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ujD=@u[D$8\$ =t(\$8\$4:D$8\$ =t \$4D$0E@D$, ËD$FD$|$t+G@D$, ËD$FËD$D$$\$X|$XD$XD$\;=%L<@==H@D$\ ËD$=]|$ |$`|$d)@D$\ ËD$T4@4D$`D$d|$` tD$dD$`D$`3C\$d\$dӻ@=\$D$\=tjE@D$\ ËD$FËD$,=;D$t2G@D$\ ËD$FËD$,=M;=BE|$0:A@D$\ = C@D$\ \$ "%E $ ", nt > " " > " end$" > "%E $ ", rez.end form -q %wrm %w FORM computed  term(s).Because the equation with % terms computed by FORM is too big, itIis not read into REDUCE but also not computed in REDUCE --> different try. $$$$$|$|D$ \$hL$T$l$$=|$l=|$=|$=|$ =$$$<$$|$@|$L|$T|$t|$x|$d|$|$|$H|$D|$X|$p|$P|$\|$`$$|$(|$0|$,|$<|$4|$8`=V95Wr 5VƉw`@w5`\$D$@=t$t|$$ |$$3@a@aD$|3a@a$3@a@a$3a@a$3CD$ b@b$ \$D$c@c$;=d(eD$=f|$@;=ft!g@h@hi@ijk@kD$x6 l =ulmD$no@on6 ‹L$x\$6 pD$L6 nL$x\$6 pD$T6 nL$x\$6 pD$t;=qt+r@6 sD$8tD$4@Bu D$Lv@vi@iw@5@ D$<^w@5@x@x6 6 l =u" l7 7 p$7 y@D$  $y@D$$ Ë$4@4D$($$t\$ t\$0D$0\$,4@4D$,$$땋\$0z4@4{@{D$0D$,{@{D$,87 |$,txL7 |$,$$tB$$\7 $$$미d7 l7 |$0$$tB$$\7 $$$미d7 x7 D$h|@|7 7 3ۋ$C|@|7 7 D$l|@|7 7 3ۋ$C|@|7 7 7 7 7 7 7 7 D$T8 7 D$T8 7 D$T(8 7 D$T8 7 D$t<8 7 D$t8 7 D$t(8 7 D$t8 P8 ;=qtD$4tD$8} D$L~@~i@iL$Indirect separation of <+Indirect separation of equations of any size3Alternative indirect separation of non-lin equations/Differentiate pdes with nonlinear leading derivs&Algebraic length reduction of equationsSolving an algebraic equation.4Solving equations for fnct.s or deriv.s algebraicallyStop batch mode-The parametric solution of underdetermined ODE-The parametric solution of underdetermined PDEChanging the list of priorities/Find and drop linear dependent general equations1Find and drop linear dependent bi-linear equations$Find factorizable bi-linear equations0Find a linear dependent equation with only 1 term&Triangularize a linear algebraic system%Drop redundant functions and constants Solve a subset of equations firstDelete redundant equationsIntegrate an identity+Differentiate indirectly separable equations2Consider a given expression to be zero and non-zero2Find symmetries and then first integrals for an ODE*Find sub-systems with 2 non-flin_ functions*Find sub-systems with 3 non-flin_ functions*Find sub-systems with 4 non-flin_ functions*Find sub-systems with 5 non-flin_ functions&Find sub-systems with 2 flin_ functions&Find sub-systems with 3 flin_ functions&Find sub-systems with 4 flin_ functions&Find sub-systems with 5 flin_ functions#Do one high priority decoupling stepPerform a user defined operation*Computation of the algebraic Groebner basis*Check whether a given solution is contained1Find a transformation to integrate a 1st order PDEReorder modules 6 and 20Reorder modules 27, 8 and 16Reorder modules 30, 47 and 215Split into cases whether most frequ. function vanishes Save backup.Algebraic length reduction of a single equation-Checking whether inequalities are contradicted,Separation that may lead to case distinctions$Doing a pre-determined case splitting+Inserting a case-splitting if system too big6Check whether a given solution contradicts inequalities&Read an additional equation from a file#Reading new equation if not too many3Split into cases whether most frequ. factor vanishesLimited shorteningsTry other orderingPerform a thorough factorization1Solve a homogeneous alg. equation in two variables@H@H@H@H@H@H@@H@H@ H@(H@0H@8H@@H@HH @PH @XH @`H @hH @pH@xH @Hy@H @H @H @H@H @H @H @H @ȝH @НH @؝H<@H @H @H @H @H6@H @H@@H @ H@(H @0H5@8H @@H @HH @PH @XH @`H @@@hH @pH @xH @H @H @H @H @H @H @H @H @H/@ȞH! @ОH" @؞H# @H$ @H% @H& @H' @H( @H @H) @H* @ H+ @(H @0H, @8H- @@H. @HH/ @PH0 @XH1 @`H2 @hH" @pH3 @xH4 @H5 @H& @H6 @H7 @H8 @H) @H9 @H: @H@ȟH, @ПH; 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@x@x     " @ @x@Fx4  @0 @x@xh  @k @x@x F @O @x@xܓ  @ @x@x  @E @x@x,  @W @x@xh  @\ @x@x|  @= @x@x F @? @x@x  @Z @x@x  @ @x@x@  @d @x@xx  @f @x@x  @! @x@xܕ F @% @x@x  @` @x@x4  @b @x@x\  @@x@x|  @@x@x  @@ @x@x̖ F @@x@x  @j @x@x<  @l @x@xl  @m @x@x  @n @x@x̗  @o @x@x F @p @x@x(  @q @x@xT  @r @x@x  @s @x@x  @t @x@xؘ  @ @x@x F @v @x@x0  @w @x@x`  @^ @x@x  @x @x@x  @y @x@xܙ  @z @x@x F @| @x@x<  @} @x@xL  @~ @x@x  @ @x@x  @3 @x@x  @ @x@x F @ @x@xH  @{ @x@x  @ @x@x  @ @x@xܛ  @ @x@x  @ @x@x0 F @B @x@xH  @ @x@xp  @ @x@x  @% @LH@TH@\H@dH@lH@tH@|H@H@H@H@H@HHHHļH̼HԼHܼHHHHHH H @HH$H @,H @4H @<H @DH @LH @TH @\H @dH @lH @tH @|HHH @HHHHHHĽH̽HԽHܽHHHHHH HHH$H,H4H<HDHLHTHA @\HdHlHtH @|HHH@\Hx5@HH5@HHe @HHA @HľH: @̾HԾH] @̾HܾHa @̾HH> @̾HHc @HHc @H HH6@H @$H,H4H<H @DHLHTH\HdH@lHtHdH@|HH6@H6@HH@H @H @H@HĿH @̿HԿH6@̿H~HܿHHH5@H @HH HH @H$H,H] @\H4H> @\H<HDH6@LHTH\HdH\H @lHtH|HHHDH @HHH6@HH5@HHeHHH5@H @HHHH @H HHH5@$HHdH@,H4H@,H\H<HDH\HLHTH @\HHdHc @HlHe @He\H5@tH|H5@HHHHHH5@H @HHHH @HHH5@HLHdHH] @H @H: @Ha @HH HHc @HedH@$HH\H5@,Hc @4H5@<HDHeLHTH\H5@dH5@lHtH|HH @HHHDHdHDHH> @HH~HyxHHdHeeH5@HH5@HHHLHHH5@H5@HHHH @ HHH$H5@,H4He<H6@DHHH5@LHTHy\HHdH5@lH5@tH|HHH @HHH5@HHdH6@HH|HHH<H6@HHHyHHH5@H5@HH HH @HH$H4HdHH,H6@4Hz<H\H6@DH@HLHTHy\HHdH5@lH5@tH|HH @HHHHH,HR @dHHdH@\H @HHHyHHH5@HHHH @HH~H6@ HHH6@H@H$H,HeH5@4H5@<HDHLHH @THH\HdHdH@\H6@lHytHeH5@|H5@HHH6@H6@HH,HHyHeH @HH<H,HxH @HdHH\HR @HHH6@H<HHdH@HH\HH-@-ø @ @ @ H-@-ø @ @ H-@-ø @ @ H-@-ø @ @ H-@-ø @ @ H-@-ø @ @ H-@-ø @ @ $H-@-ø @ @ ,H-@-ø @ @ 4H-@-ø @ @ <H-@-ø @ @F DH-@-ø @ @ ÍF  5 # @% 0Please give the name of the file in double quotes0(no ;) from which the session is to be restored: ! @tH|HH@|HH|HH|HWW;= uh=|$ @ 4 !@!$ø# @ |$=lH-@%-The global variable % has an incorrect value, please check! $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~3This is CRACK - a solver for overdetermined partial differential equationsEnter `h' for help. ./form_start < formin > formout &CRACK needed :  ms GC time :  msThis is the end of the CRACK runp|$l|$h|$d|$`|$\|$X|$T|$P|$H|$D|$@|$<|$8|$4|$0|$$|$ |$|$|$|$ |$|$$|$(|$,|$L|$% @% É\$lt0D D$l`  & @& ;=0tK;= t>    ;=u;;= u.;=0u0 ' @' ( ) z* + , -  eD$. @. D$$-@-D$$-@-D$ $@D$$J@D$|$ t$@;|$ |$h\$\$$u u@;u @D$hD$ D$$D$$=u D$ D$$ u@;u @t$hD$h뒋|$L u)t$L@;uD$D$L@D$|$ ut$@;u D$ D$D$|$ ut$@;u D$ D$D$|$  ut$ @;u D$  D$ D$ D$/ @/ \$4@4z\$9tD$0 @0 D$;=8u|$=11|$D$1 @1 D$\$l|$D$1 @1 ËD$ =ti\$8.@.=tD$\$lD$lD$11D$D$_1D$l112 1á83 @3 8;=dt;=qt 0 4 5 |$ |$h|$|$$;D$$D$$D$019@9=u*3ۋD$0Ctø6 @6 3ۋD$0CÉ\$h\$ D$$D$$=u D$ D$$D$019@9=u*3ۋD$0Ctø6 @6 3ۋD$0Ct$hD$h`D$|$ |$$|$$D$$D$  S@;BD$ =u3@D$l|$ D$ @=t\$ \$ ċD$ 19@9=t\$ D$l돋D$ \$lD$l\$ \$ R|$l|$lu( tt$l;t5 D$l5 n\$lt6 @6 G u@;u a@atø6 @6 D$$D$$'7 @7 D$ 8 ;+t 9 @9 3t==-z1 D$: @: D$;=( t@p%X  =|$|$l|$lt,D$lS @ =tD$lD$lȋ|$lt  t;=t;=8u =1=8|$|$l t|$l;= D$lD @ =t D$l @ D$|$tTD$3҉ t3CӉ=uD$D$렋|$t D$lD$lD$; |$ |$h=|$$;D$$tkD$$ @ É\$h\$ D$$D$$=uD$ +D$$ @ t$hD$h뺉ø; @< @1D$= @= D$l u~u D$l;=> u ;= tb` e\$@x . @. \$@ |$ |$h|$l|$$u D$$D$0\$(L$,T$d|$dD$dD$` u\ @;uOt@\$` @\$(D$(\$,D$,D$dD$d[|$`|$\D$0D$d=utø@É\$\\$`D$dD$d=uD$`2D$dtø@t$\D$\볉ø@D$d\$(@D$`\$,@D$\|$8|$4D$0JD$X=tutø@É\$4\$8D$XD$X=uD$82D$Xtø@t$4D$4볉D$X|$4|$TD$0\$8u|$<|$PD$@=-D$D3@a@aD$L|$D|$H|$Ht-D$H\$L@D$L\$H\$Hǹt\$L@L$PL$You can use names of equations, e.g. coeffn(e_12,df(f,x,y),2); Terminate the expression with ; 9Type in a variable from which you want to know its value: This is not a variable name. has no value = (The currently active substitution rules: The name of this session is: ""number of steps: wrong input!!!*The user will choose equations from now on.-The program will choose equations from now on."Type in a number instead of `#' to execute a specific module.4Select a method by its number that is to be executed repeatedly:5To repeat this method as often as possible, enter `;' +To repeat this method as often as possible, )but at most a number of n times, enter n :The number must be in 1 ..  . no successPrint length : (Print length must be NIL or an integer!!!More details will be printed.Fewer details will be printed.'All equation properties will be printed.&No equation properties will be printed.3Lex. ordering of functions has now highest priority.=Lex. ordering of functions is not of highest priority anymore.1From now on lexicographic ordering of derivatives.0From now on total-degree ordering of derivatives.Current variable ordering is : ,;New variable ordering : (Not all variables appear in the new list.The new list has extra ariables.New variable list: 6The current variable ordering is going to be reversed. 3The current variable ordering is going to be mixed. Current function ordering is : 9If you want to sort functions according to frequency, rare0functions first, then type: sort_by_frequency;6If you want to sort functions randomly, flin_ first and/non-vanishing last, then type: sort_randomly;else type the new list.New function ordering : Current orderings are : Functions : Variables : Equation name : $Equation name must be an identifier!!Function name : $Function name must be an identifier!!Identity name : $Identity name must be an identifier!! NAT is now on.NAT is now off.!The variable name to be assigned: 1What is the value to be assigned to that variable?'Please terminate this input with ';' : Integration only if result free of explicit integral from now on.Integration result may involve explicit integral from now on.4Integration only if result free of abs() from now on.0Integration result may involve abs() from now on./The user will confirm substitutions from now on.1No user confirmation of substitutions from now on.'Separation will be inforced from now on.