \begin{postscript} + \VCDraw{% + \begin{VCPicture}{(0,-3)(11,1)} + % states + \State[r]{(0,0)}{A} \State[j]{(4,0)}{B} \State[v]{(2,-2)}{C} + \State{(7,0)}{D} \State{(11,0)}{E} \State{(9,-2)}{F} + % initial--final + \Initial{A} \Initial{D} + % transitions + \EdgeL{A}{B}{} \EdgeR{B}{C}{} \EdgeR{C}{A}{} + \EdgeL{D}{E}{r} \EdgeR{E}{F}{j} \EdgeR{F}{D}{v} + % + \end{VCPicture}} + \end{postscript} + \end{center} + En particulier, l'automate ci-dessus permet la reconnaissance de la chaîne de caractère rjvrjvrjv\dots{} qui est une représentation de la succession des couleurs (allumées) du feu de circulation. + +\end{Solution} +\begin{Solution}{5} + On a vu qu'il y a 64 possibilités pour former un acide aminé constitué d'un triplet choisi parmi quatre nucléotides. Il peut sembler alors que l'automate correspondant doive être très complexe. Mais au contraire, en raison du fait qu'on peut mettre n'importe quel nucléotide à chaque place parmi les trois disponibles nous permet de constituer un automate particulièrement simple : + \begin{center} + \begin{postscript} + \VCDraw{% + \begin{VCPicture}{(0,-1)(8,1)} + % states + \State{(0,0)}{A} \State{(4,0)}{B} \State{(8,0)}{C} + \StateLineDouble \State{(12,0)}{D} + % initial--final + \Initial{A} + % transitions + \EdgeL{A}{B}{C|G|T|U} \EdgeR{B}{C}{C|G|T|U} \EdgeR{C}{D}{C|G|T|U} + % + \end{VCPicture}} + \end{postscript} + \end{center} + où la notation C|G|T|U signifie soit C, soit G, soit T, soit U. + + \smallskip + On constate ainsi la concision de la description de l'automate par rapport à l'exposé de tout les cas possibles. + +\end{Solution} +\begin{Solution}{6} + On doit tout d'abord faire un b, puis un nombre quelconque (voir nul) de b. Comme l'état de droite n'est pas final, il faut ensuite revenir à celui de gauche par un a. Enfin, on peut faire tout cela un nombre quelconque de fois, y compris aucune. L'expression rationnelle correspondante est : +\[(bb^*a)^*\] + +\end{Solution} +\begin{Solution}{7} + De manière très naturelle, on a : + \[[A..Z]\{2\}[0..9]\{5\}\] + +\end{Solution} +\begin{Solution}{8} + Il faut raisonner à partir de la dernière transition avant l'état final. Il s'agit toujours d'une transition b. Donc, l'expression régulière finira par un b. Puis, on a soit deux b, soit un a suivi d'un a ou b. Cela se traduit par : + \[(bb|a(a|b))b\] + +\end{Solution} diff --git a/algorithm2e.sty b/algorithm2e.sty new file mode 100644 index 0000000..4377fa8 --- /dev/null +++ b/algorithm2e.sty @@ -0,0 +1,2636 @@ +% algorithm2e.sty --- style file for algorithms +% almost everything can be customized by users. See the document for more explanations +%% Copyright 1996-2008 Christophe Fiorio +% +% This program may be distributed and/or modified under the +% conditions of the LaTeX Project Public License, either version 1.2 +% of this license or (at your option) any later version. +% The latest version of this license is in +% http://www.latex-project.org/lppl.txt +% and version 1.2 or later is part of all distributions of LaTeX +% version 1999/12/01 or later. +% +% This program consists of the files algorithm2e.sty and algorithm2e.tex and algorithm2e-compatibility.sty +% +% Report bugs and comments to: +% - algorithm2e-announce@lirmm.fr mailing list for announcement about releases^^J% +% - algorithm2e-discussion@lirmm.fr mailing list for discussion about package^^J% +% subscribe by emailing sympa@lirmm.fr with 'subscribe '^^J% +% +% $Id: algorithm2e.sty,v 4.1 2009/12/15 08:54:08 cfiorio Exp $ +% +% PACKAGES REQUIRED: +% +% - float (in contrib/supported/float) +% - ifthen (in base) +% - xspace (in packages/tools) +% - relsize (in contrib/misc/relsize.sty) +% +%%%%%%%%%%%%%%% Release 4.01 +% +% Package options: +% --------------- +% - oldcommands : to use old command names +% - french, english, german, +% portuguese, czech, italiano, +% slovak : for the name of the algorithm and some keyword code +% - onelanguage : to simply switch keyword from one language to another without changing +% keyword commands +% - boxed, boxruled, ruled, tworuled, +% algoruled, plain : layout of the algorithm +% - algo2e : environment is algorithm2e instead of algorithms and \listofalgorithmes +% instead of \listofalgorithms +% - slide : to use when making slides +% - noline,lined,vlined : how block are designed. +% - shortend, longend, noend : short or long end keyword as endif for e.g. +% - linesnumbered : auto numbering of the algorithm's lines +% - linesnumberedhidden : to hide autonumbered lines (show number on a line with \ShowLn +% - commentsnumbered, inoutnumbered : to autonumber comments and inout keywords (by defaut not numbered) +% - rightnl : to have line number on the right instead of on the left as default +% - algonl : line numbers preceded by algo number +% - scright, scleft : right or left justified side comments +% - fillcomment, nofillcomment : end mark of comment is flushed to the right so comments fill the line +% - dotocloa : add an entry in the toc for list of algorithms (require tocbibind package) +% - endfloat : add algoendfloat environment pushing algorithm so written to the end of document +% - resetcount, noresetcount : start value of line numbers. +% - algopart,algochapter,algosection : algo numbering within part, chapter or section +% - titlenumbered,titlenotnumbered : numbering of title set by \Titleofalgo +% - figure : algorithms are figures, numbered as figures, and put in the list of figures. +% - procnumbered : procedure or function are numbered as algorithm +% - nokwfunc : procedure or function name doens't become a command +% - norelsize : don't use relsize package (useful if it breaks the compatibily) +% +% defaults are; english,plain,resetcount,titlenotnumbered +% +%%%%%%%%%%%%%% +% +% Short summary +% ------------- +% +% algorithm is an environment for writing algorithm in LaTeX2e. +% Almost all is customizable. You can add keywords, change style, change the layout, ... +% It provide macros that allow you to create differents sorts of key words, therefore a set of predefined key +% word is gived. +% +% IT should be used as follows +% +% \begin{algorithm} +% ... +% ... +% \end{algorithm} +% +% +% IMPORTANT : each line MUST end with \; +% +% Note that if you define macros outside algorithm environment they +% are avaible in all the document and particulary you can use then +% inside all algorithms without re-define them. +% +% an example: +% +% \begin{algorithm} +% \SetAlgoLined +% \KwIn{this text} +% \KwOut{how to write algorithm with \LaTeX2e } +% +% initialization\; +% \While{not at end of this document}{ +% read current section\; +% \eIf{understand}{ +% go to next section\; +% current section becomes this one\; +% }{ +% go back to the beginning of current section\; +% } +% } +% \caption{How to write algorithm} +% \end{algorithm} +% +% +%%%%%%%%%%%%%% predefined keywords +% +% \KwIn{input} +% \KwOut{output} +% \KwData{input} +% \KwResult{output} +% \KwTo % a simple keyword +% \KwFrom % a simple keyword +% \KwRet{[value]} +% \Return{[value]} +% \Begin{block inside} +% \eIf{condition}{Then Block}{Else block} % in blocks +% \If{condition}{Then block} % in a block +% \uIf{condition}{Then block} % in a block unended +% \lIf{condition}{Else text} % on the same line +% \Else{inside Else} % in a block +% \lElse{Else text} % on the same line +% \uElse{Else text} % in a block unended +% \ElseIf{inside Elseif} % in a block +% \lElseIf{Elseif text} % on the same line +% \uElseIf{Elseif text} % in a block unended +% \Switch{Condition}{Switch block} +% \Case{a case}{case block} % in a block +% \lCase{a case}{case text} % on the same line +% \Other{otherwise block} % in a block +% \lOther{otherwise block} % on the same line +% \For{condition}{text loop} % in a block +% \lFor{condition}{text} % on the same line +% \ForEach{condition}{text loop} % in a block +% \lForEach{condition}{text} % on the same line +% \ForPar{condition}{text loop} % in a block +% \lForPar{condition}{text} % on the same line +% \While{condition}{text loop} % in a block +% \lWhile{condition}{text loop} % on the same line +% \Repeat{End condition}{text loop} % in a block +% \lRepeat{condition}{text} % on the same line +% +%%%%%%%%%%%%%% +% +% History: +% +% - december 14 2009 - revision 4.01 +% * ADD : new command \SetKwHangingKw{Name}{text} (hanging indent with keyword): This creates a +% hanging indent much like \texttt{SetKwInput}, except that it removes the trailing `:' +% and does not reset numbering. +% +% - november 17 2009 - revision 4.00 - +% +% * CHANGE : IMPORTANT : some commands have been renamed to have consistent naming (CamlCase +% syntax) and old commands are no more available. If you doesn't want to change +% your mind or use old latex files, you can use oldcommands option to enable old +% commands back. +% text. Here are these commands: +% - \SetNoLine becomes \SetAlgoNoLine +% - \SetVline becomes \SetAlgoVlined +% - \Setvlineskip becomes \SetVlineSkip +% - \SetLine becomes \SetAlgoLined +% - \dontprintsemicolon becomes \DontPrintSemicolon +% - \printsemicolon becomes \PrintSemicolon +% - \incmargin becomes \IncMargin +% - \decmargin becomes \DecMargin +% - \setnlskip becomes \SetNlSkip +% - \Setnlskip becomes \SetNlSkip +% - \setalcapskip becomes \SetAlCapSkip +% - \setalcaphskip becomes \SetAlCapHSkip +% - \nlSty becomes \NlSty +% - \Setnlsty becomes \SetNlSty +% - \linesnumbered becomes \LinesNumbered +% - \linesnotnumbered becomes \LinesNotNumbered +% - \linesnumberedhidden becomes \LinesNumberedHidden +% - \showln becomes \ShowLn +% - \showlnlabel becomes \ShowLnLabel +% - \nocaptionofalgo becomes \NoCaptionOfAlgo +% - \restorecaptionofalgo becomes \RestoreCaptionOfAlgo +% - \restylealgo becomes \RestyleAlgo +% - gIf macros and so on do no more exist +% * NEW: - Compatibily with other packages improven by changing name of internal +% macros. Algorithm2e can now be used with arabtex for example, if this last is +% loaded after algorithm2e package. +% * ADD: - OPTION endfloat: endfloat packages doesn't allow float environment inside other +% environment. So using it with figure option of algorithm2e makes error. This +% option enables a new environment algoendfloat to be used instead of algorithm +% environment that put algorithm at the end. algoendfloat environment make +% algorithm acting as endfloat figures. This option requires endfloat packages. +% * ADD: - OPTION norelsize: starting from this release (v4.00), algorithm2e package uses +% relsize package in order to get relative size for lines numbers; but it seems +% that some rare classes (such as inform1.cls) are not compatible with relsize; to +% have algorithm2e working, this option makes algorithm2e not to load relsize +% package and go back to previous definition by using \scriptsize font for lines +% numbers. +% * ADD: - OPTION onelanguage: allow, if using standard keywords listed below, to switch +% from one language to another without changing keywords by using appropriate +% language option : +% . KwIn, KwOut, KwData, KwResult +% . KwTo KwFrom +% . KwRet, Return +% . Begin +% . Repeat +% . If, ElseIf, Else +% . Switch, Case, Other +% . For, ForPar, ForEach, ForAll, While +% . +% * ADD: - OPTION rightnl: put lines numbers to the right of the algorithm instead of left. +% * ADD: new commands \setRightLinesNumbers and \setLeftLinesNumbers which sets the lines +% numbers to the right or to the left of the algorithm. +% * ADD: - new kind of keywords : KwArray used to define arrays: +% \SetKwArray{Kw}{array} defines an array keywords Kw called array and printed in +% DataSty style when call with \Kw. It can be used with one argument which +% denotes the element index: \Kw{n} prints array[n] with array in DataSty and n in +% ArgSty. +% * ADD/FIX: rules of ruled, algoruled, tworuled styles used rules of different sizes! This +% is now fixed. Moreover size of the rules is now controlled by a length and so +% can be customized by the user. +% \algoheightrule is the height of the rules and can be changed via \setlength +% \algoheightruledefault is the default height of he rules (0.8pt) +% \algotitleheightrule is the height of the rule that comes just after the +% caption in ruled and algoruled style; it can be changed via \setlength +% \algotitleheightruledefault is the default height of this rules (0.8pt) +% Thanks to Philippe Dumas who reports the bug and make the suggestion. +% * ADD: - \SetAlgoCaptionSeparator which sets the separator between Algorithm 1 and the +% title. By default it's ':' and caption looks like "Algorithm 2: title" but now +% you can change it by using for example \SetAlgoCaptionSeparator{.} which will +% give "Algorithm 3. title" +% * ADD: - \SetAlgoLongEnd and \SetAlgoShortEnd and \SetAlgoNoEnd commands which act as +% corresponding package options +% * ADD: - OPTIONS italiano and slovak as new language (thanks to Roberto Posenato and +% Miroslav Binas) +% * CHANGE: - Fnt and Sty macro to have consistent use and naming (see below) +% * ADD: - \AlCapSty, \AlCapNameSty, \AlCapFnt, \AlCapNameFnt, \ProcSty, \ProcFnt, +% \ProcNameSty, \ProcNameFnt, \ProcArgSty, ProcArgFnt and corresponding "set macro" +% \SetAlCapSty, \SetAlCapNameSty, \SetAlCapFnt, \SetAlCapNameFnt, \SetProcSty, +% \SetProcFnt, \SetProcNameSty, \SetProcNameFnt, \SetProcArgSty, \SetProcArgFnt which +% control the way caption is printed. Sty macro use command taking one parameter as +% argument, Fnt macros use directly command. In Fact caption is printed as follow : +% \AlCapSty{\AlCapFnt Algorithm 1:}\AlCapNameSty{\AlCapNameFnt my algorithm} +% By default, \AlCapSty is textbf and \AlCapFnt is nothing. \AlCapNameSty keep text +% as it is, and \AlCapNameFnt do nothing also. +% You can redefine \AlCapFnt and \AlCapNameFnt by giving macro to \Set commands. For +% example, you can do \SetAlCapFnt{\large} to see Algorithm printed in \large font. +% You can redefine \AlCapSty, \AlCapFnt, \AlCapNameSty and \AlCapNameFnt with the +% corresponding \Set command. For the Sty commands, you have to give in parameter +% name of a macro (whithout \) which takes one argument. For example, +% \SetAlCapFnt{textbf} defines the default behaviour. If you want to do more +% complicated thing, you should define your own macro and give it to \SetAlCapFnt or +% \SetAlCapNameFnt. Here are two examples: +% - \newcommand{\mycapsty}[1]{\tiny #1}\SetAlCapNameSty{mycapsty} +% - \newcommand{\mycapsty}[1]{\textsl{\small #1}}\SetAlCapNameSty{mycapsty} +% Or you can combine the two, for the last example you can also do: +% \SetAlCapNameSty{textsl}\SetAlCapNameFnt{\small} +% Thanks to Jan Stilhammer who gives me the idea of \AlCapNameFnt. +% * CHANGE \AlTitleFnt to match definition of all other Fnt macros and add a \AlTitleSty +% macro (see below) . Now you set \AlTitleFnt by calling \SetAlTitleFnt with +% directly a macro without parameter in argument: +% Example: \SetAlTitleFnt{\small} to set title in small font. +% * ADD: - \AlTitleSty and \SetAlTitleSty commands to set a style for title. These commands +% are defined from a macro taking the text in argument, as \textbf for example. +% To set the TitleSty you have to give name of the macro (without the '\') +% to \SetAlTitleSty. For example \SetAlTitleSty{textbf} to set \textbf style. +% * ADD: - new command \SetAlgorithmName{algorithmname}{list of algorithms name} which +% redefines name of the algorithms and the sentence list of algorithms. Second +% argument is the name that \autoref, from hyperref package, will use. Example: +% \SetAlgorithmName{Protocol}{List of protocols} if you prefer protocol than +% algorithm. +% * ADD: - new \SetAlgoRefName{QXY} which change the default ref (number of the algorithm) by +% the name given in parameter (QXY in the example). +% * ADD: - new command \SetAlgoRefRelativeSize{-2} which sets the output size of refs, defined +% by \SetAlgoRefName, used in list of algorithms. +% * ADD: - two dimensions to control the layout of caption in ruled, algoruled and boxruled +% algorithms: +% - interspacetitleruled (2pt by defaut) which controls the vertical space between +% rules and title in ruled and algoruled algorithms. +% - interspaceboxruled (2\lineskip by default) which controls the vertical space +% between rules and title in boxruled algorithms. +% These two dimensions can be changed by using \setlength command. +% * ADD: - With the fix (see below) of procedure and function environments, a new feature has +% been added: the name of the procedure or function set in caption is automatically +% defined as a KwFunction and so can be used as a macro. For example, if inside a +% procedure environment you set \caption{myproc()}, you can use \myproc macro in you +% main text. Beware that the macro is only defined after the \caption! +% * ADD: - OPTION nokwfunc to unable the new feature described above in function and +% procedure environment. Useful if you use name of procedure or function that cannot +% be a command name as a math display for example. +% * ADD: - \SetAlgoNlRelativeSize{number} command which sets the relative size of line +% numbers. By default, line numbers are two size smaller than algorithm text. Use +% this macro to change this behavior. For example, \SetAlgoNlRelativeSize{0} sets it +% to the same size, \SetAlgoNlRelativeSize{-1} to one size smaller and +% \SetAlgoNlRelativeSize{1} to one size bigger +% * ADD: - \SetAlgoProcName{aname} command which sets the name of Procedure printed by +% procedure environment (the environment prints Procedure by default). Second +% argument is the name that \autoref, from hyperref package, will use. +% * ADD: - \SetAlgoFuncName{aname} command which sets the name of Function printed by +% procedure environment (the environment prints Function by default). Second +% argument is the name that \autoref, from hyperref package, will use. +% * ADD: - \SetAlgoCaptionLayout{style} command which sets style of the caption; style must +% be the name of a macro taking one argument (the text of the caption). Examples +% below show how to use it: +% . \SetAlgoCaptionLayout{centerline} to have centered caption +% . \SetAlgoCaptionLayout{textbf} to have bold caption +% If you want to apply two styles in the same time, such as centered bold, you have +% to define you own macro and then use \SetAlgoCaptionLayout with its name. +% * ADD: - OPTION procnumbered: which makes the procedure and function to be numbered as +% algorithm +% * ADD: - OPTIONS tworuled and boxruled +% these are two new layouts: tworuled acts like ruled but doesn't put a line after +% caption ; boxruled surround algorithm by a box, puts caption above and add a line +% after caption. +% * REMOVE: - SetKwInParam has been deleted since not useful itself because of different +% macros which can do the same in a better and a more consistent way as +% SetKwFunction or SetKw. +% * FIX: - line number is now correctly vertically aligned with math display. +% * FIX: - references with hyperref. No more same identifier or missing name error. BUT