+Separation will not be inforced from now on.What is the new line length? ;Equation names will be re-used once the equation is dropped.?Equation names will not be re-used once the equation is dropped./Function names will be re-used once the function is substituted.3Function names will not be re-used once the functionThe time-limit has expired.(The current CPU time limit for automatic execution to stop is: hours and minutes. $There is no time-limit set currently.'Do you want to impose a CPU time-limit? After time has expired,* shall CRACK go into interactive mode (1)- or shall CRACK terminate with error (2) ? How many hours? How many minutes? 6Please type your comment in " " for the history_ list: You can either6- give the name (terminated by ;) of a rule list to be = activated that has been defined before the call of CRACK, or6- give the name (terminated by ;) of an equation which % is to be converted to a LET rule, or*- type in the new LET-rule in the form like/ sqrt(e)**(-~x*log(~y)/~z) => y**(-x/z/2); : #The new list of user defined rules: /Warning: Changes of equations based on LET-rules,are not recorded in the history of equations.%These are all the user defined rules: 3- give the number of a rule above to be dropped, or /- give the name of a rule list activated before - the call of CRACK which should be disabled: Rule list  has been disabled.This number is too big. Is the system homogeneous? (y/n) *Should solutions be stored as files? (y/n) 0Should FORM be used for long computations? (y/n) ;Should Singular be used for computing Groebner bases? (y/n) (Use reverse total degree ordering? (y/n) %Pure lexicographical ordering is used.3Should the GB package of J.C.Faugere be used? (y/n) #The REDUCE Groebner package is used.BWill the computation involve 1000's of steps, so that the recording> of the history of the computation should be limited (y/n)? IAbout how many last steps shall statistical data be stored? (at least 50) @H#proc_list_ has been changed, see p1. no successtr_main is now on.tr_main is now off.tr_gensep is now on.tr_gensep is now off.tr_genint is now on.tr_genint is now off.tr_decouple is now on.tr_decouple is now off.tr_redlength is now on.tr_redlength is now off.tr_short is now on.tr_short is now off.tr_orderings is now on.tr_orderings is now off.B @/Please type the name of the procedure to trace: 1This is Standard Lisp. Return to Reduce by Ctrl D.The function name: ,The argument list in the form {arg1,...}; : Result: -Please give the name of the file to be read indouble quotes (no ;) : @Please type in a list of procedure names, like: gcdf, .., reval;which should be profiled: 5Please give the name of the switch to be switched ON: 6Please give the name of the switch to be switched OFF: c%d#Duplicating process under new xterm.$Duplicating process under new screen. Duplicating process as batch job.Duplicating process under PVM.6### Currently is collect_sol=t. Therefore parallel case5### solving is not enabled because solutions would not7### be collected. You could set collect_sol to nil using!### 'as collect_sol nil;' command.sol_list%w%w4From now on parallel case solving with extra xterm's.#Shall xterms start as icons (Y/N) ? 4From now on parallel case solving with extra screens.4From now on parallel case solving by submitting jobs.+From now on parallel case solving under PVM."PVM is not active on this computer.BThe counter of additional parallel REDUCE processes which is stored in the file  is set to zero.touch %w7The new maximal number of parallel processes (currently ): *The directory name for storing case files: 7Do you want to interface with FORM through pipes (Y/N)? &Shall the temporary FORM directory be " " (Y/N)? EPlease input the directory for temporary FORM computations in "..." : #The temporary FORM directory is now 2What is the maximal number of terms of an equations/computed by FORM that shall be read into REDUCE?(Its current value is .) : illegal input: ''>>>>>>>>> Solution of level Not all conditions are solved." --> RESTART with extra conditions end of the main procedure@H@H @Hx @H @Hy @H9 @H @H @Hz @H; @H@H, @H\ @$$$|$||$x|$t|$p|$l|$h|$d|$`|$\|$X$\$|$|$ |$|$|$|$|$ |$$|$(|$,|$0|$4|$8|$<|$@|$D|$H|$L|$PD$T 3҉ t3CӉD$< 3҉ t3CӉD$H ; t;0td = |$0;=( t & @& ( \$$C @C * D @ E @E ;=to@o\$$C @C z\$$F @F z;= t<eD$ = u)á @=t & @& ;=ua;=uT|$(uH@=; t\$ @=;=t\;= tO5 4 D$X\$1 @ ËD$z\$X$G @G ;= t4\$1 @ ËD$Ë$H @H D$Lo@o$I @I ;=0t:  ;= tD$(=%|$,;t5@;t%D$P 3҉ t3CӉËD$P@=\$PD$;=0;= t @ É\$  uD\$X|$Xt'D$X=tD$XD$X͸  D$ $ z\$\$J D$ ;= ;=( uI=tl3҉ t3CӉ3C@=t* @D$ \$L< < = ~uL 6@6á '@'á@=u K @K ;= ;=< tw=< |$|$t-|$~tD$D$Nj|$tD$L  L$ 3҉ t3CӉ=ut$ ;u ( t|$ ;=( utL$ 3҉ t3CӉ=u\$ \$!\$ $L$ L$D$Pw;=( t D$P]D$Po@oD$P;=0t;= t  3҉ t3CӉËD$P@=;= D$t|$,=%D<;=0t9;= t,\$ @=t$ t=|$$d @;=0u ;= u!@!D$=ht=M @t=?t=uN @N ;=O @uP @P ;=Q @uR @R :=S @uT @T :=U @uV @V :=W @u" @ =tX @X :|$Y @uZ @Z w:|$[ @u3 \ @  =t] @] 8:|$^ @u_ @_ :|$euA;=%t!3C$@` @` 9$` @` 9|$a @=0|$$0<$|$X|$XtWD$XD$\D$\ D$\@D$\b @b D$XD$X띋|$$=029|$c @u!5 4 d @d 9|$fu=1D$ @ 1ze @e 8|$vu;\$1 @ ËD$Ë$f @f u8|$suC3@g @g \$1 @ ËD$Ë$h @h &8|$i @uij  @ =u  p k @k  7|$l @ @   @D$ <$|$X|$XtiD$XD$\ËD$ .@.=u3C@D$\ tø@L$ \$\m D$ D$XD$X닋|$|$X|$XtxD$XD$\ uM @;u@t@\$\L$ m D$ D$XD$X|D$ @@6|$n @u!8 o @o 5|$p @8  @!@!D$ ux 5q @q =uD$ e5D$ D$r @r o @o 15|$s @u$t @t 5|$u @u$v @v 4|$w @uA  i@ix @y @y 4|$z @u${ @{ 4|$| @u< } @} f4|$~ @u<  @ E4|$ @u @ )4|$ @u, j 3|$ @u$ @ 3|$wu\$$ @ 3|$au D$(t3|$g @!@!D$d @|$~u)@\$@3, 2|$tuK=%tt %=t @  p =%|$,2|$ @u @ 2|$ @u @ n2|$#u!  A2|$lt|$~ |$l t;=0u ;= u1 @ ( !@!D$;=0u ;= u38 t  !@! ~t  t3ۋD$@=t\$HD$@=t<;=0C D$H D$8; t4\$1 @ ËD$Ë$H @H D$Lo@o$I @I ;=0t'  ;=tD$8@D$8;=t@\$ D$ $ z\$\$ J D$ ;= t.=u;( t\$LD$< < = ~uQ|$8tJ D$86@6á '@'ËD$8@=u K @K ;= ;=< tw=< |$$|$$t-|$$~tD$$D$$Nj|$$tD$$L  L$ 3҉ t3CӉ=ut$ ;u ( t|$ ;=( D$8o@oD$8L$ 3҉ t3CӉ=u\$ \$!\$ $L$ L$;=0tD @ E @E ;=t(= ~u o@o = u ;= = ~u @ ;= t4\$1 @ ËD$Ë$H @H D$Lo@o$I @I ;=0th  ?;=( u2|$8u+;=0u ;= u ;= t]|$uQ|$ tE;=( u8;= eD$$á @= +|$ @ut\$$ @ y+|$# @t\$$ @ D$ $L$ 3҉ t3CӉo@oD$<;= *@*|$ @u$ @ $*|$ @u$@$*|$xt ËD$ =ttM*|$quA; t*;`t"@ @;=`t`}`@ @* t*|$ @u` @!@!D$=t~u 0)$ )|$ @u=; tt  9t T  x K)|$ @u=; tt  9t  )|$; @u @ (|$ @uZ; tt  9t  4  z1$ @ $(|$ @uZ; tt  9t x   z1$ @ $(|$ @ =z|$|$t2D$D$D$=tϸ ¸  @@D$@=q'áz @ =t< B'zD$@ @ =tl '|$@=z<$|$X|$XtPD$XD$\ @ z3 @3 @D$\x@xD$XD$X뤋 z1$ @ $;= u& zT&|$ @ zz<$|$X|$XtPD$XD$\ @ z3 @3 @D$\x@xD$XD$X뤋 z1$ @ $;= % zq%|$ @` =z|$z|$t}L$3҉ t3CӉ@o@oËD$D$ \$D$zD$ zw<$|$X|$XtPD$XD$\ @ z3 @3 @D$\x@xD$XD$X뤋 z1$ @ $;= "$ z$|$ @$ =1|$|$t2D$D$D$=tϸ ¸ ;=zuUH    4 P @@D$@;=zu6=t/ @;u"$t @t @ D$@y;=z#= @;=1|$T 13҉ t3CӉD$3CD$@=toD$@o@oËD$D$ \$TT\$D$ D$D$@D$x\$3@TT @ D$@E=t>á1 @ =u1D$@ @ =t D$@|$@5! z$D$@ @ !|$ @uNp  1 z |$ @u[ @!@!D$=tu s  W |$ @u[ @!@!D$=tu  |$ @u[< @!@!D$=tu T |$ @u:;ftt f9t S C|$ @ @!@!D$  @@D$ |$ @u=u; t @D$ @ËD$ |$ @ub; tK <$|$X|$XdD$X @x@xD$XD$X t&|$ @uP; tt  9t8 `   |$ @u:; tt  9t  |$ @u:; tt  9tD N| >|$ @u:; tt  9t  |$ @u>  !@!D$ ~uu|$ @u:; tt  9t< x h|$ @uP; tt  9t  '   |$ @@;= &eá @D$ @=tT `\$ 6@ø @i@iD$ t  <\$ 6@ø @i@iD$3ۿ@=tD$ L$< @\$ @i@i   @  =-< X  !@! =t=u !@!D$ !@!D$ |$~ tD$~ tD$ 6\$ @D$X`\$ @D$\e‹L$\\$X@ø @i@i@ G =|$ @u@ @ !@!D$ |$ @ @( < x   $ T @D$  tF$=tD$ @ D$  @ i@it ø@ i@i @y @y  i@i @y @y   [|$ @S @$  i@ix @y @y ( P   !@!D$ ~LtG @ i@i D$  ]  3҉ t3CӉ@ËD$ @=t  D$ 3CD$ @=t@D$ @D$  \$ D$ 맋 @ @ i@iD$ ø@ i@ix @y @y |$ @;  @< !@!D$ =yum>t<$|$X|$Xt[D$XD$\C@  @ ?@D$\x@xD$XD$X뤻>d !@!=yu$;= t @  t !@!=yu dt d !@!D$ =yuC !@!D$ =yu @@  @wl !@!D$ =yu@ !@!D$ =yu @.@  @  @ ;= uh  !@!D$ =nu  t*` !@! ~uָ B    P; |$\|$`=|$X;D$XtkD$X @ É\$`\$\D$XD$X=uD$\+D$X @ t$`D$`뺉ø; @<       H |$ru: z1$ @ 1$|$ @u: z1$ @ 1${|$nu$ @ \|$ @u$ @ $:|$ @u$ @ $|$cu$ @ |$ @uE\$$ @ D$ =$L$|$ @u$ @ $|$iu" @ =t @ X|$ @u: @ =t($ @ É\$ $|$ @u% @ =t$ @ |$ @u" @ =t @ |$ @u" @ =t @ |$ @u% @ =t$ @ T|$ @u% @ =t$ @ #|$ @u] @ =tKz 1$ @ É\$ t$  |$ @u$ @ |$ @u\<$|$X|$Xx D$XD$\@ \$\ @@D$XD$X뫋|$ @u:; tt  9t   |$ @u:; tt  9t$  @  |$ @u:; tt  9t\ q x a |$ @u:;tt 9t +  |$ @u:;tt 9t   |$ @u:; tt  9t   |$@u:; tt  9t< Y X I |$ @ujxH @ i@i @ !@!D$ D$  @ |$ @uO @ !@!D$ D$  @ x |$ @u"  J |$ @ @!@!D$ @@D$   @; |$ D$ ø<  |$ @u` @L  !@!D$ D$  @ / |$ @u\$$ @ |$ @uQ @  @D$ |$ @u @ |$ @ @ !@!D$ * D$XD$  ËD$X r @r D$= \$   D$ @ |$ @ @@ !@!D$ * D$XD$  ËD$X r @r D$=I\$   D$ @ |$ @t(|$ @t|$ @t|$ @[ @ =D$To@oD$Tø| p  = |$  |$ @u+ 3A\$$ @ |$ @u* \$$ @ s|$ @u* \$$ @ = @ =t+ L$$D$D @ D$D|$ =  |$ @t(|$ @t|$ @t|$ @; tI( d    @ = @ @  j pD$  =u @|$ @u~ $  @` !@!D$=yt=nu=yu t( |$ @u   |$ @u    @ ;= t   8 r|$ @ui`      p @ @ |$ @uK @  $ !@! |$ @u*, @!@! p|$ @u  U|$ @3dt @\ !