now +% you must NOT use naturalnames option of hyperref packages if you do PdfLaTeX +% * FIX: - autoref with hyperref package (thanks to Jörg Sommer who notices the problem). +% * FIX: - titlenumbered was not working! fixed. +% * FIX: - Else(){} acted like uElse. Corrected. +% * FIX: - noend management: when a block was inside another and end of block was following +% each other, a blank line was added: it's now corrected. +% * FIX: - Function and Procedure environment was no more working as defined originally: the +% label was no more name of the procedure, it acts always as if procumbered option +% has been used. +% * FIX: - line numbers had a fixed size which can be bigger than algorithm text accordingly +% to \AlFnt set (see also new command \SetAlgoNlRelativeSize above) +% * FIX: - semicolon in comments when dontprintsemicolon is used. +% * FIX: - listofalgorithms adds a vertical space before first algo of a chapter as for +% listoffigures or listoftables +% * FIX: - listofalgorithms with twocolumns mode and some classes which don't allow onecolumn +% and so don't define \if@restonecol as prescribed in LaTeX (sig-alternate for +% example) +% * FIX: - algorithm2e now works with elsart cls and some more classes. +% * FIX: - blocks defined by SetKwBlock act now as other blocks (if for instance) and don't +% write end in vlined mode, instead they print a small horizontal line as required +% by the option. +% * FIX: - underfull hbox warning at each end of algorithm environment removed. +% +% * INTERNAL CHANGE: - short end keyword are deduce from long end keyword by keeping the +% first one. Allows to avoid double definition. +% * INTERNAL CHANGE: - procedure, function and algorithm are now resolved by the same +% environment to avoid code duplication. +% +% - October 04 2005 - revision 3.9 - +% * ADD: - \setalcaphskip command which sets the horizontal skip before Algorithm: in caption +% when used in ruled algorithm. +% * ADD: - \SetAlgoInsideSkip command which allows to add an extra vertical space before and +% after the core of the algorithm (ie: \SetAlgoInsideSkip{bigskip}) +% * CHANGE: - caption, when used with figure option, is no more controlled by algorithm2e +% package and so follows the exact behaviour of figures. The drawback is that you +% cannot change the typo with AlTitleFnt or CapFnt. The avantage is that if you +% use caption package, it works. +% * FIX: - problem with numbering line and pdflatex +% * FIX: - error when algorithm2e package was used with beamer and listings together +% - February 12 2005 - revision 3.8 - +% * FIX: - extra line with noend option. +% - February 10 2005 - revision 3.7 - +% * ADD: - sidecomment: different macros allowing to put text right after code on the same +% line. They are defined in the same time comment macros are defined with a star +% after the macro name. By default comments are right justified but this can be +% change with appropriate option in the macro. Ex: +% . default: \tcc*{side comment} +% . same as previous: \tcc*[r]{side comment} +% . left justify: \tcc*[l]{side comment} +% . here: \tcc*[h]{side comment} don't put the end of line mark before +% comment (; by default) and don't end the line. +% . flushed: \tcc*[f]{side comment} same as the precedent but right +% justified +% * ADD: - OPTION scright (default): right justified side comments (side comments +% are flushed to the righr) +% * ADD: - OPTION scleft: left justified side comments (side comments are put right after the +% code line) +% * ADD: - \SetSideCommentLeft acts as scleft option +% * ADD: - \SetSideCommentRight acts as scright option +% * ADD: - block like macro side text: all macro defining a block allows now to put text right +% after key words by putting text into (). Done to be used with sidecomment macros, +% but all text can be used. +% Ex: \eIf(\tcc*[f]{then comment}){test}{then text}(else side text){else text} +% * ADD: - OPTION fillcomment (default): end mark of comment is flushed to the right so +% comments fill all the width of text. +% * ADD: - OPTION nofillcomment: end mark of comment is put right after the comment. +% * ADD: - \SetNoFillComment acts as nofillcomment option. +% * ADD: - \SetFillComment acts as fillcomment option. +% * ADD: - OPTION dotocloa: which adds an entry in the toc for the list of algorithms. This +% option load package tocbibind if not already done and so list of figures and list +% of tables are also added in the toc. If you want to control which ones of the lists +% will be added in the toc, please load package tocbibind before package algorithm +% and give it the options you want. +% * FIX: - vertical spacing for uif macro with noend option +% * FIX: - all the compatibility problems between caption and other packages +% * FIX: - typographical differences between list of algorithms and other lists when in +% report or book +% +% - January 24 2005 - revision 3.6 - +% * FIX: - vertical spacing and space characters at the beginning or end of comments. +% line numbers of comments not in the NlSty. +% Thanks to Arnaud Giersch for his comments and suggestions. +% * FIX: - Set*Sty macro: the styles defined was not protected and was modified by surrounding +% context. For example KwTo in a \For{}{} was in bold AND italic instead of just in +% bold. +% * FIX: - line number misplacement after \Indp +% +% - January 21 2005 - revision 3.5 - +% * ADD: - hidden numbering of the lines. Lines are auto-numbered but numbers are shown only +% on lines you specify: +% * linesnumberedhidden option or \LinesNumberedHidden macro activate this +% functionnality. +% * \ShowLn and \ShowLnLabel{lab} macros make the number visible on the +% line. \ShowLnLabel{lab} allows to set a label for this line. +% Thanks to Samson de Jager who makes this suggestion and provides the macros. +% * ADD: - \AlCapFnt and \SetAlCapFnt which allow to have a different font for +% caption. Works like \AlFnt and \SetAlFnt and by default is the same. +% * ADD: - \AlCapSkip skip length. This vertical space is added before caption in plain ou +% boxed mode. It allows to change distance between text and caption. +% * FIX: - caption compatible with IEEEtran class. +% * FIX: - some vertical spacing error with \uIf macros (Thanks to Arnaud Giersch) +% * FIX: - Procedure and Function: lines are also numbered like algorithms +% * FIX: - CommentSty was not used for Comments +% +% - January 10 2005 - revision 3.4 - +% * FIX: - caption compatible with new release of Beamer class. +% +% - June 16 2004 - revision 3.3 - +% * FIX: - Hyperlink references of Hyperref package works now if compiled with pdflatex +% and [naturalnames] option of hyperref package is used. +% * FIX: - algorithm[H] had problem in an list environment - corrected +% * FIX: - interline was not so regular in nested blocks - corrected +% * ADD - \SetVlineSkip macro which sets the vertical skip after the little horizontal +% rule which closes a block in Vlined mode. By default 0.8ex +% +% - June 11 2004 - revision 3.2 - AUTO NUMBERING LINES !!! +% * ADD: auto numbering of the lines (the so asked and so long awaiting feature) +% this feature is managed by 3 options and 3 commands: +% - linesnumbered option: lines of the algo are numbered except for comments and +% input/output (KwInput and KwInOut) +% - commentsnumbered option: makes comments be numbered +% - inoutnumbered option: makes data input/output be numbered +% - \nllabel{lab} labels the line so you can cite with \ref{lab} +% - \LinesNumbered make the following algorithms having auto-numbered lines +% - \linesnotnumbered make the following algorithms having no auto-numbered lines +% * Change: algo2e option renames listofalgorithms in listofalgorithmes +% * FIX: new solution for compatibility with color package, more robust and not tricky. +% Many thanks to David Carlisle for his advices +% +% - June 09 2004 - revision 3.1 - +% * Change: \SetKwSwitch command defines an additionnal macro \uCase and \Case prints end +% * Change: now macros SetKw* do a renewcommand if the keyword is already defined. So you can +% redefine default definition at your own convenience or change your definition +% without introducing a new macro and changing your text. +% * ADD: new macro \SetKwIF which do \SetKwIf and +% \SetKwIfElseIf.The following default definition has been added: +% \SetKwIF{If}{ElseIf}{Else}{if}{then}{else if}{else}{endif} +% and so you get the macros; +% \If \eIf \lIf \uIf \ElseIf \uElseIf \lElseIf \Else \uElse \lElse +% * ADD: new macro \SetAlgoSkip which allow to fix the vertical skip before and after the +% algorithms. Default is smallskip, do \SetAlgoSkip{} if you don't want an extra space +% or \SetAlgoSkip{medskip} or \SetAlgoSkip{bigskip} if you want bigger space. +% * ADD: macro \SetKwIf defines in addition a new macro \uElse (depending on wat name you +% have given in #2 arg). +% * ADD: macro \SetKwIfElseIf defines in addition a new macro \uElse and \ugElseIf (depending +% on what name you have given in #2 and #3 arg). +% * Change: baseline of algorithm is now top, so two algorithms can be put side by side. +% * FIX: Compatibility with color package solved. The problem was due to a redefinition of +% standard macros by color package. This solves compatibility problem with other +% packages as pstcol or colortbl. (notified by Dirk Fressmann, Antti Tarvainen and Koby +% Crammer) +% * Fix: extra little shift to the right with boxed style algorithm removed (notified by +% P. Tanovski) +% * Fix: algoln option was buggy (notified bye Jiaying Shen) +% * Fix: german and portuges option didn't work due to bad typo (notified by Martin Sievers, +% Thorsten Vitt and Jeronimo Pellegrini) +% +% - February 13 2004 - revision 3.0 - +% * Major revision which makes the package independent from float.sty, so now +% - algorithm* works better, in particular can be used in multicols environments +% - (known bug corrected) +% [H] works now for all sort of environment but is handled differently for classic +% environment and star environment (algorithm, figure, procedure and function). For star +% environment, H acts like for classical figure environment, so it doesn't stay here +% absolutely. +% - (known bug corrected) +% you can use now floatflt package with algorithm package and even with figure +% option. Beware that if you want to put an algorithm inside a floatingfigure, it cannot +% be floating, so [H] is required and then figure option should not be used, since +% standard figure[H] are still floating with LaTeX. +% * boxruled: a new style added. Possible now since no style no more defined by the float +% package. +% * nocaptionofalgo: dosen't print Algorithm #: in the caption for algorithm in ruled or +% algoruled style. +% note: this is just documentation of a macro which was already in the package. +% - December 14 2003 - revision 2.52 - +% * output message shorter +% * french keyword macro \PourTous was missing for longend option, it has been added. +% * TitleofAlgo prints Function or Procedure in corresponding environments. +% +% - October 27 2003 - revision 2.51 - Revision submitted to CTAN archive +% * correction of a minor which make caption in procedure +% and function to be blanck with pdfscreen package +% (thanks to Joel Gossens for the notification) +% * add two internal definition to avoid some errors when +% used with Hyperref package (Hyperref package need to +% define new counter macro from existing ones, and +% don't do it for algorithm2e package, so we do it) +% +% - October 17 2003 - revision 2.50 - first revision for CTAN archive +% * add \AlFnt and \SetAlFnt{font} macros: \AlFnt is used at the beginning of the caption and +% the body of algorithm in order to define the fonts used for typesetting algorithms. You +% can use it elsewhere you want to typeset text as algorithm. For example you can do +% \SetAlFnt{\small\sf} to have algorithms typeset in small sf font. Default is nothing so +% algorithm is typeset as the text of the document. +% * add \AlTitleFnt{text} and \SetAlTitleFnt{font} macros: The {Algorithm: } in the caption is +% typeset with \AlTitleFnt{Algorithm:}. You can use it to have text typeset as {Algorithm:} +% of captions. Default is textbf. Default can be redefined by \SetAlTitleFnt{font}, for +% example you can do \SetAlTitleFnt{emph} +% * add CommentSty typo for text comment. +% * add some compatibility with hyperref package (still an error on multiply defined refs but +% pdf correctly generated) +% * flush text to left in order to have correct indentation even with class as amsart which +% center all figures +% * add german, portugues and czech options for title of algorithms and typo. +% * add portuguese translation of predefined keywords * add czech translation of some +% predefined keywords +% +% - December 23 2002 - revision 2.40 +% * add some french keyword missing +% * add function* and procedure* environment like algorithme* environment: print in one column +% even if twocolumn option is specified for the document. +% * add a new macro \SetKwComment to define macro which writes comments in the text. First +% argument is the name of the macro, second is the text put before the comment, third is the +% text put at the end of the comment.Default are \tcc and \tcp +% * add new options to change the way algo are numbered: +% [algopart] algo are numbered within part (counter must exist) +% [algochapter] algo are numbered within chapter +% [algosection] algo are numbered within section +% +% - March 27 2002 - revision 2.39 +% * Gilles Geeraerts: added the \SetKwIfElseIf to manage +% if (c) +% i; +% else if (c) +% i; +% ... +% else +% i; +% end +% * Also added \gIf \gElseIf \gElse. +% +% - January 02 2001 - revision 2.38 +% * bugs related to the caption in procedure and function +% environment are corrected. +% * bug related to option noend (extra vertical space added +% after block command as If or For) is corrected. +% * czech option language added (thanks to Libor Bus: l.bus@sh.cvut.cz). +% +% - October 16 2000 - revision 2.37 +% * option algo2e added: change the name of environment +% algorithm into algorithm2e. So allow to use the package +% with some journal style which already define an algorithm +% environment. +% +% - September 13 2000 - revision 2.36 +% * option slide added: require package color +% * Hack for slide class in order to have correct +% margins +% +% - November 25 1999 - revision 2.35 +% * revision number match RCS number +% * Thanks to David A. Bader, a new option is added: +% noend: no end keywords are printed. +% +% - November 19 1999 - revision 2.32 +% * minor bug on longend option corrected. +% +% - August 26 1999 - revision 2.31 +% * add an option : figure +% this option makes algorithms be figure and so are numbered +% as figures, have Figure as caption and are put in +% the \listoffigures +% +% - January 21 1999 - revision 2.3 beta +% add 2 new environments: procedure and function. +% These environments works like algorithm environment but: +% - the ruled (or algoruled) style is imperative. +% - the caption now writes Procedure name.... +% - the syntax of the \caption command is restricted as +% follow: you MUST put a name followed by 2 braces like +% this ``()''. You can put arguments inside the braces and +% text after. If no argument is given, the braces will be +% removed in the title. +% - label now puts the name (the text before the braces in the +% caption) of the procedure or function as reference (not +% the number like a classic algorithm environment). +% There are also two new styles : ProcNameSty and +% ProcArgSty. These style are by default the same as FuncSty +% and ArgSty but are used in the caption of a procedure or a +% function. +% +% - November 28 1996 - revision 2.22 +% add a new macro \SetKwInParam{arg1}{arg2}{arg3}: +% it defines a macro \arg1{name}{arg} which prints name in keyword +% style followed byt arg surrounded by arg2 and arg3. The main +% application is to a function working as \SetKwInput to be used +% in the head of the algorithm. For example +% \SetKwInParam{Func}{(}{)} allows +% \Func{functionname}{list of arguments} which prints: +% \KwSty{functioname(}list of arguments\KwSty{)} +% +% +% - November 27 1996 - revision 2.21 : +% minor bug in length of InOut boxes fixed. +% add algorithm* environment. +% +% - July 12 1996 - revision 2.2 : \SetArg and \SetKwArg macros removed. +% +% \SetArg has been removed since it never has been +% documented. +% \SetKwArg has been removed since \SetKw can now +% take an argument in order to be consistent with +% \SetKwData and \SetKwFunction macros. +% +% - July 04 1996 - revision 2.1 : still more LaTeX2e! Minor compatibility break +% +% Macros use now \newcommand instead of \def, use of \setlength, +% \newsavebox, ... and other LaTeX2e specific stuff. +% The compatibility break: +% - \SetData becomes \SetKwData to be more consistent. So the old +% \SetKwData becomes \SetKwInput +% - old macros \titleofalgo, \Freetitleofalgo and \freetitleofalgo +% from LaTeX209 version which did print a warning message and call +% \Titleofalgo in version 2.0 are now removed! +% +% - March 13 1996 - revision 2.0: first official major revision. +% +% +%%%%%%%%%%%%%% +% +% Known bugs: +% ----------- +% - no more known bugs... all are corrected! +% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% +% for more complete informations you can see algorithm2e.tex +% +% +%%%%%%%%%%%%%%%%%%%%%%%% Identification Part %%%%%%%%%%%%%%%%%%%%%%%%%%%% +% +\NeedsTeXFormat{LaTeX2e}[1994/12/01] +% +\ProvidesPackage{algorithm2e}[2008/00/00 v3.10 algorithms environments] +% +% +%%%%%%%%%%%%%%%%%%%%%%%%%%% Initial Code %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% +\@makeother\*% some package redefined it as a letter (as color.sty) +\def\@firstword#1 #2\@nil{#1}% an useful fonction +% +% definition of commands which can be redefined in options of the package. +% +\newcounter{AlgoLine}% +\setcounter{AlgoLine}{0}% +% +\newcommand{\algocf@algocfref}{\relax}% +\newcommand{\listalgorithmcfname}{}% +\newcommand{\algorithmcfname}{}% +\@ifundefined{algorithmautorefname}{\newcommand{\algorithmautorefname}{algorithm}}{\renewcommand{\algorithmautorefname}{algorithm}}% +\newcommand{\algorithmcflinename}{}% +\newcommand{\algocf@typo}{}% +\newcommand{\@algocf@procname}{}\newcommand{\procedureautorefname}{}% +\newcommand{\SetAlgoProcName}[2]{\renewcommand{\@algocf@procname}{#1}\renewcommand{\procedureautorefname}{#2}}% +\newcommand{\@algocf@funcname}{}\newcommand{\functionautorefname}{}% +\newcommand{\SetAlgoFuncName}[2]{\renewcommand{\@algocf@funcname}{#1}\renewcommand{\functionautorefname}{#2}}% +\newcommand{\@algocf@titleofalgoname}{\algorithmcfname}% +\newcommand{\@algocf@algotitleofalgo}{% + \renewcommand{\@algocf@titleofalgoname}{\algorithmcfname}}% +\newcommand{\@algocf@proctitleofalgo}{% + \renewcommand{\@algocf@titleofalgoname}{\algocf@procname}}% +% +\newcommand{\algocf@style}{plain}% +\newcommand{\@ResetCounterIfNeeded}{}% +\newcommand{\@titleprefix}{}% +% +\newcommand{\algocf@numbering}[1]{\newcommand{\algocf@within}{#1}}% +% +\newcommand{\defaultsmacros@algo}{\algocf@defaults@shortend}% +% +\newcommand{\algocf@list}{loa}% +\newcommand{\algocf@float}{algocf}% +% +\newcommand{\algocf@envname}{algorithm}% +\newcommand{\algocf@listofalgorithms}{listofalgorithms}% +% +% +%% redefine chapter so that it adds a vspace in the loa as the original does for lof and lot +\let\algocf@original@chapter=\chapter% +\def\chapter{\expandafter\addtocontents{loa}{\protect\addvspace{10\p@}}\algocf@original@chapter}% +% +%% if@restonecol is defined in article and book but some other classes don't define it and we need it, so we do +\ifx\if@restonecol\relax\else\newif\if@restonecol\fi% +% +% +%%%%%%%%%%%%%%%%%%%%%% Declaration of Options %%%%%%%%%%%%%%%%%%%%%%%%%%% +% +\RequirePackage{ifthen}% +% +\newboolean{algocf@nokwfunc}\setboolean{algocf@nokwfunc}{false}% +\DeclareOption{nokwfunc}{% + \setboolean{algocf@nokwfunc}{true}% +}% +% +\newboolean{algocf@oldcommands}\setboolean{algocf@oldcommands}{false}% +\DeclareOption{oldcommands}{% + \setboolean{algocf@oldcommands}{true}% +}% +% +\newboolean{algocf@leftlinenumber}\setboolean{algocf@leftlinenumber}{true}% +\newcommand{\setLeftLinesNumbers}{\setboolean{algocf@leftlinenumber}{true}}% +\newcommand{\setRightLinesNumbers}{\setboolean{algocf@leftlinenumber}{false}}% +\DeclareOption{rightnl}{% + \setRightLinesNumbers% +}% +% +\newboolean{algocf@endfloat}\setboolean{algocf@endfloat}{false}% +\DeclareOption{endfloat}{% + \setboolean{algocf@endfloat}{true}% + \newcounter{postalgo}\setcounter{postalgo}{0}% +}% +% +\newboolean{algocf@procnumbered}\setboolean{algocf@procnumbered}{false}% +\DeclareOption{procnumbered}{% + \setboolean{algocf@procnumbered}{true}% +}% +% +\DeclareOption{algo2e}{% + \renewcommand{\algocf@envname}{algorithm2e}% + \renewcommand{\algocf@listofalgorithms}{listofalgorithmes}% +}% +% +\newboolean{algocf@slide}\setboolean{algocf@slide}{false}% +\DeclareOption{slide}{% + \setboolean{algocf@slide}{true}% +}% +% +\DeclareOption{figure}{% +\renewcommand{\algocf@list}{lof}% +\renewcommand{\algocf@float}{figure}% +}% +% +\newboolean{algocf@optonelanguage}\setboolean{algocf@optonelanguage}{false}% +\DeclareOption{onelanguage}{\setboolean{algocf@optonelanguage}{true}}% +% +\newcommand{\algocf@languagechoosen}{english}% +% +\DeclareOption{english}{% +\renewcommand{\listalgorithmcfname}{List of Algorithms}% +\renewcommand{\algorithmcfname}{Algorithm}% +\renewcommand{\algorithmautorefname}{algorithm}% +\renewcommand{\algorithmcflinename}{line}% +\renewcommand{\algocf@typo}{}% +\renewcommand{\@algocf@procname}{Procedure}% +\renewcommand{\@algocf@funcname}{Function}% +\renewcommand{\procedureautorefname}{procedure}% +\renewcommand{\functionautorefname}{function}% +\renewcommand{\algocf@languagechoosen}{english}% +}% +% +\DeclareOption{french}{% +\renewcommand{\listalgorithmcfname}{Liste des Algorithmes}% +\renewcommand{\algorithmcfname}{Algorithme}% +\renewcommand{\algorithmautorefname}{algorithme}% +\renewcommand{\algorithmcflinename}{ligne}% +\renewcommand{\algocf@typo}{\ }% +\renewcommand{\@algocf@procname}{Procédure}% +\renewcommand{\@algocf@funcname}{Fonction}% +\renewcommand{\procedureautorefname}{procédure}% +\renewcommand{\functionautorefname}{fonction}% +\renewcommand{\algocf@languagechoosen}{french}% +}% +% +\DeclareOption{czech}{% +\renewcommand{\listalgorithmcfname}{Seznam algoritm\v{u}}% +\renewcommand{\algorithmcfname}{Algoritmus}% +\renewcommand{\algorithmautorefname}{\algorithmcfname}% +\renewcommand{\algorithmcflinename}{Radek}% +\renewcommand{\algocf@typo}{}% +\renewcommand{\@algocf@procname}{Procedura}% +\renewcommand{\@algocf@funcname}{Funkce}% +\renewcommand{\procedureautorefname}{\@algocf@procname}% +\renewcommand{\functionautorefname}{\@algocf@funcname}% +\renewcommand{\algocf@languagechoosen}{czech}% +}% +% +\DeclareOption{german}{% +\renewcommand{\listalgorithmcfname}{Liste der Algorithmen}% +\renewcommand{\algorithmcfname}{Algorithmus}% +\renewcommand{\algorithmautorefname}{\algorithmcfname}% +\renewcommand{\algorithmcflinename}{Zeile}% +\renewcommand{\algocf@typo}{\ }% +\renewcommand{\@algocf@procname}{Prozedur}% +\renewcommand{\@algocf@funcname}{Funktion}% +\renewcommand{\procedureautorefname}{\@algocf@procname}% +\renewcommand{\functionautorefname}{\@algocf@funcname}% +\renewcommand{\algocf@languagechoosen}{german}% +}% +% +\DeclareOption{portuguese}{% +\renewcommand{\listalgorithmcfname}{Lista de Algoritmos}% +\renewcommand{\algorithmcfname}{Algoritmo}% +\renewcommand{\algorithmautorefname}{algoritmo}% +\renewcommand{\algorithmcflinename}{linha}% +\renewcommand{\algocf@typo}{}% +\renewcommand{\@algocf@procname}{Procedimento}% +\renewcommand{\@algocf@funcname}{Fun\c{c}\~{a}o}% +\renewcommand{\procedureautorefname}{procedimento}% +\renewcommand{\functionautorefname}{fun\c{c}\~{a}o}% +\renewcommand{\algocf@languagechoosen}{portuguese}% +}% +% +\DeclareOption{italiano}{% +\renewcommand{\listalgorithmcfname}{Elenco degli algoritmi}% +\renewcommand{\algorithmcfname}{Algoritmo}% +\renewcommand{\algorithmautorefname}{algoritmo}% +\renewcommand{\algorithmcflinename}{riga}% +\renewcommand{\algocf@typo}{}% +\renewcommand{\@algocf@procname}{Procedura}% +\renewcommand{\@algocf@funcname}{Funzione}% +\renewcommand{\procedureautorefname}{procedura}% +\renewcommand{\functionautorefname}{funzione}% +\renewcommand{\algocf@languagechoosen}{italiano}% +}% +\DeclareOption{slovak}{% +\renewcommand{\listalgorithmcfname}{Zoznam algoritmov}% +\renewcommand{\algorithmcfname}{Algoritmus}% +\renewcommand{\algorithmautorefname}{\algorithmcfname}% +\renewcommand{\algorithmcflinename}{Radek}% +\renewcommand{\algocf@typo}{}% +\renewcommand{\@algocf@procname}{Proced\'{u}ra}% +\renewcommand{\@algocf@funcname}{Funkcia}% +\renewcommand{\procedureautorefname}{\@algocf@procname}% +\renewcommand{\functionautorefname}{\@algocf@funcname}% +\renewcommand{\algocf@languagechoosen}{slovak}% +}% +% +% OPTIONs plain, boxed, ruled, algoruled & boxruled +% +\newcommand{\algocf@style@plain}{\renewcommand{\algocf@style}{plain}}% +\newcommand{\algocf@style@boxed}{\renewcommand{\algocf@style}{boxed}}% +\newcommand{\algocf@style@ruled}{\renewcommand{\algocf@style}{ruled}}% +\newcommand{\algocf@style@algoruled}{\renewcommand{\algocf@style}{algoruled}}% +\newcommand{\algocf@style@boxruled}{\renewcommand{\algocf@style}{boxruled}}% +\newcommand{\algocf@style@tworuled}{\renewcommand{\algocf@style}{tworuled}}% +\newcommand{\RestyleAlgo}[1]{\csname algocf@style@#1\endcsname}% +\DeclareOption{plain}{\algocf@style@plain}% +\DeclareOption{boxed}{\algocf@style@boxed}% +\DeclareOption{ruled}{\algocf@style@ruled}% +\DeclareOption{algoruled}{\algocf@style@algoruled}% +\DeclareOption{boxruled}{\algocf@style@boxruled}% +\DeclareOption{tworuled}{\algocf@style@tworuled}% +% +% OPTIONs algopart,algochapter & algosection +% +\DeclareOption{algopart}{\algocf@numbering{part}}% %algo part numbered +\DeclareOption{algochapter}{\algocf@numbering{chapter}}% %algo chapter numbered +\DeclareOption{algosection}{\algocf@numbering{section}}% %algo section numbered +% +% OPTIONs resetcount & noresetcount +% +\DeclareOption{resetcount}{\renewcommand{\@ResetCounterIfNeeded}{\setcounter{AlgoLine}{0}}}% +\DeclareOption{noresetcount}{\renewcommand{\@ResetCounterIfNeeded}{}}% +% +% OPTION linesnumbered +% +\newboolean{algocf@linesnumbered}\setboolean{algocf@linesnumbered}{false}% +\newcommand{\algocf@linesnumbered}{\relax}% +\DeclareOption{linesnumbered}{% + \setboolean{algocf@linesnumbered}{true}% + \renewcommand{\algocf@linesnumbered}{\everypar={\nl}}% +}% +% +% OPTION linesnumberedhidden +% +\DeclareOption{linesnumberedhidden}{% + \setboolean{algocf@linesnumbered}{true}% + \renewcommand{\algocf@linesnumbered}{\everypar{\stepcounter{AlgoLine}}}% +}% +% +% OPTION commentsnumbered inoutnumbered +% +\newboolean{algocf@commentsnumbered}\setboolean{algocf@commentsnumbered}{false}% +\DeclareOption{commentsnumbered}{\setboolean{algocf@commentsnumbered}{true}}% +\newboolean{algocf@inoutnumbered}\setboolean{algocf@inoutnumbered}{false}% +\DeclareOption{inoutnumbered}{\setboolean{algocf@inoutnumbered}{true}}% +% +% OPTIONs titlenumbered & titlenotnumbered +% +\DeclareOption{titlenumbered}{% + \renewcommand{\@titleprefix}{% + \refstepcounter{\algocf@float}% + \AlTitleSty{\AlTitleFnt\@algocf@titleofalgoname\ \expandafter\csname the\algocf@float\endcsname\algocf@typo: }% + }% +}% +% +\DeclareOption{titlenotnumbered}{\renewcommand{\@titleprefix}{% + \AlTitleSty{\AlTitleFnt\@algocf@titleofalgoname\algocf@typo: }}% +}% +% +% OPTIONs algonl +% line numbered with the counter of the algorithm +% +\DeclareOption{algonl}{\renewcommand{\theAlgoLine}{\expandafter\csname the\algocf@float\endcsname.\arabic{AlgoLine}}}% +% +% OPTIONs lined, vlined & noline +% +\DeclareOption{lined}{\AtBeginDocument{\SetAlgoLined}}% \SetAlgoLined +\DeclareOption{vlined}{\AtBeginDocument{\SetAlgoVlined}}% \SetAlgoVlined +\DeclareOption{noline}{\AtBeginDocument{\SetAlgoNoLine}}%\SetAlgoNoLine (default) +% +% OPTIONs longend, shotend & noend +% +\DeclareOption{longend}{\AtBeginDocument{\SetAlgoLongEnd}}% \SetAlgoLongEnd +\DeclareOption{shortend}{\AtBeginDocument{\SetAlgoShortEnd}}%\SetAlgoShortEnd +\DeclareOption{noend}{\AtBeginDocument{\SetAlgoNoEnd}}% \SetAlgoNoEnd +% +% OPTION dotoc +% +\newboolean{algocf@dotocloa}\setboolean{algocf@dotocloa}{false}% +\DeclareOption{dotocloa}{% + \setboolean{algocf@dotocloa}{true}% +} +% +% OPTION comments +% +\newboolean{algocf@optfillcomment}\setboolean{algocf@optfillcomment}{true}% +\DeclareOption{nofillcomment}{% + \setboolean{algocf@optfillcomment}{false}% +}% +\DeclareOption{fillcomment}{% + \setboolean{algocf@optfillcomment}{true}% +}% +% +% OPTION sidecommments +% +\newboolean{algocf@scleft}\setboolean{algocf@scleft}{false}% +\DeclareOption{scleft}{% + \setboolean{algocf@scleft}{true}% +}% +\DeclareOption{sright}{% default + \setboolean{algocf@scleft}{false}% +}% +% +% OPTION norelsize +% +\newboolean{algocf@norelsize}\setboolean{algocf@norelsize}{false}% +\DeclareOption{norelsize}{% + \setboolean{algocf@norelsize}{true}% +}% +% +% +%%%%%%%%%%%%%%%%%%%%%%% Execution of Options %%%%%%%%%%%%%%%%%%%%%%%%%%%% +% +\ExecuteOptions{english,plain,resetcount,titlenotnumbered,lined,shortend}% +% +\ProcessOptions% +% +\@algocf@algotitleofalgo% fix name for \TitleOfAlgo to \algorithmcfname by default +% +%%%%%%%%%%%%%%%%%%%%%%%%%% Package Loading %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% +% +\RequirePackage{xspace}% +% +\ifthenelse{\boolean{algocf@endfloat}}{% + \RequirePackage{endfloat}% +}{\relax}% +% +\ifthenelse{\boolean{algocf@norelsize}}{% + \newcommand{\relsize}[1]{\scriptsize}% +}{% + \RequirePackage{relsize}% +}% +% +\ifthenelse{\boolean{algocf@slide}}{\RequirePackage{color}}{}% +% + +\AtEndOfPackage{% + \ifthenelse{\boolean{algocf@dotocloa}}{% + \renewcommand{\listofalgorithmes}{\tocfile{\listalgorithmcfname}{loa}}% + }{\relax}% +}% +% +% if loa in toc required, load tocbibind package if not already done. +\ifthenelse{\boolean{algocf@dotocloa}}{% + \ifx\@tocextra\undefined% + \RequirePackage{tocbibind}% + \fi% +}% +% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Main Part %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% +\newcommand{\algocf@name}{algorithm2e}% +\newcommand{\algocf@date}{december 14 2009}% +\newcommand{\algocf@version}{Release 4.01}% +\newcommand{\algocf@id}{\algocf@version\space -- \algocf@date\space --}% +\typeout{********************************************************^^JPackage `\algocf@name'\space\algocf@id^^J% + - algorithm2e-announce@lirmm.fr mailing list for announcement about releases^^J% + - algorithm2e-discussion@lirmm.fr mailing list for discussion about package^^J% + subscribe by emailing sympa@lirmm.fr with 'subscribe '^^J% + - Author: Christophe Fiorio (fiorio@lirmm.fr)^^J********************************************************}% +%% +%% +%% +%% +%% +%% +%%%% hyperref compatibility tricks: Hyperref package defines H counters from + % standard counters (i.e \theHpage from \thepage) and check some particular + % counters of some packages, unfortunately it doesn't do the same for + % algorithm2e package but act as Hcounter was defined. To avoid errors we + % defined \theHalgocf ourself +%%%% +% +\@ifundefined{theHalgocf}{\def\theHalgocf{\thealgocf}}{}% +\@ifundefined{theHAlgoLine}{\def\theHAlgoLine{\theAlgoLine}}{}% +\@ifundefined{theHalgocfproc}{\def\theHalgocfproc{0}}{}% +\@ifundefined{theHalgocffunc}{\def\theHalgocffunc{0}}{}% +\@ifundefined{toclevel@algocf}{\def\toclevel@algocf{0}}{}% +% +% autoref from hyperref needs an autorefname, so we give it. +\def\AlgoLineautorefname{\algorithmcflinename}% +\def\algocfautorefname{\algorithmautorefname}% +\def\algocfprocautorefname{\procedureautorefname}% +\def\algocffuncautorefname{\functionautorefname}% +%% +%% +%% +\newcommand{\@defaultskiptotal}{0.5em}% +\newskip\skiptotal\skiptotal=0.5em% +\newskip\skiplinenumber\skiplinenumber=\hsize\advance\skiplinenumber by-\skiptotal% +\newskip\skiprule% +\newskip\skiphlne% +\newskip\skiptext% +\newskip\skiplength% +\newskip\algomargin% +\newskip\skipalgocfslide\skipalgocfslide=1em% +\newdimen\algowidth% +\newdimen\inoutsize% +\newdimen\inoutline% +\newdimen\interspacetitleruled\setlength{\interspacetitleruled}{2pt}% +\newdimen\interspacealgoruled\setlength{\interspacealgoruled}{2pt}% +\newdimen\interspacetitleboxruled\setlength{\interspacetitleboxruled}{2\lineskip}% +% +\newcommand{\@algoskip}{\smallskip}% +\newcommand{\SetAlgoSkip}[1]{\renewcommand{\@algoskip}{\csname#1\endcsname}}% +\newcommand{\@algoinsideskip}{\relax}% +\newcommand{\SetAlgoInsideSkip}[1]{\renewcommand{\@algoinsideskip}{\csname#1\endcsname}}% +% +\newsavebox{\algocf@inoutbox}% +\newsavebox{\algocf@inputbox}% +%% +%% +\newcommand{\arg@e}{}% +\newcommand{\arg@space}{\ }% +\newcommand{\BlankLine}{\vskip 1ex}% +%% +\newcommand{\vespace}{1ex}% +\newcommand{\SetInd}[2]{% +\skiprule=#1% +\skiptext=#2% +\skiplength=\skiptext\advance\skiplength by \skiprule\advance\skiplength by 0.4pt}% +\SetInd{0.5em}{1em} +\algomargin=\leftskip\advance\algomargin by \parindent% +\newcommand{\IncMargin}[1]{\advance\algomargin by #1}% +\newcommand{\DecMargin}[1]{\advance\algomargin by -#1}% +\newcommand{\SetNlSkip}[1]{% + \renewcommand{\@defaultskiptotal}{#1}% + \setlength{\skiptotal}{#1}}% +%% +\newskip\AlCapSkip\AlCapSkip=0ex% +\newskip\AlCapHSkip\AlCapSkip=0ex% +\newcommand{\SetAlCapSkip}[1]{\setlength{\AlCapSkip}{#1}}% +\newcommand{\SetAlCapHSkip}[1]{\setlength{\AlCapHSkip}{#1}}% +\SetAlCapHSkip{.5\algomargin}% +%% +%% +\newcommand{\Indentp}[1]{\advance\leftskip by #1}% +\newcommand{\Indp}{\advance\leftskip by 1em}% +\newcommand{\Indpp}{\advance\leftskip by 0.5em}% +\newcommand{\Indm}{\advance\leftskip by -1em}% +\newcommand{\Indmm}{\advance\leftskip by -0.5em}% +%% +%% +%% Line Numbering +%% +%% +% number line style +\newcommand{\algocf@nlrelsize}{-2}\newcommand{\SetAlgoNlRelativeSize}[1]{\renewcommand{\algocf@nlrelsize}{#1}}% +\newcommand{\NlSty}[1]{\textnormal{\textbf{\relsize{\algocf@nlrelsize}#1}}}% default definition +\newcommand{\SetNlSty}[3]{\renewcommand{\NlSty}[1]{\textnormal{\csname#1\endcsname{\relsize{\algocf@nlrelsize}#2##1#3}}}}% +% +% nl definitions +% +\newsavebox{\algocf@nlbox}% +\newcommand{\algocf@printnl}[1]{% + \ifthenelse{\boolean{algocf@leftlinenumber}}{% + \skiplinenumber=\skiptotal\advance\skiplinenumber by\leftskip% + \strut\raisebox{0pt}{\llap{\NlSty{#1}\kern\skiplinenumber}}\ignorespaces% + }{% + \sbox\algocf@nlbox{\NlSty{#1}}% + \skiplinenumber=\hsize\advance\skiplinenumber by-\leftskip\advance\skiplinenumber by-\skiptext% + \advance\skiplinenumber by\algomargin\advance\skiplinenumber by.3em\advance\skiplinenumber by-\wd\algocf@nlbox% + \strut\raisebox{0pt}{\rlap{\kern\skiplinenumber\NlSty{#1\ignorespaces}}}\ignorespaces% + }% +}% +\newcommand{\algocf@nl@sethref}[1]{% + \renewcommand{\theHAlgoLine}{\thealgocfproc.#1}% + \hyper@refstepcounter{AlgoLine}\gdef\@currentlabel{#1}% +}% +\newcommand{\nl}{% + \@ifundefined{hyper@refstepcounter}{% if not hyperref then do a simple refstepcounter + \refstepcounter{AlgoLine}% + }{% else if hyperref, do the anchor so 2 lines in two differents algorithms cannot have the same href + \stepcounter{AlgoLine}\algocf@nl@sethref{\theAlgoLine}% + }% now we can do the line numbering + \algocf@printnl{\theAlgoLine}% +}% +% +\newcommand{\nllabel}[1]{\label{#1}}% +% +\newcommand{\enl}{% + \@ifundefined{hyper@refstepcounte}{% if not hyperref then do a simple refstepcounter + \refstepcounter{AlgoLine}% + }{% else if hyperref, do the anchor so 2 lines in two differents algorithms cannot have the same href + \stepcounter{AlgoLine}\algocf@nl@sethref{\theAlgoLine}% + }% now we can do the line numbering + \skiplinenumber=\hsize\advance\skiplinenumber by-\leftskip% + \strut\raisebox{0pt}{\rlap{\kern\skiplinenumber\strut\NlSty{\theAlgoLine}}}\ignorespaces% +} +%% nlset +\newcommand{\nlset}[1]{% + \@ifundefined{hyper@refstepcounter}{\protected@edef\@currentlabel{#1}}{\algocf@nl@sethref{#1}}\algocf@printnl{#1}% +}% +% +%% lnl definitions +\newcommand{\lnl}[1]{\nl\label{#1}}% +% +%% lnlset +\newcommand{\lnlset}[2]{\nlset{#2}\label{#1}}% +% +% set char put at end of each line +% +\newcommand{\algocf@endline}{\string;} +\newcommand{\SetEndCharOfAlgoLine}[1]{\renewcommand{\algocf@endline}{#1}} +% +% end of line definition +% +\newcommand{\@endalgocfline}{\algocf@endline}% default definition: printsemicolon +\newcommand{\DontPrintSemicolon}{\renewcommand{\@endalgocfline}{\relax}}% +\newcommand{\PrintSemicolon}{\renewcommand{\@endalgocfline}{\algocf@endline}}% +\newcommand{\@endalgoln}{\@endalgocfline\hfill\strut\par}% +% +% line numbering +% +\newcommand{\LinesNumbered}{\setboolean{algocf@linesnumbered}{true}\renewcommand{\algocf@linesnumbered}{\everypar={\nl}}}% +\newcommand{\LinesNotNumbered}{% + \setboolean{algocf@linesnumbered}{false}% + \renewcommand{\algocf@linesnumbered}{\relax}% +}% +% +\newcommand{\LinesNumberedHidden}{% + \setboolean{algocf@linesnumbered}{true}\renewcommand{\algocf@linesnumbered}{\everypar{\stepcounter{AlgoLine}}}}% +\newcommand{\ShowLn}{\nlset{\theAlgoLine}\ignorespaces}% display the line number on this line (without labelling) +\newcommand{\ShowLnLabel}[1]{\lnlset{#1}{\theAlgoLine}\ignorespaces}% display the line number and label this line +% +%% +% +%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% +% Styling text commands +% +\newcommand{\AlFnt}{\relax}% default definition +\newcommand{\SetAlFnt}[1]{\renewcommand{\AlFnt}{#1}}% +\newcommand{\AlTitleFnt}{\relax}% default definition +\newcommand{\SetAlTitleFnt}[1]{\renewcommand{\AlTitleFnt}{#1}}% +% +\newcommand{\AlCapFnt}{\relax}% default definition +\newcommand{\SetAlCapFnt}[1]{\renewcommand{\AlCapFnt}{#1}}% +\newcommand{\AlCapNameFnt}{\relax}% default definition +\newcommand{\SetAlCapNameFnt}[1]{\renewcommand{\AlCapNameFnt}{#1}}% +% +\newcommand{\ProcFnt}{\relax}% default definition +\newcommand{\SetProcFnt}[1]{\renewcommand{\ProcFnt}{#1}}% +\newcommand{\ProcNameFnt}{\relax}% default definition +\newcommand{\SetProcNameFnt}[1]{\renewcommand{\ProcNameFnt}{#1}}% +\newcommand{\ProcArgFnt}{\relax}% default definition +\newcommand{\SetProcArgFnt}[1]{\renewcommand{\ProcArgFnt}{#1}}% +% +\newcommand{\AlTitleSty}[1]{\textbf{#1}\unskip}% default definition +\newcommand{\SetAlTitleSty}[1]{\renewcommand{\AlTitleSty}[1]{\textnormal{\csname#1\endcsname{##1}}\unskip}}% +\newcommand{\AlCapSty}[1]{\textnormal{\textbf{#1}}\unskip}% default definition +\newcommand{\SetAlCapSty}[1]{\renewcommand{\AlCapSty}[1]{\textnormal{\csname#1\endcsname{##1}}\unskip}}% +\newcommand{\AlCapNameSty}[1]{\textnormal{#1}\unskip}% default definition +\newcommand{\SetAlCapNameSty}[1]{\renewcommand{\AlCapNameSty}[1]{\textnormal{\csname #1\endcsname{##1}}\unskip}}% +% +\newcommand{\ProcSty}[1]{\AlCapSty{#1}}% +\newcommand{\SetProcSty}[1]{\renewcommand{\ProcSty}[1]{\textnormal{\csname#1\endcsname{##1}}\unskip}}% +\newcommand{\ProcNameSty}[1]{\AlCapNameSty{#1}}% +\newcommand{\SetProcNameSty}[1]{\renewcommand{\ProcNameSty}[1]{\textnormal{\csname#1\endcsname{##1}}\unskip}}% +\newcommand{\ProcArgSty}[1]{\AlCapNameSty{#1}}% +\newcommand{\SetProcArgSty}[1]{\renewcommand{\ProcArgSty}[1]{\textnormal{\csname#1\endcsname{##1}}\unskip}}% +% +\newcommand{\KwSty}[1]{\textnormal{\textbf{#1}}\unskip}% default definition +\newcommand{\SetKwSty}[1]{\renewcommand{\KwSty}[1]{\textnormal{\csname#1\endcsname{##1}}\unskip}}% +\newcommand{\ArgSty}[1]{\textnormal{\emph{#1}}\unskip}%\SetArgSty{emph} +\newcommand{\SetArgSty}[1]{\renewcommand{\ArgSty}[1]{\textnormal{\csname#1\endcsname{##1}}\unskip}}% +\newcommand{\FuncSty}[1]{\textnormal{\texttt{#1}}\unskip}%\SetFuncSty{texttt} +\newcommand{\SetFuncSty}[1]{\renewcommand{\FuncSty}[1]{\textnormal{\csname#1\endcsname{##1}}\unskip}}% +\newcommand{\DataSty}[1]{\textnormal{\textsf{#1}}\unskip}%%\SetDataSty{textsf} +\newcommand{\SetDataSty}[1]{\renewcommand{\DataSty}[1]{\textnormal{\csname#1\endcsname{##1}}\unskip}}% +\newcommand{\CommentSty}[1]{\textnormal{\texttt{#1}}\unskip}%%\SetDataSty{texttt} +\newcommand{\SetCommentSty}[1]{\renewcommand{\CommentSty}[1]{\textnormal{\csname#1\endcsname{##1}}\unskip}}% +\newcommand{\TitleSty}[1]{#1\unskip}%\SetTitleSty{}{} +\newcommand{\SetTitleSty}[2]{\renewcommand{\TitleSty}[1]{% +\csname#1\endcsname{\csname#2\endcsname##1}}\unskip}% +% +%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% +% Block basic commands +% +\newcommand{\algocf@push}[1]{\advance\skiptotal by #1\moveright #1}% +\newcommand{\algocf@pop}[1]{\advance\skiptotal by -#1}% +\newcommand{\algocf@addskiptotal}{\advance\skiptotal by 0.4pt\advance\hsize by -0.4pt}% 0.4 pt=width of \vrule +\newcommand{\algocf@subskiptotal}{\advance\skiptotal by -0.4pt\advance\hsize by 0.4pt}% 0.4 pt=width of \vrule +% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% +%% group of instructions definition +% +\skiphlne=.8ex% +\newcommand{\SetVlineSkip}[1]{\skiphlne=#1}% +% +%% block with a vertical line end by a little horizontal line +\newcommand{\algocf@Vline}[1]{% no vskip in between boxes but a strut to separate them, + \strut\par\nointerlineskip% then interblock space stay the same whatever is inside it + \algocf@push{\skiprule}% move to the right before the vertical rule + \hbox{\vrule% + \vtop{\algocf@push{\skiptext}%move the right after the rule + \vtop{\algocf@addskiptotal\advance\hsize by -\skiplength #1}\Hlne}}\vskip\skiphlne% inside the block + \algocf@pop{\skiprule}%\algocf@subskiptotal% restore indentation + \nointerlineskip}% no vskip after +% +%% block with a vertical line +\newcommand{\algocf@Vsline}[1]{% no vskip in between boxes but a strut to separate them, + \strut\par\nointerlineskip% then interblock space stay the same whatever is inside it + \algocf@push{\skiprule}% move to the right before the vertical rule + \hbox{\vrule% the vertical rule + \vtop{\algocf@push{\skiptext}%move the right after the rule + \vtop{\algocf@addskiptotal\advance\hsize by -\skiplength #1}}}% inside the block + \algocf@pop{\skiprule}}% restore indentation +% +\newcommand{\algocf@Hlne}{\hrule height 0.4pt depth 0pt width .5em}% +% +%% block without line +\newcommand{\algocf@Noline}[1]{% no vskip in between boxes but a strut to separate them, + \strut\par\nointerlineskip% then interblock space stay the same whatever is inside it + \algocf@push{\skiprule}% + \hbox{% + \vtop{\algocf@push{\skiptext}% + \vtop{\advance\hsize by -\skiplength #1}}}% inside the block + \algocf@pop{\skiprule}% + % \nointerlineskip% no vskip after +}% +% +%% default=NoLine +% +\newcommand{\algocf@group}[1]{\algocf@Noline{##1}}% group: set of instruction depending from another (ex: then part of the If) +\newcommand{\algocf@@block}[2]{\algocf@Noline{##1}\KwSty{##2}\par}% block: group with a end keyword. +\newcommand{\algocf@block}[2]{\algocf@@block{#1}{#2}}% command that will be used and redefined accordingly to noend option +\newcommand{\algocf@setBlock}{% + \ifthenelse{\boolean{algocf@optnoend}}{% if no end option + \renewcommand{\algocf@block}[2]{\algocf@group{##1}}% block will be a group + }{% else + \renewcommand{\algocf@block}[2]{\algocf@@block{##1}{##2}}% block stays a block + }% +}% +% +\newcommand{\Hlne}{}% little hrizontal line ending a block in vline mode +% +\newcommand{\@algocf@endoption}[1]{#1}% +\newboolean{algocf@optnoend}\setboolean{algocf@optnoend}{false}% +% +\newcommand{\SetAlgoLongEnd}{%%%%%%%%%%%%%%%%%%%%%%%%% Long End + \setboolean{algocf@optnoend}{false}% + \renewcommand{\@algocf@endoption}[1]{##1}% + \algocf@setBlock}% +% +\newcommand{\SetAlgoShortEnd}{%%%%%%%%%%%%%%%%%%%%%%%% ShortEnd + \setboolean{algocf@optnoend}{false}% + \renewcommand{\@algocf@endoption}[1]{\@firstword##1 \@nil}% + \algocf@setBlock}% +% +\newcommand{\SetAlgoNoEnd}{%%%%%%%%%%%%%%%%%%%%%%%%%%% NoEnd + \setboolean{algocf@optnoend}{true}% + \renewcommand{\@algocf@endoption}[1]{}% + \algocf@setBlock}% +% +\newcommand{\SetAlgoNoLine}{%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Noline +\renewcommand{\algocf@@block}[2]{\algocf@Noline{##1}\KwSty{##2}\strut\par}% +\renewcommand{\algocf@group}[1]{\algocf@Noline{##1}}% +\renewcommand{\Hlne}{}}% +% +\newcommand{\SetAlgoVlined}{%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Vline +\renewcommand{\algocf@@block}[2]{\algocf@Vline{##1}}% +\renewcommand{\algocf@group}[1]{\algocf@Vsline{##1}\ifthenelse{\boolean{algocf@optnoend}}{\relax}{\strut\ignorespaces}}% +\renewcommand{\Hlne}{\algocf@Hlne}}% +% +\newcommand{\SetAlgoLined}{%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Line +\renewcommand{\algocf@@block}[2]{\strut\algocf@Vsline{##1}\KwSty{##2}\strut\par}% no skip after a block so garantie at least a line +\renewcommand{\algocf@group}[1]{\algocf@Vsline{##1}\ifthenelse{\boolean{algocf@optnoend}}{\relax}{\strut\ignorespaces}}% +\renewcommand{\Hlne}{}}% +% +\newcommand{\SetNothing}{%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Noline +\renewcommand{\algocf@@block}[2]{\algocf@Noline{##1}\par}% +%\long +\renewcommand{\algocf@group}[1]{\algocf@Noline{##1}}% +\renewcommand{\Hlne}{}}% +% +%% +%% +% +% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% +% ``Input :'''s like command +% +%%% +% text staying at the right of the longer keyword of KwInOut commands +% (text of KwInOut commands are all vertically aligned) +% +\newcommand{\algocf@newinout}{\par\parindent=\wd\algocf@inoutbox}% to put right indentation after a \\ in the KwInOut +\newcommand{\SetKwInOut}[2]{% + \sbox\algocf@inoutbox{\KwSty{#2}\algocf@typo:}% + \expandafter\ifx\csname InOutSizeDefined\endcsname\relax% if first time used + \newcommand\InOutSizeDefined{}\setlength{\inoutsize}{\wd\algocf@inoutbox}% + \else% else keep the larger dimension + \ifdim\wd\algocf@inoutbox>\inoutsize\setlength{\inoutsize}{\wd\algocf@inoutbox}\fi% + \fi% the dimension of the box is now defined. + \@ifundefined{#1}{\let\algocf@mkcmd=\newcommand}{\let\algocf@mkcmd=\renewcommand}% + \expandafter\algocf@mkcmd\csname#1\endcsname[1]{% + \ifthenelse{\boolean{algocf@inoutnumbered}}{\relax}{\everypar={\relax}}% + {\let\\\algocf@newinout\hangindent=\wd\algocf@inoutbox\hangafter=1\parbox[t]{\inoutsize}{\KwSty{#2}\algocf@typo\hfill:}~##1\par}% + \algocf@linesnumbered% reset the numbering of the lines + }}% +% +%% allow to ajust the skip size of InOut +%% +\newcommand{\ResetInOut}[1]{% + \sbox\algocf@inoutbox{\hbox{\KwSty{#1}\algocf@typo:\ }}% + \setlength{\inoutsize}{\wd\algocf@inoutbox}% + }% +% +% +%%% +% text staying at the right of the keyword. +% +\newcommand{\algocf@newinput}{\par\parindent=\wd\algocf@inputbox}% to put right indentation after a \\ in the KwInput +\newcommand{\SetKwInput}[2]{% + \@ifundefined{#1}{\let\algocf@mkcmd=\newcommand}{\let\algocf@mkcmd=\renewcommand}% + \expandafter\algocf@mkcmd\csname#1\endcsname[1]{% + \sbox\algocf@inputbox{\hbox{\KwSty{#2}\algocf@typo: }}% + \ifthenelse{\boolean{algocf@inoutnumbered}}{\relax}{\everypar={\relax}}% + {\let\\\algocf@newinput\hangindent=\wd\algocf@inputbox\hangafter=1\unhbox\algocf@inputbox##1\par}% + \algocf@linesnumbered% reset the numbering of the lines + }}% +\newcommand{\SetKwData}[2]{% + \@ifundefined{#1}{\let\algocf@mkcmd=\newcommand}{\let\algocf@mkcmd=\renewcommand}% + \expandafter\algocf@mkcmd\csname @#1\endcsname[1]{\DataSty{#2(}\ArgSty{##1}\DataSty{)}}% + \expandafter\algocf@mkcmd\csname#1\endcsname{% + \@ifnextchar\bgroup{\csname @#1\endcsname}{\DataSty{#2}\xspace}}% + }% +% +% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% tallent: +% +% Add following macros: +% \SetKwHangingKw: [kw] ------------ <= hanging determined by [kw] +% ------------ +% Should act like a combination of \SetKwInput and \SetKw. +% Based on \SetKwInput: +% - remove ':' at end of keyword +% - do not reset numbering +% - use separate savebox +\newsavebox{\algocf@hangingbox} +\newcommand{\algocf@newhanging}{\par\parindent=\wd\algocf@hangingbox}% to put right indentation after a \\ in the KwInput +\newcommand{\SetKwHangingKw}[2]{% + \@ifundefined{#1}{\let\algocf@mkcmd=\newcommand}{\let\algocf@mkcmd=\renewcommand}% + \expandafter\algocf@mkcmd\csname#1\endcsname[1]{% + \sbox\algocf@hangingbox{\hbox{\KwSty{#2}\algocf@typo\ }}% + {\let\\\algocf@newhanging\hangindent=\wd\algocf@hangingbox\hangafter=1\unhbox\algocf@hangingbox##1\;}% + }% +}% +% +% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% +% Comments macros +% +%%%% +% comment in the text, first argument is the name of the macro, second is +% the text put before the comment, third is the text put at the end of the +% comment. +% +% first side comment justification +\newcommand{\SetSideCommentLeft}{\setboolean{algocf@scleft}{true}}% +\newcommand{\SetSideCommentRight}{\setboolean{algocf@scleft}{false}}% +\newcommand{\SetNoFillComment}{\setboolean{algocf@optfillcomment}{false}}% +\newcommand{\SetFillComment}{\setboolean{algocf@optfillcomment}{true}}% +% +% next comment and side comment +% +\newcommand{\algocf@endmarkcomment}{\relax}% +\newcommand{\algocf@fillcomment}{% + \ifthenelse{\boolean{algocf@optfillcomment}}{\hfill}{\relax}}% +% +\newcommand{\algocf@startcomment}{% + \hangindent=\wd\algocf@inputbox\hangafter=1\usebox\algocf@inputbox}% +\newcommand{\algocf@endcomment}{\algocf@fillcomment\algocf@endmarkcomment\ignorespaces\par}% +\newcommand{\algocf@endstartcomment}{\algocf@endcomment\algocf@startcomment\ignorespaces}% +% +\newboolean{algocf@sidecomment}% +\newboolean{algocf@altsidecomment}\setboolean{algocf@altsidecomment}{false}% +\newcommand{\algocf@scpar}{\ifthenelse{\boolean{algocf@altsidecomment}}{\relax}{\par}}% +\newcommand{\algocf@sclfill}{\ifthenelse{\boolean{algocf@scleft}}{\algocf@fillcomment}{\relax}}% +\newcommand{\algocf@scrfill}{\ifthenelse{\boolean{algocf@scleft}}{\relax}{\hfill}}% +\newcommand{\algocf@startsidecomment}{\usebox\algocf@inputbox}% +\newcommand{\algocf@endsidecomment}{\algocf@endmarkcomment\algocf@scpar}% +\newcommand{\algocf@endstartsidecomment}{% + \algocf@sclfill\algocf@endsidecomment% + \algocf@scrfill\algocf@startsidecomment\ignorespaces}% +% +\newcommand{\SetKwComment}[3]{% + % newcommand or renewcommand ? + \@ifundefined{#1}{\let\algocf@mkcmd=\newcommand}{\let\algocf@mkcmd=\renewcommand}% + %%% comment definition + \expandafter\algocf@mkcmd\csname algocf@#1\endcsname[1]{% + \sbox\algocf@inputbox{\CommentSty{\hbox{#2}}}% + \ifthenelse{\boolean{algocf@commentsnumbered}}{\relax}{\everypar={\relax}}% + {\renewcommand{\algocf@endmarkcomment}{#3}% + \let\\\algocf@endstartcomment% + \algocf@startcomment\CommentSty{% + \strut\ignorespaces##1\strut\algocf@fillcomment#3}\par}% + \algocf@linesnumbered% reset the numbering of the lines + }% + %%% side comment definitions + % option or not? + \expandafter\algocf@mkcmd\csname algocf@#1@star\endcsname{% + \@ifnextchar [{\csname algocf@#1@staropt\endcsname}{\csname algocf@#1@sidecomment\endcsname}% + }% + % manage option + \expandafter\def\csname algocf@#1@staropt\endcsname[##1]##2{% + \ifthenelse{\boolean{algocf@scleft}}{\setboolean{algocf@sidecomment}{true}}{\setboolean{algocf@sidecomment}{false}}% + \ifx##1h\setboolean{algocf@altsidecomment}{true}\SetSideCommentLeft\fi% + \ifx##1f\setboolean{algocf@altsidecomment}{true}\SetSideCommentRight\fi% + \ifx##1l\setboolean{algocf@altsidecomment}{false}\SetSideCommentLeft\fi% + \ifx##1r\setboolean{algocf@altsidecomment}{false}\SetSideCommentRight\fi% + \csname algocf@#1@sidecomment\endcsname{##2}% call sidecomment + \ifthenelse{\boolean{algocf@sidecomment}}{\setboolean{algocf@scleft}{true}}{\setboolean{algocf@scleft}{false}}% + \setboolean{algocf@altsidecomment}{false}% + }% + % side comment + \expandafter\algocf@mkcmd\csname algocf@#1@sidecomment\endcsname[1]{% + \sbox\algocf@inputbox{\CommentSty{\hbox{#2}}}% + \ifthenelse{\boolean{algocf@commentsnumbered}}{\relax}{\everypar={\relax}}% + {% + \renewcommand{\algocf@endmarkcomment}{#3}% + \let\\\algocf@endstartsidecomment% + % here is the comment + %\ifthenelse{\boolean{algocf@altsidecomment}}{\relax}{\algocf@endline\ }% + \ifthenelse{\boolean{algocf@altsidecomment}}{\relax}{\@endalgocfline\ }% + \algocf@scrfill\algocf@startsidecomment\CommentSty{% + \strut\ignorespaces##1\strut\algocf@sclfill#3}\algocf@scpar% + }% + \algocf@linesnumbered% reset the numbering of the lines + } + \expandafter\algocf@mkcmd\csname#1\endcsname{\@ifstar{\csname algocf@#1@star\endcsname}{\csname algocf@#1\endcsname}}% +}% +% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% +% Kw +% +\newcommand{\SetKw}[2]{% + \@ifundefined{#1}{\let\algocf@mkcmd=\newcommand}{\let\algocf@mkcmd=\renewcommand}% + \expandafter\algocf@mkcmd\csname @#1\endcsname[1]{\KwSty{#2} \ArgSty{##1}}% + \expandafter\algocf@mkcmd\csname#1\endcsname{% + \@ifnextchar\bgroup{\csname @#1\endcsname}{\KwSty{#2}\xspace}}% + }% +% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% +% KwFunction +% +\newcommand{\SetKwFunction}[2]{% +%%% use of gdef since newcommand doesn't manage to define the macro when SetKwFunction is used in \algocf@caption@proc + \expandafter\gdef\csname @#1\endcsname##1{\FuncSty{#2(}\ArgSty{##1}\FuncSty{)}}% + \expandafter\gdef\csname#1\endcsname{% + \@ifnextchar\bgroup{\csname @#1\endcsname}{\FuncSty{#2}\xspace}}% +}% +% +% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% +% KwTab +% +\newcommand{\SetKwArray}[2]{% +%%% use of gdef since newcommand doesn't manage to define the macro when SetKwFunction is used in \algocf@caption@proc + \expandafter\gdef\csname @#1\endcsname##1{\DataSty{#2[}\ArgSty{##1}\DataSty{]}}% + \expandafter\gdef\csname#1\endcsname{% + \@ifnextchar\bgroup{\csname @#1\endcsname}{\DataSty{#2}\xspace}}% +}% +% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% +% KwBlock +% +\newcommand{\SetKwBlock}[3]{% +\@ifundefined{algocf@#1}{\let\algocf@mkcmd=\newcommand}{\let\algocf@mkcmd=\renewcommand}% +% side text or not? +\expandafter\def\csname#1\endcsname{%Begin + \@ifnextchar({\csname algocf@#1opt\endcsname}{\csname algocf@#1\endcsname}}% +% with side text +\expandafter\def\csname algocf@#1opt\endcsname(##1)##2{% \Begin(){} + \KwSty{#2} ##1\algocf@block{##2}{\@algocf@endoption{#3}}% + \@ifnextchar({\csname algocf@#1end\endcsname}{\par}}% +% without side text at the beginning +\expandafter\algocf@mkcmd\csname algocf@#1\endcsname[1]{% \Begin{} + \KwSty{#2}\algocf@block{##1}{\@algocf@endoption{#3}}\@ifnextchar({\csname algocf@#1end\endcsname}{\par}}% +% side text at the end +\expandafter\def\csname algocf@#1end\endcsname(##1){% \Begin{} + \ ##1\par}% +}% +% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% +% For Switch +% +\newcommand{\SetKwSwitch}[8]{% #1=\Switch #2=\Case #3=\Other #4=swicth #5=case #6=do #7=otherwise #8=endsw +%%%% Switch +\@ifundefined{algocf@#1}{\let\algocf@mkcmd=\newcommand}{\let\algocf@mkcmd=\renewcommand}% +% side text or not? +\expandafter\def\csname#1\endcsname{%Switch + \@ifnextchar({\csname algocf@#1opt\endcsname}{\csname algocf@#1\endcsname}}% +% with side text +\expandafter\def\csname algocf@#1opt\endcsname(##1)##2##3{% \Switch(){}{} + \KwSty{#4} \ArgSty{##2} \KwSty{#5} ##1\algocf@block{##3}{\@algocf@endoption{#8}}}% +% without side text +\expandafter\algocf@mkcmd\csname algocf@#1\endcsname[2]{% \Switch{}{} + \KwSty{#4} \ArgSty{##1} \KwSty{#5}\algocf@block{##2}{\@algocf@endoption{#8}}}% +% side text at the end +\expandafter\def\csname algocf@#1end\endcsname(##1){% \Switch{}{}() +}% +% +%%%% Case +\@ifundefined{algocf@#2}{\let\algocf@mkcmd=\newcommand}{\let\algocf@mkcmd=\renewcommand}% +% side text or not? +\expandafter\def\csname#2\endcsname{%Case + \@ifnextchar({\csname algocf@#2opt\endcsname}{\csname algocf@#2\endcsname}}% +\expandafter\def\csname u#2\endcsname{%uCase + \@ifnextchar({\csname algocf@u#2opt\endcsname}{\csname algocf@u#2\endcsname}}% +\expandafter\def\csname l#2\endcsname{%lCase + \@ifnextchar({\csname algocf@l#2opt\endcsname}{\csname algocf@l#2\endcsname}}% +% with side text +\expandafter\def\csname algocf@#2opt\endcsname(##1)##2##3{% \Case(){}{} + \KwSty{#6} \ArgSty{##2} ##1\algocf@block{##3}{\@algocf@endoption{#8}}}% +\expandafter\def\csname algocf@u#2opt\endcsname(##1)##2##3{% \uCase(){}{} + \KwSty{#6} \ArgSty{##2} ##1\algocf@group{##3}}% +\expandafter\def\csname algocf@l#2opt\endcsname(##1)##2##3{% \lCase(){}{} + \KwSty{#6} \ArgSty{##2} ##3\algocf@endline\ ##1\par}% +% without side text +\expandafter\algocf@mkcmd\csname algocf@#2\endcsname[2]{% \Case{}{} + \KwSty{#6} \ArgSty{##1}\algocf@block{##2}{\@algocf@endoption{#8}}}% +\expandafter\algocf@mkcmd\csname algocf@u#2\endcsname[2]{% \uCase{}{} + \KwSty{#6} \ArgSty{##1}\algocf@group{##2}}% +\expandafter\algocf@mkcmd\csname algocf@l#2\endcsname[2]{% \lCase{}{} + \KwSty{#6} \ArgSty{##1} ##2}% +%%%% Other +\@ifundefined{algocf@#3}{\let\algocf@mkcmd=\newcommand}{\let\algocf@mkcmd=\renewcommand}% +% side text or not? +\expandafter\def\csname#3\endcsname{%Other + \@ifnextchar({\csname algocf@#3opt\endcsname}{\csname algocf@#3\endcsname}}% +\expandafter\def\csname l#3\endcsname{%Other + \@ifnextchar({\csname algocf@l#3opt\endcsname}{\csname algocf@l#3\endcsname}}% +% with