@!D$=yt=nu=yu qt q l !@!D$=yt=nu=nu1 !@!l$ lP    !@!|$ @u d|$ @u] @ !@!D$ =yu @@  @ |$ @uI @ !@!D$ =ytQ@ >|$ @u  @& D$ D$$@;= t ;= t t<$|$0|$4t tD$0`;=( u |$t D$0<ø` @ D$4t;=( u}|$uq;=* ud|$uX<$|$0;=   =u@ @;=`;=( Z|$J;=0u ;= u];=* uP ;= t!  k@k \$$C @C 1D$ @ D$;= u=0|$0d5 4 D$X\$1 @ ËD$z\$X$G @G ;= u |$=0;=* 3 ;=u3@< 3҉ t3CӉo@oD$  @ =;= ~|$\|$`<$|$Xu D$XC@ tø@D$`D$\D$XD$X=uD$\=D$XC@ tø@t$`D$`먉ø@D$X|$`|$d1D$ @ D$\=D$ht@\$h @É\$d\$`D$\D$\=uD$`cD$\D$ht@\$h @t$dD$d낉ø@D$\1@D$`|$h|$l=4 |$duwD$dtø@D$lD$hD$dD$d=uD$h2D$dtø@t$lD$l볉D$d;=5 u |$l|$p=5 |$hu ID$h|$t$D$x=|$|$$=tø@É$\$|$$=uD$|;$tø@$$뤉ø@É$\$tD$xD$x=u D$tD$x|$|$$=tø@É$\$|$$=uD$|;$tø@$$뤉ø@$$ø@D$pD$lD$hD$h=u D$lD$h|$t$D$x=|$|$$=tø@É$\$|$$=uD$|;$tø@$$뤉ø@É$\$tD$xD$x=u D$tD$x|$|$$=tø@É$\$|$$=uD$|;$tø@$$뤉ø@$$ø@t$pD$pø@ËD$dø@l$ ‹L$`\$\D$X"ø @-@-D$ \$|$#t$;;=<$|$X|$XD$XD$\C@ tT$\1 @ D$C@D$\x@xL$@D$\x@xD$XD$Xl<$|$ D$D$X|$XD$X-@-D$ u@;u a@aD$\3<$=t==-z1 D$\@$@$\$X\$X!\$ $ @ =tU;=0t:0 T \$ $ @ D$-;=j;= D$|$|$X|$Xt?D$X u @;t\$D$D$XD$X뵿|$\|$`<$|$XuD$XC@ @D$`D$\D$XD$X=uD$\6D$XC@ @t$`D$`믉D$X|$`|$d\$D$ @ D$\=D$h@\$h @É\$d\$`D$\D$\=uD$`^D$\D$h@\$h @t$dD$d뇉D$\1D$4@4D$`|$h|$l=4 |$duiD$d@D$lD$hD$dD$d=uD$h+D$d@t$lD$l뺉D$d|$l|$p=5 |$hu D$h|$t$D$x=|$|$$=ty@É$\$|$$=uD$|4$@$$É$\$tD$xD$x=u D$tD$x|$|$$=ty@É$\$|$$=uD$|4$@$$$$"D$pD$lD$hD$h=u D$lD$h|$t$D$x=|$|$$=ty@É$\$|$$=uD$|4$@$$É$\$tD$xD$x=u D$tD$x|$|$$=ty@É$\$|$$=uD$|4$@$$$$"t$pD$pŋT$dL$`\$\D$X @ <$ $3҉ t3CӉ3C@=;= tMj D$Xk@k D$XD$j D$\$søuËD$ D$\$$ @ ;= u3@|$\|$`<$|$XuiD$XC@ D$`D$\D$XD$X=uD$\+D$XC@ t$`D$`뺉D$X\$1 @ -5 4 \$D$X"D$;= t$;=0t|  3҉ t3CӉD$X\$<3@@ËD$X@D$ D$X\$XD$ @=;= |$u3i;= u#|$ u|$~4t/L$3҉ t3CӉD$ @ D$Xo@oD$X5D$X @\$X@=u<\$X  @ $@D$Xo@oD$X랋|$|$X|$XtHD$X u  D$XD$X묋D$Č  : |$|$ |$<$D$D$|$D$D$= <$D$$D$ =u!D$o@oD$$$ŋ<$uD$ D$@=t t` D$|`  @D$ D$ u7D$ |$ t1D$ =tD$ D$ ͋D$D$D$,;|$$D$D$D$ D$|$D$D$D$ o@oD$ D$<$|$|$t>\$D$ =u#D$o@oD$\$\$붋|$tW3CD$ @=t a D$ @=tD$ D$D$D$a )Please type in a list of the numbers 1 .. , like 1,2,5,4,..,15; which9will be the new priority list of procedures done by CRACK.'Numbers stand for the following actions:The list so far was: The new list: Error: # is not one of the possible numbers.:The first number must be 1 (to do most urgent steps first).%The procedure list is still unchanged.|$|$|$<$|$|$|$ Lc  3҉ t3CӉ|c c c   @  d   @ =|$(d @@$|$==u D$ t<$<$~uN 3҉ t3CӉË$@=t@ functionsfunctions > derivatives&od : Toggle ordering of derivatives to total-degree lexicographic/oi : Interactive change of ordering on variables!or : Reverse ordering on variables&om : Mix randomly ordering on variables/of : Interactive change of ordering on functionsop : Print current ordering1ne : Root of the name of new generated equations (5nf : Root of the name of new functions and constants ((ni : Root of the name of new identities (na : Change output to OFF NATON NATas : Input of an assignmentke : Do not keepKeep# a partitioned copy of each equationfi : Allow unresolved integralsForbid unresolved integralsfa : !Allow solutions of ODEs with ABS()"Forbid solutions of ODEs with ABS()cs : 7No confirmation of intended substitutions/factorizations4Confirmation of intended substitutions/factorizationsfs : Do not enforce direct separationEnforce direct separationll : change of the line lengthre : Do not re-cycle equation names.Do re-cycle equation names.rf : Do not re-cycle function names.Do re-cycle function names.3st : Setting a CPU time limit for un-interrupted run)cm : Adding a comment to the history_ listlr : Adding a LET-rulecr : Clearing a LET-rule3ap : Adapting the setting to the system to be solved@s 0|s s ;= ts s s s ;= ts s s Ft 8t ;= tXt tt t ;= tt t t u Du pu u Fu  |s v  |s tu|$|$)@$ D$D$|$ tD$D$D$3C\$\$ӻ@=t f;=8t*)@$ 89@9=/|$|$ D$D$\$ ؿ1 @1 D$@$ ËD$@=t\$\$y\$\$t$uD$|$J\$D$  =|$t+\$D$ =uD$D$D${D$@H @ factorized to 5 @$|$ |$|$$\$|$|$ |$|$<$t-$4 @4 D$=u$$ȋ|$D$D$ 5 @-@-D$طH-@-ËD$6 @6 =tt-@- -@-D$7 @7 =tH @ -@-$C@ tø@D$\$3҉ t3CӉ=uD$\$@ËD$1 @1 D$D$7 @7 =t! @ @ -@-|$Zt$@;EL$3҉ t3CӉD$D$ o@oËD$ =3@a@aD$ \$\$|$D$D$ @;a@aD$ D$a@a@ËD$ @\$D$\$ @D$ \$D$D$4D$|$|$ |$$D$ @@ D$ @D$  D$8 @D$  |$=====l$T$ \$D$@D$;=0tD$  D$|$t\$D$ $|$|$$D$\$ L$;=%t23CD$ @D$@+ @D$+ JD$D$9 @9 D$=\$ D$ |$|$|$t-D$\$ @D$ \$\$Nj|$|$|$t>D$ø) @\$\$붋\$D$ D$=u-;t%<$|$@D$+ @D$+ D$|$|$|$$D$\$ L$;=%t23CD$ @D$@+ @D$+ JD$t D$: @: D$=\$ D$ |$|$|$t-D$\$ @D$ \$\$Nj|$|$|$t^D$D$ËD$ .@.=u(D$ø) @D$D$떋\$D$ D$=u-;t%<$|$@D$+ @D$+ D$|$|$|$$D$\$ L$;=%t23CD$ @D$@+ @D$+ JD$ D$: @: D$=\$ D$ |$|$|$t-D$\$ @D$ \$\$NjD$D$|$t^D$D$ËD$ .@.