side text +\expandafter\def\csname algocf@#3opt\endcsname(##1)##2{% \Other(){}{} + \KwSty{#7} ##1\algocf@block{##2}{\@algocf@endoption{#8}}}% +\expandafter\def\csname algocf@l#3opt\endcsname(##1)##2{% \Other(){}{} + \KwSty{#7} ##2\algocf@endline\ ##1\par}% +% without side text +\expandafter\algocf@mkcmd\csname algocf@#3\endcsname[1]{% default + \KwSty{#7}\algocf@block{##1}{\@algocf@endoption{#8}}}% +\expandafter\algocf@mkcmd\csname algocf@l#3\endcsname[1]{% ldefault + \KwSty{#7} ##1}% +}% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% +% If macros +% +\newcommand{\SetKwIF}[8]{% #1=\If #2=\ElseIf #3=\Else #4=if #5=then #6=elseif si #7=else #8=endif +% +% common text +\@ifundefined{#1@ifthen}{\let\algocf@mkcmd=\newcommand}{\let\algocf@mkcmd=\renewcommand}% +\expandafter\algocf@mkcmd\csname #1@ifthen\endcsname[1]{% + \KwSty{#4} \ArgSty{##1} \KwSty{#5}}% +\expandafter\algocf@mkcmd\csname #1@endif\endcsname[1]{\algocf@block{##1}{\@algocf@endoption{#8}}}% +\expandafter\algocf@mkcmd\csname #1@noend\endcsname[1]{\algocf@group{##1}}% +\expandafter\algocf@mkcmd\csname #1@else\endcsname[1]{\algocf@group{##1}\KwSty{#7}}% +\@ifundefined{#2@elseif}{\let\algocf@mkcmd=\newcommand}{\let\algocf@mkcmd=\renewcommand}% +\expandafter\algocf@mkcmd\csname #2@elseif\endcsname[1]{% + \KwSty{#6} \ArgSty{##1} \KwSty{#5}}% +\@ifundefined{#3@else}{\let\algocf@mkcmd=\newcommand}{\let\algocf@mkcmd=\renewcommand}% +\expandafter\algocf@mkcmd\csname #3@else\endcsname{\KwSty{#7}}% +%%%% If then { } endif +% +\@ifundefined{algocf@#1}{\let\algocf@mkcmd=\newcommand}{\let\algocf@mkcmd=\renewcommand}% +% side text or not? +\expandafter\def\csname#1\endcsname{% + \@ifnextchar({\csname algocf@#1opt\endcsname}{\csname algocf@#1\endcsname}}% +% with side text +\expandafter\def\csname algocf@#1opt\endcsname(##1)##2##3{% \If(){}{} + \csname #1@ifthen\endcsname{##2} ##1\csname #1@endif\endcsname{##3}}% +% without side text +\expandafter\algocf@mkcmd\csname algocf@#1\endcsname[2]{% \If{}{} + \csname #1@ifthen\endcsname{##1}\csname #1@endif\endcsname{##2}}% +% +%%%% If then {} else {} endif +% +% side text or not? +\expandafter\def\csname e#1\endcsname{% + \@ifnextchar({\csname algocf@e#1thenopt\endcsname}{\csname algocf@e#1then\endcsname}}% +% with side text after if +\expandafter\def\csname algocf@e#1thenopt\endcsname(##1)##2##3{% \eIf() + \csname #1@ifthen\endcsname{##2} ##1\csname #1@else\endcsname{##3}% + \csname algocf@e#1thenelse\endcsname}% +% without side text after if +\expandafter\def\csname algocf@e#1then\endcsname##1##2{% \eIf() + \csname #1@ifthen\endcsname{##1}\csname #1@else\endcsname{##2}% + \csname algocf@e#1thenelse\endcsname}% +% side text after else or not ? +\expandafter\def\csname algocf@e#1thenelse\endcsname{% + \@ifnextchar({\csname algocf@e#1elseopt\endcsname}{\csname algocf@e#1else\endcsname}}% +% else with a side text +\expandafter\def\csname algocf@e#1elseopt\endcsname(##1)##2{% + ##1\csname #1@endif\endcsname{##2}}% +% else without side text +\expandafter\algocf@mkcmd\csname algocf@e#1else\endcsname[1]{% + \csname #1@endif\endcsname{##1}}% +% +%%%% If then +% +% side text or not? +\expandafter\def\csname l#1\endcsname{% lif + \@ifnextchar({\csname algocf@l#1opt\endcsname}{\csname algocf@l#1\endcsname}}% +\expandafter\def\csname u#1\endcsname{% uif + \@ifnextchar({\csname algocf@u#1opt\endcsname}{\csname algocf@u#1\endcsname}}% +% with side text +\expandafter\def\csname algocf@l#1opt\endcsname(##1)##2##3{% \lIf(){}{} + \csname #1@ifthen\endcsname{##2} ##3\algocf@endline\ ##1\par}% +\expandafter\def\csname algocf@u#1opt\endcsname(##1)##2##3{% \uIf(){}{} + \csname #1@ifthen\endcsname{##2} ##1\csname#1@noend\endcsname{##3}}% +% without side text +\expandafter\algocf@mkcmd\csname algocf@l#1\endcsname[2]{% \lIf{}{} + \csname #1@ifthen\endcsname{##1} ##2}% +\expandafter\algocf@mkcmd\csname algocf@u#1\endcsname[2]{% \uIf{}{} + \csname #1@ifthen\endcsname{##1}\csname#1@noend\endcsname{##2}}% +% +%%%% ElseIf {} endif +% +\@ifundefined{algocf@#2}{\let\algocf@mkcmd=\newcommand}{\let\algocf@mkcmd=\renewcommand}% +% side text or not? +\expandafter\def\csname#2\endcsname{% ElseIf + \@ifnextchar({\csname algocf@#2opt\endcsname}{\csname algocf@#2\endcsname}}% +% with side text +\expandafter\def\csname algocf@#2opt\endcsname(##1)##2##3{% \ElseIf(){}{} + \csname #2@elseif\endcsname{##2} ##1\csname #1@endif\endcsname{##3}}% +% without side text +\expandafter\algocf@mkcmd\csname algocf@#2\endcsname[2]{% \ElseIf{}{} + \csname #2@elseif\endcsname{##1}\csname #1@endif\endcsname{##2}}% +% +%%%% ElseIf +% +% side text or not? +\expandafter\def\csname l#2\endcsname{% lElseIf + \@ifnextchar({\csname algocf@l#2opt\endcsname}{\csname algocf@l#2\endcsname}}% +\expandafter\def\csname u#2\endcsname{% uElseIf + \@ifnextchar({\csname algocf@u#2opt\endcsname}{\csname algocf@u#2\endcsname}}% +% with side text +\expandafter\def\csname algocf@l#2opt\endcsname(##1)##2##3{% \lElseIf(){}{} + \csname #2@elseif\endcsname{##2} ##3\algocf@endline\ ##1\par}% +\expandafter\def\csname algocf@u#2opt\endcsname(##1)##2##3{% \uElseIf(){}{} + \csname #2@elseif\endcsname{##2} ##1\csname #1@noend\endcsname{##3}}% +% without side text +\expandafter\algocf@mkcmd\csname algocf@l#2\endcsname[2]{% \lElseIf{}{} + \csname #2@elseif\endcsname{##1} ##2}% +\expandafter\algocf@mkcmd\csname algocf@u#2\endcsname[2]{% \uElseIf{}{} + \csname #2@elseif\endcsname{##1}\csname #1@noend\endcsname{##2}}% +% +%%%% Else {} endif +% +\@ifundefined{algocf@#3}{\let\algocf@mkcmd=\newcommand}{\let\algocf@mkcmd=\renewcommand}% +% side text or not? +\expandafter\def\csname#3\endcsname{% Else + \@ifnextchar({\csname algocf@#3opt\endcsname}{\csname algocf@#3\endcsname}}% +% with side text +\expandafter\def\csname algocf@#3opt\endcsname(##1)##2{% \Else(){} + \csname #3@else\endcsname\ ##1\csname #1@endif\endcsname{##2}}% +% without side text +\expandafter\algocf@mkcmd\csname algocf@#3\endcsname[1]{% \Else{} + \csname #3@else\endcsname\csname #1@endif\endcsname{##1}}% +% +%%%% Else +% +% side text or not? +\expandafter\def\csname l#3\endcsname{% lElse + \@ifnextchar({\csname algocf@l#3opt\endcsname}{\csname algocf@l#3\endcsname}}% +\expandafter\def\csname u#3\endcsname{% uElse + \@ifnextchar({\csname algocf@u#3opt\endcsname}{\csname algocf@u#3\endcsname}}% +% with side text +\expandafter\def\csname algocf@l#3opt\endcsname(##1)##2{% \lElse(){} + \csname #3@else\endcsname\ ##2\algocf@endline\ ##1\par}% +\expandafter\def\csname algocf@u#3opt\endcsname(##1)##2{% \uElse(){} + \csname #3@else\endcsname\ ##1\csname #1@noend\endcsname{##2}}% +% without side text +\expandafter\algocf@mkcmd\csname algocf@l#3\endcsname[1]{% \lElse{} + \csname #3@else\endcsname\ ##1}% +\expandafter\algocf@mkcmd\csname algocf@u#3\endcsname[1]{% \uElse{} + \csname #3@else\endcsname\csname #1@noend\endcsname{##1}}% +}% +% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% +% For macros +% +\newcommand{\SetKwFor}[4]{% +\@ifundefined{algocf@#1}{\let\algocf@mkcmd=\newcommand}{\let\algocf@mkcmd=\renewcommand}% +% side text or not? +\expandafter\def\csname#1\endcsname{%For + \@ifnextchar({\csname algocf@#1opt\endcsname}{\csname algocf@#1\endcsname}}% +\expandafter\def\csname l#1\endcsname{%For + \@ifnextchar({\csname algocf@l#1opt\endcsname}{\csname algocf@l#1\endcsname}}% +% with side text +\expandafter\def\csname algocf@#1opt\endcsname(##1)##2##3{% \For(){}{} + \KwSty{#2} \ArgSty{##2} \KwSty{#3} ##1\algocf@block{##3}{\@algocf@endoption{#4}}}% +\expandafter\def\csname algocf@l#1opt\endcsname(##1)##2##3{% \lFor(){}{} + \KwSty{#2} \ArgSty{##2} \KwSty{#3} ##3\algocf@endline\ ##1\par}% +% without side text +\expandafter\algocf@mkcmd\csname algocf@#1\endcsname[2]{% \For{}{} + \KwSty{#2} \ArgSty{##1} \KwSty{#3}\algocf@block{##2}{\@algocf@endoption{#4}}}% +\expandafter\algocf@mkcmd\csname algocf@l#1\endcsname[2]{% \lFor{}{} + \KwSty{#2} \ArgSty{##1} \KwSty{#3} ##2}% +}% +% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% +% Repeat macros +% +\newcommand{\SetKwRepeat}[3]{% +\@ifundefined{algocf@#1}{\let\algocf@mkcmd=\newcommand}{\let\algocf@mkcmd=\renewcommand}% +% side text or not? +\expandafter\def\csname#1\endcsname{% Repeat + \@ifnextchar({\csname algocf@#1opt\endcsname}{\csname algocf@#1\endcsname}}% +\expandafter\def\csname l#1\endcsname{% lRepeat + \@ifnextchar({\csname algocf@l#1opt\endcsname}{\csname algocf@l#1\endcsname}}% +% with side text +\expandafter\def\csname algocf@#1opt\endcsname(##1)##2##3{% \Repeat(){}{} + \KwSty{#2} ##1\algocf@group{##3}\KwSty{#3} \ArgSty{##2}% + \@ifnextchar({\csname algocf@#1optopt\endcsname}{\@endalgoln}% +}% +\expandafter\def\csname algocf@#1optopt\endcsname(##1){% \Repeat(){}{}() + ##1\@endalgoln}% +\expandafter\def\csname algocf@l#1opt\endcsname(##1)##2##3{% \lRepeat(){}{} + \KwSty{#2} ##3 \KwSty{#3} \ArgSty{##2}\algocf@endline\ ##1\par}% +% without side text +\expandafter\algocf@mkcmd\csname algocf@#1\endcsname[2]{% \Repeat{}{} + \KwSty{#2}\algocf@group{##2}\KwSty{#3} \ArgSty{##1}% + \@ifnextchar({\csname algocf@#1optopt\endcsname}{\@endalgoln}% +}% +\expandafter\algocf@mkcmd\csname algocf@l#1\endcsname[2]{% \lRepeat{}{} + \KwSty{#2} ##2 \KwSty{#3} \ArgSty{##1}}% +}% +% +% +% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%% Environments definitions %%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% +%% +%% Caption management +%% +% for the following macros: +% #1 is given by caption and is equal to fnum@algocf +% #2 is the text given in argument by the user in the \caption macro +% +%%%%% text of caption +\newcommand{\algocf@captionlayout}[1]{#1}% +\newcommand{\SetAlgoCaptionLayout}[1]{% + \renewcommand{\algocf@captionlayout}[1]{\csname #1\endcsname{##1}}}% +\newcommand{\algocf@capseparator}{:}% +\newcommand{\SetAlgoCaptionSeparator}[1]{\renewcommand{\algocf@capseparator}{#1}}% +\newcommand{\algocf@captiontext}[2]{% + \algocf@captionlayout{\AlCapSty{\AlCapFnt #1\algocf@typo\algocf@capseparator}\nobreakspace% + \AlCapNameSty{\AlCapNameFnt{}#2}}}% text of caption +% +%%%%% default caption of algorithm: used if no specific style caption is defined +\newcommand{\algocf@makecaption}[2]{% + \addtolength{\hsize}{\algomargin}% + \sbox\@tempboxa{\algocf@captiontext{#1}{#2}}% + \ifdim\wd\@tempboxa >\hsize% % if caption is longer than a line + \hskip .5\algomargin% + \parbox[t]{\hsize}{\algocf@captiontext{#1}{#2}}% then caption is not centered + \else% + \global\@minipagefalse% + \hbox to\hsize{\hfil\box\@tempboxa\hfil}% else caption is centered + \fi% + \addtolength{\hsize}{-\algomargin}% +}% +% +\newsavebox\algocf@capbox% +\newcommand{\algocf@makecaption@plain}[2]{% + \global\sbox\algocf@capbox{\algocf@makecaption{#1}{#2}}}% +\newcommand{\algocf@makecaption@boxed}[2]{% + \addtolength{\hsize}{-\algomargin}% + \global\sbox\algocf@capbox{\algocf@makecaption{#1}{#2}}% + \addtolength{\hsize}{\algomargin}% + }% +% +\newcommand{\algocf@makecaption@tworuled}[2]{\algocf@makecaption@ruled{#1}{#2}}% +\newcommand{\algocf@makecaption@algoruled}[2]{\algocf@makecaption@ruled{#1}{#2}}% +\newcommand{\algocf@makecaption@boxruled}[2]{\algocf@makecaption@ruled{#1}{#2}}% +\newcommand{\algocf@makecaption@ruled}[2]{% + \global\sbox\algocf@capbox{\hskip\AlCapHSkip% .5\algomargin% + \parbox[t]{\hsize}{\algocf@captiontext{#1}{#2}}}% then caption is not centered +}% +% +\newlength{\algoheightruledefault}\setlength{\algoheightruledefault}{0.8pt}% +\newlength{\algoheightrule}\setlength{\algoheightrule}{\algoheightruledefault}% +\newlength{\algotitleheightruledefault}\setlength{\algotitleheightruledefault}{0.8pt}% +\newlength{\algotitleheightrule}\setlength{\algotitleheightrule}{\algotitleheightruledefault}% +\newcommand{\algocf@caption@plain}{\vskip\AlCapSkip\box\algocf@capbox}% +\newcommand{\algocf@caption@boxed}{\vskip\AlCapSkip\box\algocf@capbox}% +\newcommand{\algocf@caption@ruled}{\box\algocf@capbox\kern\interspacetitleruled\hrule height\algotitleheightrule depth0pt\kern\interspacealgoruled}% +\newcommand{\algocf@caption@tworuled}{\box\algocf@capbox\kern\interspacetitleruled}% +\newcommand{\algocf@caption@algoruled}{\algocf@caption@ruled}% +\newcommand{\algocf@caption@boxruled}{% + \addtolength{\hsize}{-0.8pt}% + \hbox to\hsize{% + \vrule%\hskip-0.35pt% + \vbox{% + \hrule\vskip\interspacetitleboxruled% + \hbox to\hsize{\unhbox\algocf@capbox\hfill}\vskip\interspacetitleboxruled% + }% + %\hskip-0.35pt% + \vrule% + }\nointerlineskip% + \addtolength{\hsize}{0.8pt}% +}% +% +% +%%%% set caption for the environment +\newcommand{\algocf@captionref}{% + \renewcommand{\fnum@algocf}[1]{\AlCapSty{\AlCapFnt\algorithmcfname\nobreakspace\algocf@algocfref}}% + \addtocounter{algocf}{-1}% \caption do a refstepcounter, so we restore the precedent value + \let\old@thealgocf=\thealgocf\renewcommand{\thealgocf}{{\relsize{\algocf@refrelsize}\algocf@algocfref}}% + \gdef\@currentlabel{\algocf@algocfref}% let the label use the new ref +}% +% +% Unfortunatly, we also need our own caption to set some specific stuff for special references. But after these +% settings, we call the original caption. +% +\long\def\algocf@caption@algo#1[#2]#3{% + \ifthenelse{\equal{\algocf@algocfref}{\relax}}{}{\algocf@captionref}% + \@ifundefined{hyper@refstepcounter}{\relax}{% if hyper@refstepcounter undefind, no hyperref, else... + \ifthenelse{\equal{\algocf@algocfref}{\relax}}{\renewcommand{\theHalgocf}{\thealgocf}}{% take algocf as Href + \renewcommand{\theHalgocf}{\algocf@algocfref}}%else if SetAlgoRefName done, take this name as ref. + \hyper@refstepcounter{algocf}%set algocf as category of ref + }% + \algocf@latexcaption{#1}[#2]{#3}% call original caption +}% +% +% beamer define is own caption overrinding latex caption! +% as we need it, we have put here the original definition +% to handle manual ref, unfortunately we have to add one line to handle algocf@algocfref +\long\def\algocf@latexcaption#1[#2]#3{% original definition of caption + \par% + \addcontentsline{\csname ext@#1\endcsname}{#1}% + {\protect\numberline{\csname the#1\endcsname}{\ignorespaces #2}}% + \begingroup% + \@parboxrestore% + \if@minipage% + \@setminipage% + \fi% + \normalsize% + \@makecaption{\csname fnum@#1\endcsname}{\ignorespaces #3}\par% + \endgroup% +}% +% +% \ifx\beamer@makecaption\undefined% +% \else% beamer detected +\ifx\@makecaption\undefined% +\newcommand{\@makecaption}[2]{\relax}% +\fi% +%% + +% +% more and more packages redefine \@caption instead of just \@makecaption which makes algorithm2e +% caption not works since based on standard \@caption. So we force the definition of \@caption to be +% the standard one (the one from LaTeX) inside algorithm environment. +% +% unfortunately, makecaption is called with \ignorespace #3 so +% we can't do the @currentlabel definition inside \algocf@captionproctext +\long\def\algocf@caption@proc#1[#2]#3{% + \ifthenelse{\boolean{algocf@nokwfunc}}{\relax}{% + \SetKwFunction{\algocf@captname#3@}{\algocf@captname#3@}% + }% + % we tell hyperref to use algocfproc as category and to take the appropriate ref. + \ifthenelse{\boolean{algocf@func}}{\def\@proc@func{algocffunc}}{\def\@proc@func{algocfproc}}% + \@ifundefined{hyper@refstepcounter}{\relax}{% if hyper@refstepcounter undefind, no hyperref, else... + \ifthenelse{\boolean{algocf@procnumbered}}{% + \expandafter\def\csname theH\@proc@func\endcsname{\algocf@captname#3@}%if procnumbered, take \thealgocf as ref + }{% + \expandafter\def\csname theH\@proc@func\endcsname{\algocf@captname#3@}%else take procedure or function name + }% + \hyper@refstepcounter{\@proc@func}% + }% + \ifthenelse{\boolean{algocf@procnumbered}}{\relax}{% + \addtocounter{algocf}{-1}% \caption do a refstepcounter, so we restore the precedent value + \gdef\@currentlabel{\algocf@captname#3@}% let the label be the name of the function, not the counter + }% + \ifthenelse{\equal{\algocf@captparam#2@}{\arg@e}}{% if no paramater, we remove the () + \algocf@latexcaption{#1}[\algocf@procname\nobreakspace\algocf@captname#2@]{#3}% + }{% else we give the complete name + \algocf@latexcaption{#1}[\algocf@procname\nobreakspace#2]{#3}% + }% +}% +%% +%%% setcaption +\newcommand{\algocf@setcaption}{% + \ifthenelse{\boolean{algocf@procenvironment}}{% if proc environment, caption text must be changed + \let\algocf@oldcaptiontext=\algocf@captiontext% + \renewcommand{\algocf@captiontext}[2]{% + \algocf@captionproctext{##1}{##2}% + }% + }{}% + \let\algocf@savecaption=\@caption% + \ifthenelse{\boolean{algocf@procenvironment}}{\let\@caption=\algocf@caption@proc}{\let\@caption=\algocf@caption@algo}% + \let\algocf@oldmakecaption=\@makecaption% + \renewcommand{\@makecaption}[2]{% + \expandafter\csname algocf@makecaption@\algocf@style\endcsname{##1}{##2}% + }% +}% +% +%%%%% reset caption +% +% since we have force the LaTeX caption for algorithm environment, we must go back to the caption +% used in the text. +\newcommand{\algocf@resetcaption}{% + \ifthenelse{\boolean{algocf@procenvironment}}{% if proc environment + \let\thealgocf=\old@thealgocf% restore normal counter printing + \let\algocf@captiontext=\algocf@oldcaptiontext% restore normal caption text + }{}% + \let\@caption=\algocf@savecaption% now restore caption outside algo/proc/func environment + \let\@makecaption=\algocf@oldmakecaption% and restore makecaption outside outside algo/proc/func environment + \algocf@resetfnum% +}% +% +%%%%% nocaptionofalgo and restorecaptionofalgo -- +\newcommand{\NoCaptionOfAlgo}{% + \let\@old@algocf@captiontext=\algocf@captiontext% + \renewcommand{\algocf@captiontext}[2]{\AlCapNameSty{\AlCapNameFnt{}##2}}% +}% +\newcommand{\RestoreCaptionOfAlgo}{% + \let\algocf@captiontext=\@old@algocf@captiontext% +}% +% +% ---------------------- algocf environment +% +\newcounter{algocfline}% % new counter to make lines numbers be internally +\setcounter{algocfline}{0}% % different in different algorithms +\newcounter{algocfproc}% counter to count all algo environment (proc, func), just used by hyperref to avoir "same +\setcounter{algocfproc}{0}% identifier" error caused by algocf being set to '-' for procedure or function or not + % changed if no caption is given. +% +\expandafter\ifx\csname algocf@within\endcsname\relax% if \algocf@within doesn't exist +\newcounter{algocf}% % just define a new counter +\renewcommand{\thealgocf}{\@arabic\c@algocf}% and the way it is printed +\else% else +\newcounter{algocf}[\algocf@within]% % counter is numbered within \algocf@within +\renewcommand\thealgocf{\csname the\algocf@within\endcsname.