=u(D$ø) @D$D$떋\$D$ D$=u-;t%<$|$@D$+ @D$+ D$(|$$|$ $|$|$ |$|$|$|$\$t43҉ t3CӉ=u $D$|$|$ |$ D$ D$$I @ =D @D$$ =ulC@D$$ @\$D$@@D$$ \$4@4D$ @D$$ \$4@4D$D$ D$ 7|$|$) |$====T$-L$\$D$; @; D$ =D$< @< D$\$ D$< @< D$ D$/ @/ ËD$ 4@4D$l$>¹D$ \$= @= D$ =) |$ |$ t0D$ 1# @# \$ 1\$ ċ\$) 4@4D$) á1< @< D$ 3|$=t==l$T$ L$ @\$@D$\$\$D$ \$\$D$ (|$$3ҋ1@D$=t$D$|$$3틈t1@FD$=t$D$ÍF|$$3틈1@FD$=t$D$ÍF|$|$$\$L$|$ |$ D$|$t^D$D$\$ @ D$ =t \$\$D$D$D$떋D$\$$x@x |$|$$5+ ;85+ ;!+ 3҉ t3CӉ2@=<$|$|$D$D$E@$> @> G@$D$> @> M @D$ Ë$ @ M @D$x@xD$D$p+ á+ +  FhP<$td$@ @==t$$ˋ@F$\$D$$$둋D$%|$|$ |$|$<$= ~ "$=< |$ $@=t-|$t!$o@o$\$\$븋|$t$|$|$ =|$uiD$ @ D$ D$D$D$=uD$+D$ @ t$ D$ 뺉ø; @t$differentiating  w.r.t.  we get the new equations : <|$8|$4|$0|$,|$(|$$|$ |$|$|$$|$|$ |$\$;=%t3CD$@D$ JD$ |$ |$|$D$D$ @==])@D$ D$|$:D$D$ ËD$A @A 3C@=;=0tK D$ D$ @B @B  D$ @D$$|$$CD$$D$(C@D$ \$(@D$,@@D$ D$0 @D$ D$4 @ø@ø@D$83|$=t|$8=l$4T$0 D$,@D$\$D$(C @C ;=0tD$ \$D$@D$\$$\$$ @D$@$D$D$D$D$D$D$\D$<differentiating  w.r.t.  we get the new equations : 8|$4|$0|$,|$(|$$|$ |$|$$|$|$ |$|$\$;=%t3CD$@D$ JD$ |$ |$|$D$D$ @==v|$fI @D$ =L;=%t! @D$ 0 @0  @D$ D$;=0t@4 D$L D$B @B \ |$|$ |$ {D$ D$$C@D$ \$$@D$(@@D$ D$, @D$ D$0@ø@D$48 @D$ |$=t|$4=l$0T$, D$(@D$I @ =u*D$@+ D$@+ L$\$D$$C @C ;=0tD$ \$D$@D$\$ \$ u @D$@$D$D$D$D$CD$8!Situation before call of Groebner:.An algebraic Groebner basis computation yields a contradiction.!a single new system of conditions."All previous equations are dropped.The new equations are:* cases. All previous equations are dropped.)CRACK is now called with a case resulting #from a Groebner Basis computation : ,|$(|$$|$ |$|$|$|$|$ |$|$$; 3҉ t3CӉ3C@= $3҉ t3CӉD$$3҉ t3CӉD$$\$ D$|$ uD$8D$ @ \$@D$\$ \$ 붉D$ $\$D$|$uD$8D$&@ \$@D$\$\$붉ŋT$ L$\$ "áø;@= |$|$ |$|$|$$\$ @;= !1@D$|$$|$(|$|$ u D$ C@ tø@@D$(D$$D$ D$ =uD$$HD$ C@ tø@@t$(D$(띉ø@D @\$E @ø@-@-1F @;= !1@D$|$$|$(|$|$ u D$ C@ tø@@D$(D$$D$ D$ =uD$$HD$ C@ tø@@t$(D$(띉ø@G @\$E @ø@-@- @;= !1@D$|$$|$(|$|$ u D$ C@ tø@@D$(D$$D$ D$ =uD$$HD$ C@ tø@@t$(D$(띉ø@H @\$E @ø@-@- @;= !1@D$|$$|$(|$|$ u D$ C@ tø@@D$(D$$D$ D$ =uD$$HD$ C@ tø@@t$(D$(띉ø@D @\$I @ø@-@- @;= 1@D$|$$|$(|$|$ u D$ C@ tø@@D$(D$$D$ D$ =uD$$HD$ C@ tø@@t$(D$(띉ø@H @\$I @ø@-@-l $1 @ Ë$ËD$h @h D$J @J D$ ;=0t |$ u @3C@ø@ËD$  =t,;=0t 0 ( t|$t \$@D$ԋD$ D$ ;D$ ;=0t3H p  3t==-z1L$ : @: D$B @B ;=( t $D$;=0tFL$ 3҉ t3CӉ D$$D$ËD$ @ |$  D$o@oD$|$ |$$\$ \$u@3۸ @D$$D$ D$D$=uD$ :D$@3۸ @t$$D$$뫉ËD$K @K ;=0t"  , 3t==-zL$ 1: @: D$\$ \$ \$L @L D$=t ;=( u\$M @M D$( ;D$ \$D$ @ D$\$L$N @N D$,1Type in the expression for which its vanishing and"non-vanishing should be considered.You can use names of pds, e.g.: 4coeffn(e_12,df(f,x,2),1); or df(e_12,df(f,x,2));$According to the known inequalities, this expression can not vanish! --> Back to main menu.:If you first want to consider this expression to vanish and(afterwards it to be non-zero then input t- otherwise input nil : The case8contradicts inequalities and is not further investigated.+CRACK is now called with the assumption 0 =  : "CRACK is now called with assuming  to be nonzero. BAccording to the system of equations, this expression must be zero! --> Next case. --> Case splitting completed.CThis completes the investigation of all cases of a case-distinction.H|$D|$@|$<|$8|$4|$0|$,|$(|$$|$ |$|$|$|$|$ |$|$<$;=( |$|$|$8|$@|$4|$|$$|$|$0|$D|$|$ <$|$<\$(L$,;t JD$\$(D$ =D$ th    =|$, @@a@aD$|$(|$ |$ tbD$ D$C@ tø@ËD$ËD$' @' D$\$ \$ 뒋=0<$0\$1< @< tËD$O @O D$3C3@ËD$ =t ( t<$=0;=( ( ;\$ tA4 `  |$=|$|$ |$ tRD$ D$3C3@ËD$ =u\$D$(6 @6 D$ D$ 뢹\$D$(6 @6 \$,D$(|$ tH   !@!D$8|$= D$8D$@tD$4\$,D$( @ D$D$\$\$ |$ t-D$ \$@D$\$ \$ NjD$@D$\$$D$4o@oD$4|$8"L$3۸ @ËD$4K @K ;=0t3|$(=t==-z1L$\$D$@D$$;=0F;=( t0D D$P @P T   D$$ D$P @P L$3۸Q @ËD$4K @K ;=0t1 D$@ |$|$ |$ t0D$ ËD$(6 @6 D$ D$ Ŀ;=( t;=0t  ;=( tO;=0t%|$4u X  l ( D$03 @ <, ;D$8uD$(\$(D$$@\$,L @L D$0L$,\$(D$0R @R \$(D$,;=( u L$@|$0t$;=( u\$DD$0M @M D$D( |$4u!=S |$<;D$8tt \$8\$ Enter 17or a set of functions (and then all equations containingConly these functions are selected) --> Enter 2: 8Specify a subset of equations to be solved in the form: ; Equations  are not valid.8Specify a subset of functions to be solved in the form: Functions ?There is no subset of equations containing only these functions.%Do you want an automatic substitution 'of computed functions afterwards (Y/N)? 9CRACK is now called with the following subset of equations8|$,|$ |$$|$|$|$(|$ |$|$0|$4|$|$|$$;=t C=|$ @;=uT  !@!D$=yt=nu=nu|$= @  !