\@arabic\c@algocf}% +\fi% +% +\def\fps@algocf{htbp}% % default +\def\ftype@algocf{10}% % float type +\def\ext@algocf{\algocf@list} % loa by default, lof if figure option used +\newcommand{\fnum@algocf}[1]{\AlCapSty{\AlCapFnt\algorithmcfname\nobreakspace\thealgocf}}% +\newcommand{\algocf@resetfnum}{\renewcommand{\fnum@algocf}[1]{\AlCapSty{\AlCapFnt\algorithmcfname\nobreakspace\thealgocf}}}% +\newenvironment{algocf}% % float environment for algorithms + {\@float{algocf}}% + {\end@float}% +\newenvironment{algocf*}% % float* environment for algorithms + {\@dblfloat{algocf}}% + {\end@dblfloat}% +% +\def\algocf@seclistalgo{}% +\ifx\l@chapter\undefined\let\algocf@seclistalgo=\section\else\let\algocf@seclistalgo=\chapter\fi% +\@ifundefined{if@restonecol}{\newif\if@restonecol}\relax% +\newcommand\listofalgocfs{% + \ifx\algocf@seclistalgo\chapter% + \if@twocolumn\@restonecoltrue\onecolumn\else\@restonecolfalse\fi% + \fi% + \algocf@seclistalgo*{\listalgorithmcfname}% + \@mkboth{\MakeUppercase\listalgorithmcfname}% + {\MakeUppercase\listalgorithmcfname}% + \@starttoc{loa}% + \ifx\algocf@seclistalgo\chapter% + \if@restonecol\twocolumn\fi% + \fi% +} +% +\newcommand*\l@algocf{\@dottedtocline{1}{1em}{2.3em}}% line of the list +% +% ---------------------- algorithm environment +% +%%%%%%% +%% +%% Algorithm environment definition +%% +%%%%%%% +%% +% +\newsavebox\algocf@algoframe% +\def\@algocf@pre@plain{\relax}% action to be done before printing the algo. +\def\@algocf@post@plain{\relax}% action to be done after printing the algo. +\def\@algocf@capt@plain{bottom}% where the caption should be localized. +\def\@algocf@pre@boxed{\noindent\begin{lrbox}{\algocf@algoframe}} +\def\@algocf@post@boxed{\end{lrbox}\framebox[\hsize]{\box\algocf@algoframe}\par}% +\def\@algocf@capt@boxed{under}% +\def\@algocf@pre@ruled{\hrule height\algoheightrule depth0pt\kern\interspacetitleruled}% +\def\@algocf@post@ruled{\kern\interspacealgoruled\hrule height\algoheightrule\relax}% +\def\@algocf@capt@ruled{top}% +\def\@algocf@pre@algoruled{\hrule height\algoheightrule depth0pt\kern\interspacetitleruled}% +\def\@algocf@post@algoruled{\kern\interspacealgoruled\hrule height\algoheightrule \relax}% +\def\@algocf@capt@algoruled{top}% +\def\@algocf@pre@tworuled{\hrule height\algoheightrule depth0pt\kern\interspacetitleruled}% +\def\@algocf@post@tworuled{\kern\interspacealgoruled\hrule height\algoheightrule\relax}% +\def\@algocf@capt@tworuled{top}% +\def\@algocf@pre@boxruled{\noindent\begin{lrbox}{\algocf@algoframe}}% +\def\@algocf@post@boxruled{\end{lrbox}\framebox[\hsize]{\box\algocf@algoframe}\par}% +\def\@algocf@capt@boxruled{above}% +% +\newcommand{\noalgocaption}{\def\@algocf@capt@ruled{none}} +% +%% before algocf or figure environment +\newcommand{\@algocf@init@caption}{% + \ifthenelse{\boolean{algocf@procenvironment}}{% if we are inside a procedure/function environment + \@algocf@proctitleofalgo% set Titleofalgo to Procedure: or Function: + % accordingly to the environment + \let\old@thealgocf=\thealgocf\ifthenelse{\boolean{algocf@procnumbered}}{\relax}{% + \renewcommand{\thealgocf}{-}}% + }{% else inside environment algorithm + \@algocf@algotitleofalgo% fix name for \Titleofalgo to \algorithmcfname + }% + \algocf@setcaption% set caption to our caption style +}% +% +\newcommand{\@algofloatboxreset}{\@setminipage} +\newcommand{\@algocf@init}{% + \refstepcounter{algocfline}% + \stepcounter{algocfproc}%to have a different counter for each environment and being abble to make the difference + %between href of algoline in different algorithms. + \ifthenelse{\boolean{algocf@optnoend}}{% + \renewcommand{\algocf@block}[2]{\algocf@group{##1}}% + }{% + \renewcommand{\algocf@block}[2]{\algocf@@block{##1}{##2}}% + }% +}% +%% after the end of algocf or figure environment +\newcommand{\@algocf@term@caption}{% + \algocf@resetcaption% restore original caption +}% +% +\newcommand{\@algocf@term}{% + \setboolean{algocf@algoH}{false}% no H by default + \ifthenelse{\boolean{algocf@optnoend}}{% + \renewcommand{\algocf@block}[2]{\algocf@@block{##1}{##2}}% + }{% + \renewcommand{\algocf@block}[2]{\algocf@group{##1}}% + }% + \SetAlgoRefName{\relax}% +}% +% +%%%%%%%%%%%%%%%%% +%% makethealgo: macro which print effectively the algo in its box +%% +\newsavebox\algocf@algobox% +\newcommand{\algocf@makethealgo}{% + \vtop{% + % place caption above if needed bye the style + \ifthenelse{\equal{\csname @algocf@capt@\algocf@style\endcsname}{above}}% + {\csname algocf@caption@\algocf@style\endcsname}{}% + % + % precommand according to the style + \csname @algocf@pre@\algocf@style\endcsname% + % place caption at top if needed bye the style + \ifthenelse{\equal{\csname @algocf@capt@\algocf@style\endcsname}{top}}% + {\csname algocf@caption@\algocf@style\endcsname}{}% + % + \box\algocf@algobox% the algo + % place caption at bottom if needed bye the style + \ifthenelse{\equal{\csname @algocf@capt@\algocf@style\endcsname}{bottom}}% + {\csname algocf@caption@\algocf@style\endcsname}{}% + % postcommand according to the style + \csname @algocf@post@\algocf@style\endcsname% + % place caption under if needed bye the style + \ifthenelse{\equal{\csname @algocf@capt@\algocf@style\endcsname}{under}}% + {\csname algocf@caption@\algocf@style\endcsname}{}% + }% +}% +%%%%%%%%%%%%%%%%%%% +% +%% at the beginning of algocf or figure environment +\newcommand{\@algocf@start}{% + \@algoskip% + \begin{lrbox}{\algocf@algobox}% + \setlength{\algowidth}{\hsize}% + \vbox\bgroup% save all the algo in a box + \hbox to\algowidth\bgroup\hbox to \algomargin{\hfill}\vtop\bgroup% + \ifthenelse{\boolean{algocf@slide}}{\parskip 0.5ex\color{black}}{}% + % initialization + \addtolength{\hsize}{-1.5\algomargin}% + \let\@mathsemicolon=\;\def\;{\ifmmode\@mathsemicolon\else\@endalgoln\fi}% + \raggedright\AlFnt{}% + \ifthenelse{\boolean{algocf@slide}}{\IncMargin{\skipalgocfslide}}{}% + \@algoinsideskip% +}% +% +%% at the end of algocf or figure environment +\newcommand{\@algocf@finish}{% + \@algoinsideskip% + \egroup%end of vtop which contain all the text + \hfill\egroup%end of hbox wich contains [margin][vtop] + \ifthenelse{\boolean{algocf@slide}}{\DecMargin{\skipalgocfslide}}{}% + % + \egroup%end of main vbox + \end{lrbox}% + \algocf@makethealgo% print the algo + \@algoskip% + % restore dimension and macros + \setlength{\hsize}{\algowidth}% + \lineskip\normallineskip\setlength{\skiptotal}{\@defaultskiptotal}% + \let\;=\@mathsemicolon% +}% +% +%%%%%%%%%%%%%%%%%%%% +%% basic definition of the environment algorithm +%% +% +\newboolean{algocf@procenvironment}\setboolean{algocf@procenvironment}{false}% +\newboolean{algocf@func}\setboolean{algocf@func}{false}% +\newboolean{algocf@algoH}\setboolean{algocf@algoH}{false}% +\newboolean{algocf@algostar}\setboolean{algocf@algostar}{false}% +% +%%% environment for {algorithm}[H] +\newenvironment{algocf@Here}{\noindent% + \def\@captype{algocf}% if not defined, caption exit with an error + \begin{minipage}{\hsize}% +}{% + \end{minipage}%\par% +}% +%%% real algorithm environment which manages H and * option +% \let\algocf@originalfloatboxreset=\@floatboxreset% +% \let\@floatboxreset=\@algofloatboxreset% +\newenvironment{algocf@algorithm}[1][htbp]{ + \@algocf@init% + \ifthenelse{\equal{\algocf@float}{figure}}{% if option figure set + \ifthenelse{\boolean{algocf@algostar}}{% if algorithm* with figure option + \begin{figure*}[#1]% call figure* + }{% else algorithm environment with figure option + \begin{figure}[#1]% call figure + }% + }{% else normal algorithm environment + \@algocf@init@caption% + \ifthenelse{\equal{#1}{H}}{% if [H] algorithm + \if@twocolumn\@latex@error{[H] in two columns mode is not allowed for algorithms}\fi% TODO: SCREAM if H in two colums! + \setboolean{algocf@algoH}{true}\begin{algocf@Here}% call corresponding environment + }{% else floating algorithm environment + \ifthenelse{\boolean{algocf@algostar}}{% if algorithm* + \begin{algocf*}[#1]% call algocf* + }{% else algorithm environment + \begin{algocf}[#1]% call algcf + }% + }% + }% fin test option figure ou pas + \@algocf@start% + \@ResetCounterIfNeeded% + \algocf@linesnumbered\ignorespaces% +}{% + \@algocf@finish% + \ifthenelse{\equal{\algocf@float}{figure}}{% + \ifthenelse{\boolean{algocf@algostar}}{% if algorithm* with figure option + \end{figure*}% call figure* + }{% else algorithm environment with figure option + \end{figure}% call figure + }% + }{% + \@algocf@term@caption% + \ifthenelse{\boolean{algocf@algoH}}{% if [H] algorithm + \end{algocf@Here}\par% call corresponding environment + }{% else floating algorithm environment + \ifthenelse{\boolean{algocf@algostar}}{% if algorithm* + \end{algocf*}% call algocf* + }{% else algorithm environment + \end{algocf}% call algocf + }% + }% + }% + \@algocf@term\ignorespacesafterend% +}% +% +%%% user algorithm environment +\newenvironment{\algocf@envname}[1][htbp]{% + \setboolean{algocf@algostar}{false}% + \setboolean{algocf@procenvironment}{false}\gdef\algocfautorefname{\algorithmautorefname}% + \begin{algocf@algorithm}[#1]\ignorespaces% +}{% + \end{algocf@algorithm}\ignorespacesafterend% +}% +%%% user algorithm* environment +\newenvironment{\algocf@envname*}[1][htbp]{% + \setboolean{algocf@algostar}{true}% + \setboolean{algocf@procenvironment}{false}\gdef\algocfautorefname{\algorithmautorefname}% + \begin{algocf@algorithm}[#1]\ignorespaces% +}{% + \end{algocf@algorithm}\ignorespacesafterend% +}% +% +%%%%%%%%%%%%%%%%%%%%%%% +%%% +% +\expandafter\newcommand\csname\algocf@listofalgorithms\endcsname{% + \ifthenelse{\equal{\algocf@float}{figure}}{\listoffigures}{\listofalgocfs}% +}% +%%% +%%% +% +% ---------------------- procedure and function environments +% +% +% -- new style (used in particular in the caption of function and procedure environments) +% +% three macros to extract parts of the caption +\gdef\algocf@captname#1(#2)#3@{#1} % keep characters before the first brace +\gdef\algocf@captparam#1(#2)#3@{#2} % keep character in between the braces +\gdef\algocf@captother#1(#2)#3@{#3} % keep character after the braces +% +%%% Text of caption for Procedure or Function +\newcommand{\algocf@captionproctext}[2]{% + {% + \ProcSty{\ProcFnt\algocf@procname\ifthenelse{\boolean{algocf@procnumbered}}{\nobreakspace\thealgocf\algocf@typo\algocf@capseparator}{\relax}}% + \nobreakspace\ProcNameSty{\ProcNameFnt\algocf@captname #2@}% Name of the procedure in ProcName Style. + \ifthenelse{\equal{\algocf@captparam #2@}{\arg@e}}{}{% if no argument, write nothing + \ProcNameSty{\ProcNameFnt(}\ProcArgSty{\ProcArgFnt\algocf@captparam #2@}\ProcNameSty{\ProcNameFnt)}%else put arguments in ProcArgSty: + }% endif + \algocf@captother #2@% + }% +}% +% +% +% -- procedure and function environments are defined from algocf@algorithm environment +% +\newenvironment{procedure}[1][htbp]{% + \setboolean{algocf@algostar}{false}% + \setboolean{algocf@procenvironment}{true}\setboolean{algocf@func}{false}% + \newcommand{\algocf@procname}{\@algocf@procname}\gdef\algocfprocautorefname{\procedureautorefname}% + \begin{algocf@algorithm}[#1]\ignorespaces% +}{% + \end{algocf@algorithm}\ignorespacesafterend% +}% +\newenvironment{function}[1][htbp]{% + \setboolean{algocf@algostar}{false}% + \setboolean{algocf@procenvironment}{true}\setboolean{algocf@func}{true}% + \newcommand{\algocf@procname}{\@algocf@funcname}\gdef\algocffuncautorefname{\functionautorefname}% + \begin{algocf@algorithm}[#1]\ignorespaces% +}{% + \end{algocf@algorithm}\ignorespacesafterend% +}% +% +\newenvironment{procedure*}[1][htbp]{% + \setboolean{algocf@algostar}{true}% + \setboolean{algocf@procenvironment}{true}\setboolean{algocf@func}{false}% + \newcommand{\algocf@procname}{\@algocf@procname}\gdef\algocfprocautorefname{\procedureautorefname}% + \begin{algocf@algorithm}[#1]\ignorespaces% +}{% + \end{algocf@algorithm}\ignorespacesafterend% +}% +\newenvironment{function*}[1][htbp]{% + \setboolean{algocf@algostar}{true}% + \setboolean{algocf@procenvironment}{true}\setboolean{algocf@func}{true}% + \newcommand{\algocf@procname}{\@algocf@funcname}\gdef\algocffuncautorefname{\functionautorefname}% + \begin{algocf@algorithm}[#1]\ignorespaces% +}{% + \end{algocf@algorithm}\ignorespacesafterend% +}% +% +% +%%%%%%%%%%%%%%%%%%%% +%% definition of algondfloat environment +%% +\ifthenelse{\boolean{algocf@endfloat}}{% if endfloat option then +\newcommand{\algoplace}{% macro which is used to writhe algorithm about there + \begin{center}% + [\algorithmcfname~\thepostfig\ about here.]% + \end{center}% +}% +\newcommand{\algoendfloat}{% use as a \begin{algoendfloat} environment to start scanning of line +% \immediate\openout\@mainfff\jobname.fff% + \efloat@condopen{fff} + \efloat@iwrite{fff}{\string\begin{\algocf@envname}}% + \if@domarkers% + \ifthenelse{\equal{\algocf@list}{lof}}{% + \addtocounter{postfig}{1}% + }{% + \addtocounter{postalgo}{1}% + }% + \algoplace% + \fi% + \bgroup% + \let\do\ef@makeinnocent\dospecials% + \ef@makeinnocent\^^L% and whatever other special cases + \endlinechar`\^^M \catcode`\^^M=12 \ef@xalgocfendfloat}% +}{\relax}%%%% end of endfloat option ifthenelse +%% some macros useful for endfloat option that cannot be defined inside the ifthenelse +%scan algoendfloat algorithm and write the text into .fff file +{\catcode`\^^M=12 \endlinechar=-1 % + \gdef\ef@xalgocfendfloat#1^^M{% scan the lines inside algoendfloat environment being read by latex + \def\test{#1}% test is the line being currently scan by latex + \ifx\test\ef@endalgocftest% if it is \end{algoendfloat} + \def\next{% define next as to not continue the scan and write \end{algorithm} into .fff file + \egroup\end{algoendfloat}% + \efloat@iwrite{fff}{\string\end{\algocf@envname}}% + \efloat@iwrite{fff}{\string\efloatseparator}% + \efloat@iwrite{fff}{ }% + }% + \else% else write the current line being scanned by latex and set next to continue the scan + \efloat@iwrite{fff}{#1}% + \let\next\ef@xalgocfendfloat% + \fi% endif + \next}% next is continue if it was else condition, else it does not continue the scan and write end to file +}% +% test if the scan is finish by looking at the string \end{algoendfloat} +{\escapechar=-1% + \xdef\ef@endalgocftest{\string\\end\string\{algoendfloat\string\}}% +}% +% +% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% +% +\newcommand{\TitleOfAlgo}[1]{\@titleprefix\ + \TitleSty{#1}\par\smallskip}% +% +\newcommand{\SetAlgorithmName}[3]{% + \renewcommand{\listalgorithmcfname}{#3}% + \renewcommand{\algorithmcfname}{#1}% + \renewcommand{\algorithmautorefname}{#2}% +}% +% +\newcommand{\algocf@refrelsize}{-2}\newcommand{\SetAlgoRefRelativeSize}[1]{\renewcommand{\algocf@refrelsize}{#1}}% +\newcommand{\SetAlgoRefName}[1]{% + \renewcommand{\algocf@algocfref}{#1}% +}% +% +% +% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% +% +% ------------------------- Default Definitions +% +%% +%% +% +\SetKwComment{tcc}{/* }{ */}% +\SetKwComment{tcp}{// }{}% +% +%\newcommand{\algocf@defaults@common}{ +% +% +% french keywords +% +%\SetKwInOut{AlgDonnees}{Donn\'ees}\SetKwInOut{AlgRes}{R\'esultat} +\SetKwInput{Donnees}{Donn\'ees}% +\SetKwInput{Res}{R\'esultat}% +\SetKwInput{Entree}{Entr\'ees}% +\SetKwInput{Sortie}{Sorties}% +\SetKw{KwA}{\`a}% +\SetKw{Retour}{retourner}% +\SetKwBlock{Deb}{d\'ebut}{fin}% +\SetKwRepeat{Repeter}{r\'ep\'eter}{jusqu'\`a}% +% +\SetKwIF{Si}{SinonSi}{Sinon}{si}{alors}{sinon si}{sinon}{fin si}% +\SetKwSwitch{Suivant}{Cas}{Autre}{suivant}{faire}{cas o\`u}{autres cas}{fin d'alternative}% +\SetKwFor{Pour}{pour}{faire}{fin pour}% +\SetKwFor{PourPar}{pour}{faire en parallèle}{fin pour}% +\SetKwFor{PourCh}{pour chaque}{faire}{fin pour chaque}% +\SetKwFor{PourTous}{pour tous les}{faire}{fin pour tous}% +\SetKwFor{Tq}{tant que}{faire}{fin tq}% +% +% english keywords +% +\SetKwInput{KwIn}{Input}% +\SetKwInput{KwOut}{Output}% +\SetKwInput{KwData}{Data}% +\SetKwInput{KwResult}{Result}% +\SetKw{KwTo}{to} +\SetKw{KwRet}{return}% +\SetKw{Return}{return}% +\SetKwBlock{Begin}{begin}{end}% +\SetKwRepeat{Repeat}{repeat}{until}% +% +\SetKwIF{If}{ElseIf}{Else}{if}{then}{else if}{else}{end if}% +\SetKwSwitch{Switch}{Case}{Other}{switch}{do}{case}{otherwise}{end switch}% +\SetKwFor{For}{for}{do}{end for}% +\SetKwFor{ForPar}{for}{do in parallel}{end forpar} +\SetKwFor{ForEach}{foreach}{do}{end foreach}% +\SetKwFor{ForAll}{forall the}{do}{end forall}% +\SetKwFor{While}{while}{do}{end while}% +% +% --- German keywords +% +\SetKwInput{Ein}{Eingabe}%KwIn +\SetKwInput{Aus}{Ausgabe}%KwOut +\SetKwInput{Daten}{Daten}%KwData +\SetKwInput{Ergebnis}{Ergebnis}%KwResult +\SetKw{Bis}{bis}%KwTo +\SetKw{KwZurueck}{zur\"uck}%KwRet +\SetKw{Zurueck}{zur\"uck}%Return +\SetKwBlock{Beginn}{Beginn}{Ende}%Begin +\SetKwRepeat{Wiederh}{wiederhole}{bis}%Repeat +% +\SetKwIF{Wenn}{SonstWenn}{Sonst}{wenn}{dann}{sonst wenn}{sonst}{Ende wenn}%gIf +\SetKwSwitch{Unterscheide}{Fall}{Anderes}{unterscheide}{tue}{Fall}{sonst}{Ende Unt.}%Switch +\SetKwFor{Fuer}{f\"ur}{tue}{Ende f\"ur}%For +\SetKwFor{FuerPar}{f\"ur}{tue gleichzeitig}{Ende gleichzeitig}%ForPar +\SetKwFor{FuerJedes}{f\"ur jedes}{tue}{Ende f\"ur}%ForEach +\SetKwFor{FuerAlle}{f\"ur alle}{tue}{Ende f\"ur}%ForAll +\SetKwFor{Solange}{solange}{tue}{Ende solange}%While +% +% --- Czech keywords +% +\SetKwInput{Vst}{Vstup}% +\SetKwInput{Vyst}{V\'{y}stup}% +\SetKwInput{Vysl}{V\'{y}sledek}% +% +% --- Portuguese keywords +% +\SetKwInput{Entrada}{Entrada}% +\SetKwInput{Saida}{Sa\'{i}da}% +\SetKwInput{Dados}{Dados}% +\SetKwInput{Resultado}{Resultado}% +\SetKw{Ate}{at\'{e}} +\SetKw{KwRetorna}{retorna}% +\SetKw{Retorna}{retorna}% +\SetKwBlock{Inicio}{in\'{i}cio}{fim}% +\SetKwRepeat{Repita}{repita}{at\'{e}}% +% +\SetKwIF{Se}{SenaoSe}{Senao}{se}{ent\~{a}o}{sen\~{a}o se}{sen\~{a}o}{fim se}% +\SetKwSwitch{Selec}{Caso}{Outro}{selecione}{fa\c{c}a}{caso}{sen\~{a}o}{fim selec}% +\SetKwFor{Para}{para}{fa\c{c}a}{fim para}% +\SetKwFor{ParaPar}{para}{fa\c{c}a em paralelo}{fim para} +\SetKwFor{ParaCada}{para cada}{fa\c{c}a}{fim para cada}% +\SetKwFor{ParaTodo}{para todo}{fa\c{c}a}{fim para todo}% +\SetKwFor{Enqto}{enquanto}{fa\c{c}a}{fim enqto}% +% +% --- Italian keywords +% +\SetKwInput{KwIng}{Ingresso}% +\SetKwInput{KwUsc}{Uscita}% +\SetKwInput{KwDati}{Dati}% +\SetKwInput{KwRisult}{Risultato}% +\SetKw{KwA}{a}% +\SetKw{KwRitorna}{ritorna}% +\SetKw{Ritorna}{ritorna}% +\SetKwBlock{Inizio}{inizio}{fine}% +\SetKwRepeat{Ripeti}{ripeti}{finché}% +% +\SetKwIF{Sea}{AltSe}{Altrimenti}{se}{allora}{altrimenti se}{allora}{fine se}% +\SetKwSwitch{Switch}{Case}{Other}{switch}{do}{case}{otherwise}{endsw}% +\SetKwFor{Per}{per}{fai}{fine per}% +\SetKwFor{PerPar}{per}{fai in parallelo}{fine per}% +\SetKwFor{PerCiascun}{per ciascun}{fai}{fine per ciascun}% +\SetKwFor{PerTutti}{per tutti i}{fai}{fine per tutti}% +\SetKwFor{Finche}{finché}{fai}{fine finché}% +% +% --- End +%} +% +%\algocf@defaults@common +% +% option onelanguage redefinition +% +\ifthenelse{\boolean{algocf@optonelanguage}\AND\equal{\algocf@languagechoosen}{french}}{% +\SetKwInput{KwIn}{Entr\'ees}% +\SetKwInput{KwOutSortie}{Sorties}% +\SetKwInput{KwData}{Donn\'ees}% +\SetKwInput{KwResult}{R\'esultat}% +\SetKw{KwTo}{\`a}% +\SetKw{KwRet}{retourner}% +\SetKw{Return}{retourner}% +\SetKwBlock{Begin}{d\'ebut}{fin}% +\SetKwRepeat{Repeat}{r\'ep\'eter}{jusqu'\`a}% +% +\SetKwIF{If}{ElseIf}{Else}{si}{alors}{sinon si}{sinon}{fin si}% +\SetKwSwitch{Switch}{Case}{Other}{suivant}{faire}{cas o\`u}{autres cas}{fin d'alternative}% +\SetKwFor{For}{pour}{faire}{fin pour}% +\SetKwFor{ForPar}{pour}{faire en parallèle}{fin pour}% +\SetKwFor{ForEach}{pour chaque}{faire}{fin pour chaque}% +\SetKwFor{ForAll}{pour tous les}{faire}{fin pour tous}% +\SetKwFor{While}{tant que}{faire}{fin tq}% +}{}% +\ifthenelse{\boolean{algocf@optonelanguage}\AND\equal{\algocf@languagechoosen}{german}}{% +\SetKwInput{KwIn}{Eingabe}%KwIn +\SetKwInput{KwOut}{Ausgabe}%KwOut +\SetKwInput{KwData}{Daten}%KwData +\SetKwInput{KwResult}{Ergebnis}%KwResult +\SetKw{KwTo}{bis}%KwTo +\SetKw{KwRet}{zur\"uck}%KwRet +\SetKw{Return}{zur\"uck}%Return +\SetKwBlock{Begin}{Beginn}{Ende}%Begin +\SetKwRepeat{Repeat}{wiederhole}{bis}%Repeat +% +\SetKwIF{If}{ElseIf}{Else}{wenn}{dann}{sonst wenn}{sonst}{Ende wenn}%gIf +\SetKwSwitch{Switch}{Case}{Other}{unterscheide}{tue}{Fall}{sonst}{Ende Unt.