@!D$(=t=u= $B @B  @D$$ @ D$(|$u\$=t4 D$($ D$|$|$,|$,t:D$,@@ ËD$ 4@4D$ \$,\$,뺋|$|$48 1B @B  @D$1 @ D$(|$u\$=t4x D$($ D$|$|$ $\$,|$,tQD$,D$@@ \$ @ =u\$D$D$D$,D$,룋|$u |$u   !@!D$=yt=nu=yu D$$t \$$|$=, D$$$T @T D$ t\$ D$= @= D$0\$D$,|$,D$, uh\$|$tFD$S FÉ\$(t\$D$(D$D$D$뮋D$,D$,aD$U @U D$4\$S S D$0á14@4\$01\$,|$,D$,D$3|$4=t==-z1 D$@\$4@D$4\$,\$,]D$0D$,|$,3D$,D$  @;a@a@ËD$@D$ 3|$4=t==-z1 D$ @D$(\$4@D$4|$$t(D$(ø) @D$,D$,D$0á14@4$1D$48%8;Unfortunately this module is not completely implemented yet.;=0t  $D$D$<$$I @ =t~$I @ @=tR|$t*$@ \$@=t$D$@ D$$$GD$ Separation of ' leads to one or more case distinctions. yields |$|$|$|$ |$$\$ @==u%; 1 @$==I @$ =$V @V @$ D$ @@$ D$ @$ D$D @$ D$;= tD$ŋT$L$\$D$ W @W D$=u@;=0t+h $| 3@)3҉ t3CӉ3C@=uQL$3҉ t3CӉ=ut$^|$|$|$|$ u_D$ D$D$D$ D$ =uD$&D$ t$D$뿉D$ @@$ D$ @$ D$ @D$8 @$ |$=t|$=l$T$L$ ؿ: @: D$;=0t5h $ D$D$8 @$@;= t @$@0|$,|$($\$L$|$ |$|$|$|$|$ |$$Y @Y =t$3@D$ q<$ t3C$D$ Q4$@;u$$$;D$tt L$<$ u4$ @;u$\$ $D$ D$D$|$ D$ \$Y @Y =tD$ \$D$e\$ \$$ uAc @;u43C6@\$$5@D$$|$$ t$$5@;D$$\$Y @Y =|$$ t$$@;b\$$\$(|$(tZD$(D$,\$Y @Y =t\$D$,D$\$ D$,D$ D$(D$(뚋|$tat$;t\$@D$D$D$\$$L$5@\$D$|$ t&t$ ;t\$ @D$ D$ D$ \$$L$ 5@\$D$\$D$$D$D$ D$ ;|$;= u*\$D$9@9=uÉ\$ \$\$cL$3҉ t3CӉ3C@=t\$ @ D$D$|$tgL$3҉ t3CӉ3C@=t\$ @D$\$\$|$u|$|$uE|$t#\$@3@D$ \$3@D$ |$tA|$t!\$@ËD$D$ Q\$D$D$ =|$t 3C@ËD$D$ 3CD$D$ D$ 0separation w.r.t.  . ,|$(|$$|$ $\$L$T$ |$|$|$|$<$<$ y4$6@;_L$$\$X @X D$|$u $L$\$X @X D$|$|$$$t$u\$\$#L$\$6@D$t$u \$D$D$;L$\$6@ËD$D$\$D$[ @[ D$t$D$cL$$\$X @X D$=t&$$ËD$[ @[ D$$D$ |$4L$3҉ t3CӉ3C@=|$ |$|$|$t6D$\$9@9=tD$D$뾋D$=|$ t$6@;|$$|$(|$|$ u wD$ D$(D$$D$ D$ =uD$$2D$ t$(D$(볉$|$$|$(|$|$ u_D$ D$(D$$D$ D$ =uD$$&D$ t$(D$(뿉$T$ L$\$$\ @\ D$=u D$=D$;=0t1 D$% @% 0 |$ t$6@;|$$|$(|$|$ u wD$ D$(D$$D$ D$ =uD$$2D$ t$(D$(볉$|$$|$(|$|$ u_D$ D$(D$$D$ D$ =uD$$&D$ t$(D$(뿉$D$Ë$] @] =uD$D$,|$|$|$|$ |$$\$=u3e =t3@P<$ uJ4$5@;u:$\$ =t$<$ 4$ @;|$|$|$|$$\$ 9tc\$ L$L$D$ D$ =uD$)D$ \$t$D$뼉D$Át=uD$tً|$t\$@<|$8$\$L$T$ |$|$|$|$|$ |$$|$,|$0|$4D$( 4$;|$(t |$(<$|$$|$$t0D$$\$9@9=tD$$D$$ċ|$$u3@$uD$$D$$$\$D$^ @^ D$=u D$(4<$|$ |$ |$(t |$(D$ D$\$ \$ \$^ @^ D$=uD$(띉\$@@=0|$,0a@at\$ @ D$$|$,=0\$9@9=t ( t;=( t( |$$ u D$$D$$\$ _ @_ =t#\$(3@$\$ \$$\$$D$(|$|$4|$4-|$(t\$$@ø@D$8\$4@ËD$8` @` D$(==4 |$0|$0tuD$(D$8D$0@\$(D$8m @@=uD$0D$0|$0t D$(|$(D$4D$4|$('\$$@D$(< linear independent expressions : 'Are the expressions linear independent? |$$\$D$D$D$ t|$ |$ D$\$9@9=t\$\$밋D$\$=t\$\$녋|$ ujt$5@;uSD$~u:D$\$=t\$\$ D$ |$ ;= t=0|$0';=0t <$|$|$t%D$a @a D$D$Ͽ;= t,  =t D$ tD$ |$=0D$ tD$ ÍFSP=u ؃%F =tK$D$ËD$$%$\$lË$%($\$L$T$ |$|$|$|$|$ |$$<$$D$\$L$$$u\$$D$$뚋l$L$닀Y @Y =tL|$t)L$D$\$$$1\$$D$D$$D$ËD$< @< / @/ ËD$=t5\$D$L$ËD$$$T$ L$D$\$Z @Z D$ =u$D$$Y=|$u |$tE\$D$‹\$D$ c @c $$\$ыc @c \$$D$$;D$t5\$D$L$ËD$$$\$$D$D$$jD$$(|$$\$L$T$ =tb$L$ \$D$T$ L$$\$ËD$%0|$,|$(|$$$\$L$T$ l$|$|$|$|$  6@;e @e D$ 3҉ t3CӉ3C@=t\$ @D$ \$ \$ $$ @;|$ u |$(|$,\$$uqL$ ø6@D$,D$(D$$D$$=uD$(1D$$L$ ø6@t$,D$,봉øD$A|$ uøD$"L$ ø6@øD$L$D$T$L$ \$b @b D$=|$D$D$;t)@ËD$D$D$;tX;t' @\$D$1D$3@D$\$D$\$\$D$0$\$L$D$ =@=<$ uO4$6@;u?$D$< @< / @/ \$ @ =uOʼn‹L$\$$d @d D$  u'ËD$ 4@4D$ D$ 0|$$|$$\$L$T$ l$|$|$,|$(|$|$ =t@=t33D$,6<$ up4$6@;u`$D$< @< / @/ \$ @ =t$3@D$,<$ u!4$ @;u$@$l$T$ L$\$$d @d D$,=b=t'ËD$,4@4D$D$,|$D$D$(\$\$\$\$$|$$D$$D$L$(6@@D$ \$D$\$ < @< =ua\$D$D$L$\$ @\$(@@\$(D$(D$$D$$ \$,D$(D$,D$,0 |$$\$ t %h @;u %=u $ %$\$yD$$\$cËD$ %i (SPLIT_SIMPLIFY DOES NOT HAVE 4 ARGUMENTS.Start of splitting equationsTHE :. ELEMENT OF THE INPUT LIST OF EQUATIONS IS NOT IN STANDARDCQUOTIENT FORM! THIS MAY HAVE BEEN CAUSED BY USING COMMANDS LIKE consFIN ALGEBRAIC MODE. IN THAT CASE, USE sqcons, sqrest, sqfirst, sqsecond,sqthird, sqpart INSTEAD.+ s: Determination of the splitting variables' s: Variables to be used for splitting:  s: Start of splitting equations equations result7 s: Now simplifying equations and dropping multiple ones. simplification run!Substitution of vanishing unknownsCoefficients found to be zero: & redundant equations have been dropped. s: The system is formulated.\|$X|$T|$P|$L$|$|$|$ |$|$|$|$|$ |$$|$(|$,|$0|$4|$8|$<|$@|$D|$H;=> t eD$D $3҉ t3CӉ=t ;=0t $-@-D$<$@\$@$@L$4$J@D$HD$|$<|$L|$LD$L~utp u @;tY  D$  `      D$o@oD$\$L\$LA;=> tHe\$DD$@6@6! |$|$DD$8D$|$4|$L|$LD$LD$PËD$@Y @Y =D$P\$84@4D$8=S |$T|$TD$TD$XD$@.@.=tCD$X\$P.@.=u#D$X\$4@4D$D$TD$TpD$LD$L|$D$|$<|$L|$LtaD$L u=@;u0k @k \$4@4D$D$LD$L듋|$|$L|$LD$LD$P\$9@9=t4\$4D$P9@9=u7\$@D$P9@9=tD$P\$84@4D$8D$LD$Lh|$8u\$<@\%t$8;tD$8{@{D$8\$4D$8] @] =u D$<$ ;=> e\$DD$@6@6L! 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 |$$=|$ @=1|$;=8t?  =t-8D$ @ ËD$ @ D$|$=\$${ @{ .Horizontally: function names (each vertical), Vertically: equation indices s+<|$0|$(|$$|$ |$$\$|$ |$|$|$|$,|$|$8=| |$4t!