}%Switch +\SetKwFor{For}{f\"ur}{tue}{Ende f\"ur}%For +\SetKwFor{ForPar}{f\"ur}{tue gleichzeitig}{Ende gleichzeitig}%ForPar +\SetKwFor{ForEach}{f\"ur jedes}{tue}{Ende f\"ur}%ForEach +\SetKwFor{ForAll}{f\"ur alle}{tue}{Ende f\"ur}%ForAll +\SetKwFor{While}{solange}{tue}{Ende solange}%While +}{}% +\ifthenelse{\boolean{algocf@optonelanguage}\AND\equal{\algocf@languagechoosen}{portugues}}{% +\SetKwInput{KwIn}{Entrada}% +\SetKwInput{KwOut}{Sa\'{i}da}% +\SetKwInput{KwData}{Dados}% +\SetKwInput{KwResult}{Resultado}% +\SetKw{KwTo}{at\'{e}} +\SetKw{KwRet}{retorna}% +\SetKw{Return}{retorna}% +\SetKwBlock{Begin}{in\'{i}cio}{fim}% +\SetKwRepeat{Repeat}{repita}{at\'{e}}% +% +\SetKwIF{If}{ElseIf}{Else}{se}{ent\~{a}o}{sen\~{a}o se}{sen\~{a}o}{fim se}% +\SetKwSwitch{Switch}{Case}{Other}{selecione}{fa\c{c}a}{caso}{sen\~{a}o}{fim selec}% +\SetKwFor{For}{para}{fa\c{c}a}{fim para}% +\SetKwFor{ForPar}{para}{fa\c{c}a em paralelo}{fim para} +\SetKwFor{ForEach}{para cada}{fa\c{c}a}{fim para cada}% +\SetKwFor{ForAll}{para todo}{fa\c{c}a}{fim para todo}% +\SetKwFor{While}{enquanto}{fa\c{c}a}{fim enqto}% +}{}% +\ifthenelse{\boolean{algocf@optonelanguage}\AND\equal{\algocf@languagechoosen}{italiano}}{% +\SetKwInput{KwIn}{Ingresso}% +\SetKwInput{KwOut}{Uscita}% +\SetKwInput{KwData}{Dati}% +\SetKwInput{KwResult}{Risultato}% +\SetKw{KwTo}{a}% +\SetKw{KwRet}{ritorna}% +\SetKw{Return}{ritorna}% +\SetKwBlock{Begin}{inizio}{fine}% +\SetKwRepeat{Repeat}{ripeti}{finché}% +% +\SetKwIF{If}{ElseIf}{Else}{se}{allora}{altrimenti se}{allora}{fine se}% +\SetKwSwitch{Switch}{Case}{Other}{switch}{do}{case}{otherwise}{endsw}% +\SetKwFor{For}{per}{fai}{fine per}% +\SetKwFor{ForPar}{per}{fai in parallelo}{fine per}% +\SetKwFor{ForEach}{per ciascun}{fai}{fine per ciascun}% +\SetKwFor{ForAll}{per tutti i}{fai}{fine per tutti}% +\SetKwFor{While}{finché}{fai}{fine finché}% +}{}% +% +%%%% old commands compatibility +% +\ifthenelse{\boolean{algocf@oldcommands}}{% +\newcommand{\SetNoLine}{\SetAlgoNoLine}% +\newcommand{\SetVline}{\SetAlgoVlined}% +\newcommand{\SetLine}{\SetAlgoLined}% +% +\newcommand{\dontprintsemicolon}{\DontPrintSemicolon}% +\newcommand{\printsemicolon}{\PrintSemicolon}% +\newcommand{\incmargin}[1]{\IncMargin{#1}}% +\newcommand{\decmargin}[1]{\DecMargin{-#1}}% +\newcommand{\setnlskip}[1]{\SetNlSkip{#1}}% +\newcommand{\Setnlskip}[1]{\SetNlSkip{#1}}% +\newcommand{\setalcapskip}[1]{\SetAlCapSkip{#1}}% +\newcommand{\setalcaphskip}[1]{\SetAlCapHSkip{#1}}% +\newcommand{\nlSty}[1]{\NlSty{#1}}% +\newcommand{\Setnlsty}[3]{\SetNlSty{#1}{#2}{#3}}% +\newcommand{\linesnumbered}{\LinesNumbered}% +\newcommand{\linesnotnumbered}{\LinesNotNumbered}% +\newcommand{\linesnumberedhidden}{\LinesNumberedHidden}% +\newcommand{\showln}{\ShowLn}% +\newcommand{\showlnlabel}[1]{\ShowLnLabel{#1}}% +\newcommand{\nocaptionofalgo}{\NoCaptionOfAlgo}% +\newcommand{\restorecaptionofalgo}{\RestoreCaptionOfAlgo}% +\newcommand{\restylealgo}[1]{\RestyleAlgo{#1}}% +% +\newcommand{\Titleofalgo}[1]{\TitleOfAlgo{#1}}% +\SetKwIF{gSi}{gSinonSi}{gSinon}{si}{alors}{sinon si}{sinon}{fin si}% +\SetKwIF{gIf}{gElsIf}{gElse}{if}{then}{else if}{else}{end if}% +\SetKwIF{gIf}{gElseIf}{gElse}{if}{then}{else if}{else}{end if}% +\SetKwIF{gWenn}{gSonstWenn}{gSonst}{wenn}{dann}{sonst wenn}{sonst}{Ende wenn}%gIf +\SetKwIF{gSe}{gSenaoSe}{gSenao}{se}{ent\~{a}o}{sen\~{a}o se}{sen\~{a}o}{fim se}% +\SetKwIF{gSea}{gAltSe}{gAltrimenti}{se}{allora}{altrimenti se}{allora}{fine se}% +}{% + \relax% +}% +% +% +% +%% +%%% +%%%% END \ No newline at end of file diff --git a/algorithme.sty b/algorithme.sty new file mode 100644 index 0000000..8f696ce --- /dev/null +++ b/algorithme.sty @@ -0,0 +1,661 @@ +%% Package algorithme.tex (version 2.4) +%% Envoyer toutes remarques a Nicolas.Delestre@insa-rouen.fr + +% Modifications depuis la version 2.3.8 +% * Ajout de la commande \tadOperationAvecPreconditions +% * Ajout des environnement tadPreconditions et des commandes +% \motclefTADPreconditions et \tadPrecondition +% * Ajout de la commande \champ + + +% Modifications depuis la version 2.3.7 +% * Ajout des types de base naturelNonNul, reelPositif, reelPositifNonNul (resp Negatif) +% * Elimination d'accents dans les mots cles lies à l'algorithmique +% * ajout de la commande \commentaire + + +% Modification depuis la version 2.3.6 +% * Ajout des commandes \fonctionAvecPreconditions +% \signatureFonctionAvecPreconditions, \procedureAvecPreconditions +% et \signatureProcedureAvecPreconditions +% * remplacement de structure et champStructure par enregistrement et champEnregistrement + + +% Modification depuis la version 2.3.5 +% * Suppression de la commande \programme + + +% Modification depuis la version 2.3.4 +% * Dans les commandes fonction et procedure, modification du test pour savoir si les variables locales sont presentes ou pas. +% * Suppresion de l'erreur de compilation du a la redefinition de l'environnement structure + +% Modification depuis la version 2.3.2 +% * structure est maintenant un environnement qui s'integre dans l'environnement algorithme +% * \champStructure a pris une majuscule +% * Apparition de \tadSemantiquesAxiomatiques +% * Apparition de l'environnement \tadAxiomes et et commande \tadAxiome +% * Apparition de \typePointeur, \pointeur, \pointeurNULL, \champPointee, \allouer, \desallouer +% * Elimination de l'erreur de compilation lie a la defintion de la commande \structure + + + +\NeedsTeXFormat{LaTeX2e}[1994/12/01] +\ProvidesPackage{algorithme}[2002/12/22 package algorithme] +\RequirePackage{calc} +\RequirePackage{amssymb} + +%% Definition des mots clefs +%% peuvent être redéfini à l'aide de la commande \renewcommand +% +\newcommand{\motclefAffectation}[0]{$\leftarrow$} +% +\newcommand{\motclefAlgoNom}[0]{Nom} +\newcommand{\motclefAlgoRole}[0]{Role} +\newcommand{\motclefAlgoEntree}[0]{Entrée} +\newcommand{\motclefAlgoSortie}[0]{Sortie} +\newcommand{\motclefAlgoEntreeSortie}[0]{Entrée/Sortie} +\newcommand{\motclefAlgoDeclaration}[0]{Déclaration} +% +\newcommand{\motclefAlgoDebut}[0]{debut} +\newcommand{\motclefAlgoFin}[0]{fin} +% +\newcommand{\motclefEntier}[0]{Entier} +\newcommand{\motclefNaturel}[0]{Naturel} +\newcommand{\motclefNaturelNonNul}[0]{NaturelNonNul} +\newcommand{\motclefReel}[0]{Reel} +\newcommand{\motclefReelPositif}[0]{ReelPositif} +\newcommand{\motclefReelPositifNonNul}[0]{ReelPositifNonNul} +\newcommand{\motclefReelNegatif}[0]{ReelNegatif} +\newcommand{\motclefReelNegatifNonNul}[0]{ReelNegatifNonNul} +\newcommand{\motclefBooleen}[0]{Booleen} +\newcommand{\motclefCaractere}[0]{Caractere} +\newcommand{\motclefChaine}[0]{Chaine de caracteres} +% +\newcommand{\motclefRemarque}[0]{Remarque} +\newcommand{\motclefCommentaire}[0]{// } +% +\newcommand{\motclefType}[0]{Type} +\newcommand{\motclefDeclaration}[0]{Déclaration} +\newcommand{\motclefConstante}[0]{Constante} +\newcommand{\motclefTableau}[0]{Tableau} +\newcommand{\motclefStructure}[0]{Structure} +\newcommand{\motclefFinstructure}[0]{finstructure} +% +\newcommand{\motclefSi}[0]{si} +\newcommand{\motclefAlors}[0]{alors} +\newcommand{\motclefSinon}[0]{sinon} +\newcommand{\motclefFinsi}[0]{finsi} +% +\newcommand{\motclefCasou}[0]{cas où} +\newcommand{\motclefCasouVaut}[0]{vaut} +\newcommand{\motclefFincas}[0]{fincas} +% +\newcommand{\motclefTantque}[0]{tant que} +\newcommand{\motclefTantqueFaire}[0]{faire} +\newcommand{\motclefFintantque}[0]{fintantque} +% +\newcommand{\motclefRepeter}[0]{repeter} +\newcommand{\motclefJusquaceque}[0]{jusqu'a ce que} +% +\newcommand{\motclefPour}[0]{pour} +\newcommand{\motclefPourA}[0]{à} +\newcommand{\motclefPourPas}[0]{pas de} +\newcommand{\motclefPourFaire}[0]{faire} +\newcommand{\motclefFinPour}[0]{finpour} +% +\newcommand{\motclefEcrire}[0]{ecrire} +\newcommand{\motclefLire}[0]{lire} +% +\newcommand{\motclefPreconditions}[0]{précondition(s)} +% +\newcommand{\motclefFonction}[0]{fonction} +\newcommand{\motclefRetourner}[0]{retourner} +% +\newcommand{\motclefProcedure}[0]{procédure} +\newcommand{\motclefParamEntree}[0]{E} +\newcommand{\motclefParamSortie}[0]{S} +\newcommand{\motclefParamEntreeSortie}[0]{E/S} +% +\newcommand{\motclefTADNom}[0]{Nom} +\newcommand{\motclefTADParametre}[0]{Paramètre} +\newcommand{\motclefTADUtilise}[0]{Utilise} +\newcommand{\motclefTADOperations}[0]{Opérations} +\newcommand{\motclefTADSemantiques}[0]{Sémantiques} +\newcommand{\motclefTADAxiomes}[0]{Axiomes} +\newcommand{\motclefTADPreconditions}[0]{Préconditions} +% +\newcommand{\motclefPointeurNULL}[0]{null} +\newcommand{\motclefAllouer}[0]{allouer} +\newcommand{\motclefDesallouer}[0]{desallouer} + +%% ------------------------------------------------------------ +%% Définitions privées + +%% Permet d'étendre le nombre de niveau maximale pour l'environnment list +\csname @listdepth\endcsname=-10 + + +%% L'environnement \algolist +\newenvironment{algolist}[1] +{\begin{list} + {} + {\setlength{\parsep}{0cm} + \setlength{\leftmargin}{#1} + \setlength{\itemsep}{0cm} + \setlength{\topsep}{0cm} + } +} +{\end{list}} + + +%% Définition d'un bloc d'instructions +\newcommand{\unbloc}[1]{ + \begin{algolist}{0,5cm} + #1 + \end{algolist} +} + +\newcommand{\entetealgo}[6]{ + \instruction{\textbf{\motclefAlgoNom : }#1} + \instruction{\textbf{\motclefAlgoRole : }#2} + \instruction{\textbf{\motclefAlgoEntree : }#3} + \instruction{\textbf{\motclefAlgoSortie : }#4} + \instruction{\textbf{\motclefAlgoEntreeSortie : }#5} + \instruction{\textbf{\motclefAlgoDeclaration : }#6} +} + +\newcommand{\debutfin}[1]{ + \instruction{\textbf{\motclefAlgoDebut}} + \unbloc{#1} + \instruction{\textbf{\motclefAlgoFin}} +} + +%% Des listes alignees +%% #1 : le mot de reference +%% #2 : l'espace depuis le cote gauche +%% #3 : le separateur +\newenvironment{listeAlignee}[3]% +{\begin{list}{} + {\renewcommand{\makelabel}[1]{\textsf{##1#3}\hfil} + \settowidth{\labelwidth}{\textsf{#1#3}}% + \setlength{\leftmargin}{\labelwidth+\labelsep+#2} + \setlength{\itemsep}{0cm} + } + }% +{\end{list}} + +\newenvironment{listeAligneeSanssf}[3]% +{\begin{list}{} + {\renewcommand{\makelabel}[1]{##1#3\hfil} + \settowidth{\labelwidth}{#1#3}% + \setlength{\leftmargin}{\labelwidth+\labelsep+#2} + \setlength{\itemsep}{0cm} + } + }% +{\end{list}} + +%% ------------------------------------------------------------ +%% Définitions des commandes et environnement publiques + + + +%% Commande \instruction +%% #1 : L'instruction +%% Remarque : Toute instruction doit utiliser cette commande +\newcommand{\instruction}[1]{ + \item #1 +} + +%% Commande \remarque +%% #1 : La remarque +\newcommand{\remarque}[1]{ + \instruction{\textit{\textbf{Remarque : }#1}} +} + +%% Commande \commentaire +%% #1 : Le commentaire +\newcommand{\commentaire}[1]{ + \instruction{\textit{// #1}} +} + +%% Les types de bases +\newcommand{\entier}{\textbf{\motclefEntier}} +\newcommand{\naturel}{\textbf{\motclefNaturel}} +\newcommand{\naturelNonNul}{\textbf{\motclefNaturelNonNul}} +\newcommand{\reel}{\textbf{\motclefReel}} +\newcommand{\reelPositif}{\textbf{\motclefReelPositif}} +\newcommand{\reelPositifNonNul}{\textbf{\motclefReelPositifNonNul}} +\newcommand{\reelNegatif}{\textbf{\motclefReelNegatif}} +\newcommand{\reelNegatifNonNul}{\textbf{\motclefReelNegatifNonNul}} +\newcommand{\booleen}{\textbf{\motclefBooleen}} +\newcommand{\caractere}{\textbf{\motclefCaractere}} +\newcommand{\chaine}{\textbf{\motclefChaine}} + +%% Commande \type +%% #1 : Nom du type +%% #2 : Défintion du type +\newcommand{\type}[2]{ + \instruction{\textbf{\motclefType \ }#1\textbf{ = }#2} +} + +%% Commande \variables +%% #1 : Variable(s) a declarée(s) +\newcommand{\variables}[1]{ +% \instruction{\textbf{\motclefDeclaration \ }#1} + \begin{listeAligneeSanssf}{\textbf{\motclefDeclaration \ }}{0.5cm}{} + \item[\textbf{\motclefDeclaration \ }] #1 + \end{listeAligneeSanssf} +} + +%% Commande \constante +%% #1 : Nom de la constante +%% #2 : Valeur de la constante +\newcommand{\constante}[2]{ + \instruction{\textbf{\motclefConstante \ }#1\textbf{ = }#2} +} + +%% Commande \tableauUneDimension +%% #1 : Intervalle +%% #2 : {de} ou {d'} +%% #3 : Type +\newcommand{\tableauUneDimension}[3]{ + \textbf{\motclefTableau[}#1\textbf{] #2}#3 +} + +%% Commande \tableauDeuxDimensions +%% #1 : Intervalle 1ere dimension +%% #2 : Intervalle 2eme dimension +%% #3 : {de} ou {d'} +%% #4 : Type +\newcommand{\tableauDeuxDimensions}[4]{ + \textbf{\motclefTableau[}#1\textbf{]}\textbf{[}#2\textbf{] #3} #4 +} + +%% Environnement enregistrement +%% #1 : Nom du type cree +%\newenvironment{structure}[1]{\type{#1}{\textbf{\motclefStructure}}\begin{algolist}{0.5cm}}{\end{algolist}\instruction{\textbf{\motclefFinstructure}}} +\newenvironment{enregistrement}[1] +{\type{#1}{\textbf{\motclefStructure}}\begin{algolist}{0.5cm}} +{\end{algolist}\instruction{\textbf{\motclefFinstructure}}} + +\newcommand{\champEnregistrement}[2]{ + \instruction{#1 : #2} +} + +\newcommand{\champ}[2]{ + #1.#2 +} + +%% Commande \affecter +%% #1 : Nom de la variable +%% #2 : Valeur +\newcommand{\affecter}[2]{ + \instruction{#1 \textbf{\motclefAffectation} #2} +} + +%% Commande \sialors +%% #1 : Condition +%% #2 : Instruction(s) du alors +\newcommand{\sialors}[2]{ + \instruction{\textbf{\motclefSi} #1 \textbf{\motclefAlors}} + \unbloc{#2} + \instruction{\textbf{\motclefFinsi}} +} + +%% Commande \sialorssinon +%% #1 : Condition +%% #2 : Instruction(s) du alors +%% #3 : Instruction(s) du sinon +\newcommand{\sialorssinon}[3]{ + \instruction{\textbf{\motclefSi} #1 \textbf{\motclefAlors}} + \unbloc{#2} + \instruction{\textbf{\motclefSinon}} + \unbloc{#3} + \instruction{\textbf{\motclefFinsi}} +} + +%% Commande \cas +%% #1 : Nom de la variable +%% #2 : Les cas +\newcommand{\cas}[2]{ + \instruction{\textbf{\motclefCasou} #1 \textbf{\motclefCasouVaut}} + \unbloc{#2} + \instruction{\textbf{\motclefFincas}} +} + +\newcommand{\casclausegenerale}[2]{ + \instruction{\textit{#1}: } + \unbloc{#2} +} + +\newcommand{\casclauseautre}[1]{ + \instruction{\textit{autre} : } + \unbloc{#1} +} + +%% Commande \tantque +%% #1 : Condition +%% #2 : Instruction(s) +\newcommand{\tantque}[2]{ + \instruction{\textbf{\motclefTantque} #1 \textbf{\motclefTantqueFaire}} + \unbloc{#2} + \instruction{\textbf{\motclefFintantque}} +} + +%% Commande \repeter +%% #1 : Condition +%% #2 : Instruction(s) +\newcommand{\repeter}[2]{ + \instruction{\textbf{\motclefRepeter}} + \unbloc{#1} + \instruction{\textbf{\motclefJusquaceque} #2} +} + +%% Commande \pour +%% #1 : Variable +%% #2 : Borne inférieure +%% #3 : Borne supérieure +%% #4 : Pas (si {} alors de 1 par défaut et n'apparait pas) +%% #5 : Instruction(s) +\newcommand{\pour}[5]{ + \instruction{\textbf{\motclefPour} #1 \motclefAffectation #2 \textbf{\motclefPourA} #3 \ifx#4\ \else \textbf{\motclefPourPas} #4 \fi\textbf{\motclefPourFaire}} + \unbloc{#5} + \instruction{\textbf{\motclefFinPour}} +} + +%% Commande \lire +%% #1 : Variables à lire (liste non controlée) +\newcommand{\lire}[1]{ + \instruction{\textbf{\motclefLire}(#1)} +} + +%% Commande \ecrire +%% #1 : Ce qui doit être affiché +\newcommand{\ecrire}[1]{ + \instruction{\textbf{\motclefEcrire}(#1)} +} + +%% Commande \retourner +%% #1 : Ce qui doit être retourné +\newcommand{\retourner}[1]{ + \instruction{\textbf{\motclefRetourner} #1} +} + +%% Commande algo +%% #1 : Taille de la police +%% #2 : Nom de l'algo +%% #3 : Signification de l'algo (rôle) +%% #4 : Paramètre en entrée +%% #5 : Paramètre en sortie +%% #6 : Paramètre en entrée/sortie +%% #7 : Déclaration variable locale +%% #8 : Corps de l'algo +\newcommand{\algo}[8]{ + #1 + \entetealgo{#2}{#3}{#4}{#5}{#6}{#7} + \debutfin{#8} +} + +%% Commande \fonction +%% #1 : Nom de la fonction +%% #2 : Paramètres +%% #3 : Type de retour +%% #4 : Variables locales +%% #5 : Instructions +\newcommand{\fonction}[5]{ + \instruction{\textbf{\motclefFonction} #1 \textbf{(}#2\textbf{)} \textbf{:} #3} + \ifx\empty#4\empty + \else +% \unbloc{\variables{#4}} + \variables{#4} + \fi + \debutfin{#5} +} + +%% Commande \preconditions +%% #1 : les préconditions séparées par // +\newcommand{\preconditions}[1]{ + \ifx\empty#1\empty + \else + \begin{listeAligneeSanssf}{$\lfloor$\textbf{\motclefPreconditions \ }}{0.5cm}{} + \item[$\lfloor$\textbf{\motclefPreconditions \ }] #1 + \end{listeAligneeSanssf} + \fi +} + + +%% Commande \fonctionAvecPreconditions +%% #1 : Nom de la fonction +%% #2 : Paramètres +%% #3 : Type de retour +%% #4 : Précondition(s) +%% #5 : Variable(s) locale(s) +%% #6 : Instructions +\newcommand{\fonctionAvecPreconditions}[6]{ + \instruction{\textbf{\motclefFonction} #1 \textbf{(}#2\textbf{)} \textbf{:} #3} + \preconditions{#4} + \ifx\empty#5\empty + \else + \variables{#5} + \fi + \debutfin{#6} +} + +%% Commande : signaturefonction +%% #1 : Nom de la fonction +%% #2 : paramètres +%% #3 : type de retour +\newcommand{\signaturefonction}[3]{ + \instruction{\textbf{\motclefFonction} #1 \textbf{(}#2\textbf{)} \textbf{:} #3} +} +\newcommand{\signatureFonction}[3]{ + \signaturefonction{#1}{#2}{#3} +} + + +%% Commande : signatureFonctionAvecPreconditions +%% #1 : Nom de la fonction +%% #2 : paramètres +%% #3 : type de retour +%% #4 : préconditions +\newcommand{\signatureFonctionAvecPreconditions}[4]{ + \instruction{\textbf{\motclefFonction} #1 \textbf{(}#2\textbf{)} \textbf{:} #3} + \preconditions{#4} +} + +\newcommand{\paramEntree}[1]{ + \textbf{\motclefParamEntree} #1 +} + +\newcommand{\paramSortie}[1]{ + \textbf{\motclefParamSortie} #1 +} + +\newcommand{\paramEntreeSortie}[1]{ + \textbf{\motclefParamEntreeSortie} #1 +} + +%% Commande \procedure +%% #1 : Nom de la procedure +%% #2 : Paramètres +%% #3 : Variables locales +%% #4 : Corps de la procédure +\newcommand{\procedure}[4]{ + \instruction{\textbf{\motclefProcedure} #1 \textbf{(}#2\textbf{)}} + \ifx\empty#3\empty + \else +% \unbloc{\variables{#3}} + \variables{#3} + \fi + \debutfin{#4} +} + +%% Commande \procedureAvecPreconditions +%% #1 : Nom de la procedure +%% #2 : Paramètres +%% #3 : Précondition(s) +%% #3 : Variables locales +%% #4 : Corps de la procédure +\newcommand{\procedureAvecPreconditions}[5]{ + \instruction{\textbf{\motclefProcedure} #1 \textbf{(}#2\textbf{)}} + \preconditions{#3} + \ifx\empty#4\empty + \else + \variables{#4} + \fi + \debutfin{#5} +} + +%% Commande signatureprocedure +%% #1 : Nom de la procedure +%% #2 : Paramètres +\newcommand{\signatureprocedure}[2]{ + \instruction{\textbf{\motclefProcedure} #1 \textbf{(}#2\textbf{)}} +} +\newcommand{\signatureProcedure}[2]{ + \signatureprocedure{#1}{#2} +} + +%% Commande procedure +%% #1 : Nom de la procedure +%% #2 : Paramètres +%% #3 : Préconditions +\newcommand{\signatureProcedureAvecPreconditions}[3]{ + \instruction{\textbf{\motclefProcedure} #1 \textbf{(}#2\textbf{)}} + \preconditions{#3} +} + + +%% Toutes ces commandes doivent être utilisées au sein de l'environnement algorithme +\newenvironment{algorithme}{\begin{algolist}{0cm}}{\end{algolist}} + + +%% Types abstraits de donnees (TAD ou aussi appele TDA) + +%% Commande \tadNom +%% #1 : le nom du TAD +\newcommand{\tadNom}[1]{\item[\textbf{\motclefTADNom}] #1} + +%% Commande \tadParametres +%% #1 : parametres du TAD (permettant de creer des TAD generiques) +\newcommand{\tadParametres}[1]{\item[\textbf{\motclefTADParametre}] #1} + +%% Commande \tadDependances +%% #1 : Dependance du TAD vis à vis d'autre types +\newcommand{\tadDependances}[1]{\item[\textbf{\motclefTADUtilise}] #1} + +%% Commande \tadOperation +%% #1 : Nom de l'operation +%% #2 : Parametres en entree de l'operation (utiliser \tadXXParam) +%% #3 : Parametres en sortie de l'operation(utiliser \tadXXParam) +\newcommand{\tadOperation}[3]{\item[#1] #2 $\rightarrow$ #3} + +%% Commande \tadOperationAvecPreconditions +%% #1 : Nom de l'operation +%% #2 : Parametres en entree de l'operation (utiliser \tadXXParam) +%% #3 : Parametres en sortie de l'operation(utiliser \tadXXParam) +\newcommand{\tadOperationAvecPreconditions}[3]{\item[#1] #2 $\nrightarrow$ #3} + + +%% Commandes \tadXXParams +%% Produit cartesien des types des parametres +%% Plusieurs commande car pour l'instant impossible de créer une commande avec nombre d'arguement variable +\newcommand{\tadUnParam}[1]{#1} +\newcommand{\tadDeuxParams}[2]{#1 $\times$ #2} +\newcommand{\tadTroisParams}[3]{#1 $\times$ #2 $\times$ #3} +\newcommand{\tadQuatreParams}[4]{#1 $\times$ #2 $\times$ #3 $\times$ #4} +\newcommand{\tadCinqParams}[5]{#1 $\times$ #2 $\times$ #3 $\times$ #4 $\times$ #5} +\newcommand{\tadSixParams}[6]{#1 $\times$ #2 $\times$ #3 $\times$ #4 $\times$ #5 $\times$ #6} +\newcommand{\tadSeptParams}[7]{#1 $\times$ #2 $\times$ #3 $\times$ #4 $\times$ #5 $\times$ #6 $\times$ #7} +\newcommand{\tadHuitParams}[8]{#1 $\times$ #2 $\times$ #3 $\times$ #4 $\times$ #5 $\times$ #6 $\times$ #7 $\times$ #8} +\newcommand{\tadNeufParams}[9]{#1 $\times$ #2 $\times$ #3 $\times$ #4 $\times$ #5 $\times$ #6 $\times$ #7 $\times$ #8 $\times$ #9} + +%% toutes les operations (tadOperation) doivent etre definies dans un environnement tadOperations +\newenvironment{tadOperations}[1] +{\item[\textbf{\motclefTADOperations}] + \begin{listeAlignee}{#1}{0cm}{:} +} +{\end{listeAlignee}} + +%% Commande \tadSemantique +%% #1 : Nom de l'operation +%% #2 : rôle de l'operation +\newcommand{\tadSemantique}[2]{\item[#1] #2} + + +%% toutes les semantiques d'operations (\tadSemantique) doivent etre definies dans un environnement tadSemantiques +\newenvironment{tadSemantiques}[1] +{\item[\textbf{\motclefTADSemantiques}] + \begin{listeAlignee}{#1}{0cm}{:} +} +{\end{listeAlignee}} + + +% Semantique axiomatique +%% Commande \tadSemantique +%% #1 : Nom de l'operation +%% #2 : rôle de l'operation +%\newcommand{\tadSemantique}[2]{\item[#1] #2} + + +% toutes les semantiques d'operations (\tadSemantique) doivent etre definies dans un environnement tadSemantiques +\newenvironment{tadSemantiquesAxiomatiques}[1] +{\item[\textbf{\motclefTADSemantiques}] + \begin{listeAlignee}{#1}{0cm}{ =} +} +{\end{listeAlignee}} + + +\newenvironment{axiomelist}[1] +{\begin{list} + {-} + {\setlength{\parsep}{0cm} + \setlength{\leftmargin}{#1} + \setlength{\itemsep}{0cm} + \setlength{\topsep}{0cm} + } +} +{\end{list}} + + +\newenvironment{tadAxiomes}[0] +{\item[\textbf{\motclefTADAxiomes}] + \begin{axiomelist}{0cm} +} +{\end{axiomelist}} + +\newcommand{\tadAxiome}[1]{\item #1} + +% préconditions +%% toutes les préconditions d'operations (\tadPrecondition) doivent etre definies dans un environnement tadPreconditions +\newenvironment{tadPreconditions}[1] +{\item[\textbf{\motclefTADPreconditions}] + \begin{listeAlignee}{#1}{0cm}{:} +} +{\end{listeAlignee}} + +%% Commande \tadPrecondition +%% #1 : Nom de l'operation +%% #2 : la précondition +\newcommand{\tadPrecondition}[2]{\item[#1] #2} + + + +%% La définition d'un tad ce fait dans un environnement tad +\newenvironment{tad}{\begin{listeAlignee}{\textbf{\motclefTADPreconditions}}{0cm}{:}}{\end{listeAlignee}} + + +%% Commandes \algoset +\newcommand{\algoset}[2]{\renewcommand{#1}[0]{#2}} + + +%% Pour les pointeurs :\typePointeur, \pointeur, \pointeurNULL, \champPointe, \allouer, \desallouer +\newcommand{\typePointeur}[1]{\^\ #1} +\newcommand{\pointeur}[1]{#1\^\ } +\newcommand{\pointeurNULL}[0]{\textbf{\motclefPointeurNULL}} +\newcommand{\champPointeur}[2]{#1$\rightarrow$#2} +\newcommand{\allouer}[1]{\textbf{\motclefAllouer}(#1)} +\newcommand{\desallouer}[1]{\textbf{\motclefDesallouer}(#1)} + +\endinput + diff --git a/algorithmique.sty b/algorithmique.sty new file mode 100644 index 0000000..be6d199 --- /dev/null +++ b/algorithmique.sty @@ -0,0 +1,248 @@ +%% +%% 11 juin 2010 +%% +%% algorithmique.sty +%% +%% S O F T W A R E L I C E N S E +%% ================================= +%% +%% The file pseudocode.sty +%% is referred to as `the pseudocode package' +%% or simply `the package'. +%% +%% The pseudocode package is copyright 1999 D.L. Kreher and D.R. Stinson. +%% +%% The pseudocode package and its drivers may be distributed and/or modified +%% under the conditions of the LaTeX Project Public License, either version +%% 1.3 of this license or (at your option) any later version. +%% The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3 or later is part of all distributions of LaTeX +%% version 2003/12/01 or later. +%% +%% The package has the LPPL maintenance status "maintained". +%% +%% Francisé par Alain Busser, Lycée Roland-Garros, Le Tampon +%% +%% end of software license +%% +%% +%% DESCRIPTION: +%% algorithmique.sty est un fichier de style LaTeX, comprenant: +%% l'environment pseudocode traduit en Français +%% +%% INSTALLATION: +%% À placer là où LaTeX cherche des fichiers. +%% +%% DOCUMENTATION: +%% Voir pseudocode.tex. +%% +%% COPYING: +%% Copying of part or all of this file is allowed under the following +%% conditions only: +%% (1) You may freely distribute unchanged copies of the file. Please +%% include the documentation when you do so. +%% (2) You may modify a renamed copy of the file, but only for personal +%% use or use within an organization. +%% (3) You may copy fragments from the file, for personal use or for +%% distribution, as long as credit is given where credit is due. +%% +%% You are NOT ALLOWED to take money for the distribution or use of +%% this file or modified versions or fragments thereof, except for +%% a nominal charge for copying etc. +%% +%% +%% CODE: +\typeout{** } +\typeout{** Document Style `algorithmique' } +\typeout{** by D.L. Kreher and D.R. Stinson } +\typeout{** Alain Busser } + +% +\NeedsTeXFormat{LaTeX2e} +\ProvidesPackage{algorithmique} +% +\if@twocolumn +\newlength{\pcode@width} +\setlength{\pcode@width}{\columnwidth} +\addtolength{\pcode@width}{-40pt} +\else +\newlength{\pcode@width} +\setlength{\pcode@width}{\textwidth} +\addtolength{\pcode@width}{-40pt} +\fi +% +\RequirePackage{fancybox} +\RequirePackage{ifthen} +% +\newcommand{\pcode@AF}[1]{\mbox{\textsc{#1}}} % choix de la police +\newcounter{pseudocode}[section] +\newcounter{pseudonum} +\renewcommand{\thepseudonum}{\arabic{pseudonum}} +\newcommand{\pcode@tab}[1]{\hspace*{#1ex}} +\renewcommand{\thepseudocode}{\thesection.\arabic{pseudocode}} +% +\newboolean{pcode@plain} +\newboolean{pcode@ruled} +\newboolean{pcode@disp} +\newboolean{pcode@shad} +\newboolean{pcode@dbox} +\newboolean{pcode@obox} +\newboolean{pcode@Obox} +\newboolean{pcode@fbox} +% +\newenvironment{pseudocode}[3][plain] +{% + \refstepcounter{pseudocode}% + \ifthenelse{\equal{#1}{plain}}{\setboolean{pcode@plain}{true}}{\setboolean{pcode@plain}{false}}% + \ifthenelse{\equal{#1}{ruled}}{\setboolean{pcode@ruled}{true}}{\setboolean{pcode@ruled}{false}}% + \ifthenelse{\equal{#1}{display}}{\setboolean{pcode@disp}{true}}{\setboolean{pcode@disp}{false}}% + \ifthenelse{\equal{#1}{shadowbox}}{\setboolean{pcode@shad}{true}}{\setboolean{pcode@shad}{false}}% + \ifthenelse{\equal{#1}{doublebox}}{\setboolean{pcode@dbox}{true}}{\setboolean{pcode@dbox}{false}}% + \ifthenelse{\equal{#1}{ovalbox}}{\setboolean{pcode@obox}{true}}{\setboolean{pcode@obox}{false}}% + \ifthenelse{\equal{#1}{Ovalbox}}{\setboolean{pcode@Obox}{true}}{\setboolean{pcode@Obox}{false}}% + \ifthenelse{\equal{#1}{framebox}}{\setboolean{pcode@fbox}{true}}{\setboolean{pcode@fbox}{false}}% + \setcounter{pseudonum}{0}% + \ifthenelse{\boolean{pcode@disp}}% + {% + \noindent\begin{math}% + }% + {% + \begin{Sbox}% + \begin{minipage}{\pcode@width}% + \ifthenelse{\boolean{pcode@ruled}} + { + \noindent\rule{\pcode@width}{1pt}\hfill\\ + {\bfseries Algorithme \thepseudocode:\pcode@tab{1}}\pcode@AF{#2}($#3$)\\ + \noindent\rule{\pcode@width}{1pt}\hfill\\[1ex] + } + { + {\bfseries Algorithme \thepseudocode:\pcode@tab{1}}\pcode@AF{#2}($#3$)\\[2ex] + } + \noindent\begin{math}\begin{array}{@{\pcode@tab{1}}lr@{}}% + }{}% +}% +{% + \ifthenelse{\boolean{pcode@disp}}% + {% + \end{math} + }% + {% + \ifthenelse{\boolean{pcode@ruled}} + { + \end{array}\end{math}\\[1ex] + \noindent\rule{\pcode@width}{1pt}\hfill + \end{minipage}\end{Sbox}\bigskip\noindent% + } + {\end{array}\end{math}\end{minipage}\end{Sbox}\bigskip\noindent}% + \ifthenelse{\boolean{pcode@plain}}{\TheSbox}{}% + \ifthenelse{\boolean{pcode@ruled}}{\TheSbox}{}% + \ifthenelse{\boolean{pcode@shad}}{\shadowbox{\TheSbox}}{}% + \ifthenelse{\boolean{pcode@dbox}}{\doublebox{\TheSbox}}{}% + \ifthenelse{\boolean{pcode@obox}}{\cornersize*{4ex}\ovalbox{\TheSbox}}{}% + \ifthenelse{\boolean{pcode@Obox}}{\cornersize*{4ex}\Ovalbox{\TheSbox}}{}% + \ifthenelse{\boolean{pcode@fbox}}{\fbox{\TheSbox}}{}% + \bigskip% + }% +}% +% +\newcommand{\STMTNUM}[2]{&\refstepcounter{pseudonum}\hspace*{#1}% +\mbox{(\thepseudonum)\label{#2}}} +\newcommand{\BOOL}[1]{\mbox{\bfseries variables booléennes }\pcode@tab{1}#1} +\newcommand{\INT}[1]{\mbox{\bfseries variables entières }\pcode@tab{1}#1} +\newcommand{\ENT}[1]{\mbox{\bfseries variables entières }\pcode@tab{1}#1} +\newcommand{\FLOAT}[1]{\mbox{\bfseries variables réelles }\pcode@tab{1}#1} +\newcommand{\REAL}[1]{\mbox{\bfseries variables réelles }\pcode@tab{1}#1} +\newcommand{\REEL}[1]{\mbox{\bfseries variables réelles }\pcode@tab{1}#1} +\newcommand{\STRING}[1]{\mbox{\bfseries chaînes de caractères }\pcode@tab{1}#1} +\newcommand{\CHAINE}[1]{\mbox{\bfseries chaînes de caractères }\pcode@tab{1}#1} +\newcommand{\ARRAY}[1]{\mbox{\bfseries tableaux }\pcode@tab{1}#1} +\newcommand{\TABLEAU}[1]{\mbox{\bfseries tableaux }\pcode@tab{1}#1} +\newcommand{\LOCAL}[1]{\mbox{\bfseries variables locales }\pcode@tab{1}#1} +\newcommand{\GLOBAL}[1]{\mbox{\bfseries variables globales }\pcode@tab{1}#1} +\newcommand{\EXTERNAL}[1]{\mbox{\bfseries variables externes }\pcode@tab{1}#1} +\newcommand{\EXTERNE}[1]{\mbox{\bfseries variables externes }\pcode@tab{1}#1} +\newcommand{\COMMENT}[1]% +{% + \vspace*{1ex}% + \mbox{\bfseries Commentaire: }\pcode@tab{.5} \mbox{#1} +} +\newcommand{\BEGIN}{\left\{\begin{array}{@{}lr@{}}} +\newcommand{\DEBUT}{\left\{\begin{array}{@{}lr@{}}} +\newcommand{\END}{\end{array}\right.} +\newcommand{\FIN}{\end{array}\right.} +\newcommand{\PROCEDURE}[2]% +{% + \mbox{\bfseries procédure }\pcode@AF{#1}(\ensuremath{#2})\\% + \begin{array}{@{\pcode@tab{1}}l@{}}% +} +\newcommand{\ENDPROCEDURE}{\vspace*{2ex}\end{array}\\ } +\newcommand{\FINPROCEDURE}{\vspace*{2ex}\end{array}\\ } +\newcommand{\CALL}[2]{\pcode@AF{#1}(#2)} % +\newcommand{\APPEL}[2]{\pcode@AF{#1}(#2)} % +\newcommand{\MAIN}{\mbox{\bfseries principal }\\\begin{array}{@{\pcode@tab{1}}lr@{}}} +\newcommand{\ENDMAIN}{\end{array}} +\newcommand{\RETURN}[1]{\ifthenelse{\equal{#1}{} }% +{\mbox{\bfseries fini }}% +{\mbox{\bfseries retourne }(#1)}} +\newcommand{\RETOURNE}[1]{\ifthenelse{\equal{#1}{} }% +{\mbox{\bfseries fini }}% +{\mbox{\bfseries retourne }(#1)}} +\newcommand{\INPUT}[1]{\mbox{\bfseries Entrer }(#1)} +\newcommand{\ENTRER}[1]{\mbox{\bfseries Entrer }(#1)} +\newcommand{\OUTPUT}[1]{\mbox{\bfseries Afficher }(#1)} +\newcommand{\AFF}[1]{\mbox{\bfseries Afficher }(#1)} +\newcommand{\EXIT}{\mbox{\bfseries quitte }} +\newcommand{\BREAK}{\mbox{\bfseries rompt }} +\newcommand{\IF}{\mbox{\bfseries si }} +\newcommand{\SI}{\mbox{\bfseries si }} +\newcommand{\LET}{\mbox{\bfseries soit }} +\newcommand{\SOIT}{\mbox{\bfseries soit }} +\newcommand{\THEN}{\\\pcode@tab{1}\mbox{ \bfseries alors }} +\newcommand{\ALORS}{\\\pcode@tab{1}\mbox{ \bfseries alors }} +\newcommand{\ELSE}{\\\pcode@tab{1}\mbox{ \bfseries sinon }} +\newcommand{\SINON}{\\\pcode@tab{1}\mbox{ \bfseries sinon }} +\newcommand{\ELSEIF}{\\\pcode@tab{1}\mbox{ \bfseries sinon, si }} +\newcommand{\SINONSI}{\\\pcode@tab{1}\mbox{ \bfseries sinon, si }} +\newcommand{\FOREACH}{\mbox{\bfseries pour chaque }} +\newcommand{\POURTOUT}{\mbox{\bfseries pour tout }} +\newcommand{\FORALL}{\mbox{\bfseries pour tout }} +\newcommand{\FOR}{\mbox{\bfseries pour }} +\newcommand{\POUR}{\mbox{\bfseries pour }} +\newcommand{\TO}{\mbox{ \bfseries jusqu'à }} +\newcommand{\JUSQUE}{\mbox{ \bfseries jusqu'à }} +\newcommand{\DOWNTO}{\mbox{ \bfseries descendant jusqu'à }} +\newcommand{\CASE}{\mbox{\bfseries selon }} +\newcommand{\SELON}{\mbox{\bfseries selon }} +\newcommand{\OF}{\\\pcode@tab{1}\mbox{ \bfseries parmi }\BEGIN} +\newcommand{\PARMI}{\\\pcode@tab{1}\mbox{ \bfseries parmi }\BEGIN} +\newcommand{\ENDCASE}{\END} +\newcommand{\FINSELON}{\END} +\newcommand{\AND}{\mbox{ \bfseries et }} +\newcommand{\ET}{\mbox{ \bfseries et }} +\newcommand{\OR}{\mbox{ \bfseries ou }} +\newcommand{\OU}{\mbox{ \bfseries ou }} +\newcommand{\NOT}{\mbox{ \bfseries non }} +\newcommand{\NON}{\mbox{ \bfseries non }} +\newcommand{\SUCCESS}{\mbox{ \bfseries succès }} +\newcommand{\SUCCES}{\mbox{ \bfseries succès }} +\newcommand{\FAIL}{\mbox{ \bfseries échec }} +\newcommand{\ECHEC}{\mbox{ \bfseries échec }} +\newcommand{\TRUE}{\mbox{ \bfseries vrai }} +\newcommand{\VRAI}{\mbox{ \bfseries vrai }} +\newcommand{\FALSE}{\mbox{ \bfseries faux }} +\newcommand{\FAUX}{\mbox{ \bfseries faux }} +\newcommand{\GOTO}{\mbox{\bfseries aller à }} +\newcommand{\DO}{\\\pcode@tab{1}\mbox{ \bfseries faire }} +\newcommand{\FAIRE}{\\\pcode@tab{1}\mbox{ \bfseries faire }} +\newcommand{\WHILE}{\mbox{\bfseries tant que }} +\newcommand{\TANTQUE}{\mbox{\bfseries tant que }} +\newcommand{\REPEAT}{\mbox{\bfseries répéter }\\\begin{array}{@{\pcode@tab{1}}lr@{}}} +\newcommand{\REPETE}{\mbox{\bfseries répéter }\\\begin{array}{@{\pcode@tab{1}}lr@{}}} +\newcommand{\UNTIL}{\end{array}\\\mbox{\bfseries jusqu'à }} +\newcommand{\JUSQUA}{\end{array}\\\mbox{\bfseries jusqu'à }} +\newcommand{\GETS}{\leftarrow} +\newcommand{\PREND}{\leftarrow} +\endinput +%% END pseudodcode.sty + diff --git a/tmp.mf b/tmp.mf new file mode 100644 index 0000000..12a60cf --- /dev/null +++ b/tmp.mf @@ -0,0 +1,31 @@ +% tmp.mf, generated by Mfpic, v0.9 2006/05/26, +% from TeX source "Lsystem" on 2011/03/16 at 14:10 +numeric mfpicversion; mfpicversion:=90; +if unknown mode: mode = localfont; fi +if unknown mag: mag = 1000/1000; fi +if unknown grafbase: input grafbase; fi + +if unknown grafbaseversion: grafbaseversion = 0; fi +if grafbaseversion <> mfpicversion : +GBmsg "Bad mfpic installation: mfpic and grafbase versions do not match."; +fi + + +%----------------------------<1> +unitlen := 1.0pt#; +xscale := 15; yscale := 15; +bounds (-3, 3, -3, 3); + +beginmfpic(1); % line 666 in the TeX source. + +axes (5.0pt); + +store (curpath) +drawn +function (true) (-2,2,0.1) (((x**3)-x)/2); + +endmfpic; % (1) line 669 in the TeX source. +%------------------------------------------ + + +end. diff --git a/tmp.mp b/tmp.mp new file mode 100644 index 0000000..b016c18 --- /dev/null +++ b/tmp.mp @@ -0,0 +1,50 @@ +% tmp.mp, generated by Mfpic, v0.9 2006/05/26, +% from TeX source "Lsystem" on 2011/03/16 at 14:08 +numeric mfpicversion; mfpicversion:=90; +if unknown mode: mode = localfont; fi +if unknown mag: mag = 1000/1000; fi +if unknown grafbase: input grafbase; fi + +if unknown grafbaseversion: grafbaseversion = 0; fi +if grafbaseversion <> mfpicversion : +GBmsg "Bad mfpic installation: mfpic and grafbase versions do not match."; +fi + +if unknown Apricot: input dvipsnam.mp; fi +verbatimtex +\gdef \mpxshipout {\afterassignment \middlempxshipout \setbox 0=\hbox }% +\gdef \middlempxshipout {\aftergroup \finishmpxshipout }% +\gdef \finishmpxshipout {\dimen 1=\ht 0 \dimen 2=\dp 0 +\dimen 0=\dimen 1 \advance \dimen 0\dimen 2 +\setbox 0=\hbox {\hbox {\box 0 +\ifnum \dimen 0>0 \vrule width1sp height\dimen 1 depth\dimen 2 +\else \vrule width1sp height1sp depth0sp\relax +\fi }}% +\ht 0=0pt \dp 0=0pt \shipout \hbox {\box 0}}% +\def \MFPtext #1{\vbox {\def \\{\cr }\MFPcfont \everylabel +\halign {##\hfil \cr #1\crcr }}}% + +\def \everylabel {} +etex; + + +%----------------------------<1> +unitlen := 1.0pt#; +xscale := 15; yscale := 15; +bounds (-3, 3, -3, 3); + +beginmfpic(1); % line 666 in the TeX source. +verbatimtex \bgroup \font \MFPcfont =ecrm1095\relax etex; + +axes (5.0pt); + +store (curpath) +drawn +function (true) (-2,2,0.1) (((x**3)-x)/2); + +verbatimtex \egroup etex; +endmfpic; % (1) line 669 in the TeX source. +%------------------------------------------ + + +end. diff --git a/tmp0.mp b/tmp0.mp new file mode 100644 index 0000000..1748d08 --- /dev/null +++ b/tmp0.mp @@ -0,0 +1,48 @@ +% tmp0.mp, generated by Mfpic, v0.9 2006/05/26, +% from TeX source "Lsystem" on 2011/03/16 at 13:50 +numeric mfpicversion; mfpicversion:=90; +if unknown mode: mode = localfont; fi +if unknown mag: mag = 1000/1000; fi +if unknown grafbase: input grafbase; fi + +if unknown grafbaseversion: grafbaseversion = 0; fi +if grafbaseversion <> mfpicversion : +GBmsg "Bad mfpic installation: mfpic and grafbase versions do not match."; +fi + +if unknown Apricot: input dvipsnam.mp; fi +verbatimtex +\gdef \mpxshipout {\afterassignment \middlempxshipout \setbox 0=\hbox }% +\gdef \middlempxshipout {\aftergroup \finishmpxshipout }% +\gdef \finishmpxshipout {\dimen 1=\ht 0 \dimen 2=\dp 0 +\dimen 0=\dimen 1 \advance \dimen 0\dimen 2 +\setbox 0=\hbox {\hbox {\box 0 +\ifnum \dimen 0>0 \vrule width1sp height\dimen 1 depth\dimen 2 +\else \vrule width1sp height1sp depth0sp\relax +\fi }}% +\ht 0=0pt \dp 0=0pt \shipout \hbox {\box 0}}% +\def \MFPtext #1{\vbox {\def \\{\cr }\MFPcfont \everylabel +\halign {##\hfil \cr #1\crcr }}}% + +\def \everylabel {} +etex; + +setvariable (boolean) (truebbox) true; + +%----------------------------<1> +unitlen := 1.0pt#; +xscale := 1; yscale := 1; +bounds (0, 20, 0, 20); + +beginmfpic(1); % line 665 in the TeX source. +verbatimtex \bgroup \font \MFPcfont =ecrm1095\relax etex; +draw (0,0)--(10,10)--(20,0)--(10,-10)--cycle ; +draw (0,-10)--(20,-10)--(20,10)--(0,10)--cycle ; + + +verbatimtex \egroup etex; +endmfpic; % (1) line 670 in the TeX source. +%------------------------------------------ + + +end.