} @h@h-@-D$L$3҉ t3CӉD$ D$D$0|$0D$0 D$3҉ t3CӉD$\$@=t|$|$\$D$D$\$0\$0hD$ @uD$8  D$0\$0D$@= D$$\$$D$ @=\$$D$D$3҉ t3CӉËD$0@=t \$$D$\$0D$$o@oD$$ND$0o@oD$0D$o@oo@oD$,<$|$0|$0lD$0D$$D$,@=1D$$ D$D$(L$3҉ t3CӉ\$(@=u)\$(D$D$(o@oD$(뒿|$(L$3҉ t3CӉ@D$(\$(@=u  D$(o@oD$(Ż@@D$$ D$$ @D$$ =u;( @D$$ =u%+ @D$$ =uD$$, @, |$|$(|$(D$(D$ ËD$.@.=t  V$ @\$ D$$~ @~ =u( @\$ D$$~ @~ =t    D$(D$(TD$,o@oD$,D$0D$0|$4t!} @@-@-D$8u< schrott.tmp +Do you want a quick overview on the screen? 6Here are the maximal values scaled to 1 in the diagram:max method index: max # of unknows: max # of equations: max # of terms: max # of factors/term: /100max # of free cells: !) title "method :" with lines' using ($1/60000):($2/plot ' %w%w%w%d%w!) title "unknowns :" with lines' using ($1/60000):($3/, ' %w%w%w%w%d%w!) title "equations :" with lines' using ($1/60000):($4/!) title "all terms :" with lines' using ($1/60000):($5/!) title "factors/term:" with lines' using ($1/60000):(100*$6/$5/!) title "free cells :" with lines' using ($1/60000):($7/+' using ($1/60000):(0) title "step :"%w%w%w%w.Do you want to add to the plot a graph for the - method used at each step: 1 - number of unknowns: 2 - number of pdes: 3 - number of terms: 4 - number of factors/term: 5 - number of last free cells: 6*or add no further graphs: n *What is the scaling factor for this graph? plot ,%w%w*$2) title "method :"' using ($1/60000):('*$3) title "unknowns :"*$4) title "equations :"*$5) title "all terms :"*$6/$5) title "factors/term:"*$7) title "free cells :".set title "Modules in order of their priority: %w%d%w" 7Do you want the x-range to be determined automatically?  @ H @ H2What is the minimal value of x (time in minutes) ? 2What is the maximal value of x (time in minutes) ? ]: set xrange [ %w%d%w%d%w @ H @ H.Do you want to display the plot on the screen? Do you want to print the plot?  @ H @ H4Give the file name in which to save the plot in " ": set output '%w%w%w @       0  P  set autoscale xset autoscale yset noautoscale set key Leftset output '|lpr -Pmath4'set terminal postscript eps 22h|$P|$H|$$|$ D$ |$D<$|$|$`|$@|$X|$L|$|$|$|$|$|$T|$\|$(|$,|$0|$4|$8|$<=|$d @|$ |$H|$HD$H$\$H\$H; @;uȋ|$u L$밹L$$<$u|$<$t-$D$.@.=u$$뭋|$t3\$D$.@.=u\$\$n<$u+D$\$D$\$\$8|$u($\$D$$$\$$ =t6$\$D$$$L$L$\$$.@.=t+D$\$D$\$\$d$D$.@.=t($\$D$$$D$\$Ë$D$$$L$L$D$D$`D$D|$tAD$Do@oD$D\$\$LD$L\$\$D$(D$,D$0D$4D$8D$<D$LD$LD$D$D$D D$Dv@vD$ D$P|$PD$PD$$~ ;{|$u\$D$@D$\$@=t0\$D$@ËD$@D$|$|$D$ D$$\$LFD$=t3 D$$J D$$J D$$J D$$J D$$J|$|$|$t.D$\$(@=t\$\$(D$$J\$,@=tD$$JD$,D$$J\$0@=tD$$JD$0D$$J\$4@=tD$$JD$4D$$Jød'@'D$ D$$JËD$8'@'ËD$ @=t]D$$Jød'@'D$ D$$JËD$ 6@6D$8D$$J\$<@=tD$$JD$<D$PD$PD$D~@~D$`  =tD$\t|$\ < D$(X D$,t D$0 D$4 D$8  D$< l$( L$D4 @ pP |$,=x T$D ø p |$0= T$D ø p |$4= T$D ø p8 |$8=` T$D ø p |$<= T$D ø p T$D ø pD$`  @ h     0 !@!D$H=t'=t =t=t=t =` !@!D$@~u|$`u D$`  \$` pD$`|$Hu: |$@= T$D \$` pD$`=|$Hu: |$@= T$D \$` pD$`|$Hu: |$@= T$D \$` pD$`|$Hu:4 |$@= T$D \$` pD$`z|$Hu7T |$@= T$D \$` pD$`<|$Hu5x |$@= T$D \$` pD$`|$Hn\D$X |$L|$P|$Pt8D$P \$X pD$X\$P\$P뼹 \$X pD$Xø @-@-|$\u  =t% H-@-(H-@-0 !@!D$Th !@! ź L$T  pD$XH-@-(H-@-\$X @-@-H-@-|$\  =   =t,  < H-@-D H-@-_L  !@!   p  ø @-@-D H-@-\$` @-@- Hr @r =tt -@-|$d=h.Do you want to plot statistics of this session,'i.e. since loading CRACK the last time? size_hist%w.%w5h @;=ju $t*    =u$t<$t6 @ P j` Fp @ < } @} |$=Start :  step steps%w%d 0= 0<> %w%w,  soln1Somthing is wrong with level counting in size_histn= level:0|$,|$(|$ |$|$|$|$|$$|$|$$|$ uD$,=|$(4 @g@h@h-@-< D$D$D$ D$$$<$$D$$=a=\$D$@D$ H D$ |$ uP \ |$|$D$o@oD$$JD$ D$4 $\$|$t/D$\$h pD$\$\$ŋD$|$ |$  tD$ t$ @;u4t D$  @ It$ Q @;u2| D$  @ D$ D$ =toD$4 $\$|$t2D$ \$ pD$\$\$‹D$|$ =zuLD$@D$|$ ztU $J (~u$\$|$$at|$$z|$|$$D$D$|$ tD$D$D$3C\$\$ӉËD$ =uF  D$ $|$$|$ $$>g@@-@-|$(=D$,u06Do you want to print the tree of cases of this session,'i.e. since loading CRACK the last time? size_hist%w.%w5h  @;=ju $t0  !  =u$t<$t? @F @! jP! pF @ <  @ |$=*** Mis-use of *** came before  in proc_list_ !New proc_list_ based on $\$D$ =|$D$t|$;=$ =D$.@.=us4" $H" T" $h" |$=D$+\$ D$ |$\$ $;=tËD$D$ ;=t.\$=tſ;=t-\$ D$ ƋD$ ;= t" $*** Mis-use of choose_6_209*** subst_level_45 came before choose_6_20 in proc_list_ !)proc_list_ has been automatically changed.6 is changed to 20.$|$ |$|$|$|$|$ |$|$$) @.@.=0|$|$|$|$ |$|$ ;= tC=< |$ |$ t-|$ ~tD$ D$ Nj|$  13҉ t3CӉD$$\$D$|$uD$8D$@ \$@D$\$\$붉D$4D$ JD$D$ JD$ D$@= D$@==|$D$t|$;=5x @;}5 @;u:$ $ |$=D${\$ D$ K|$;=t\$ ) @D$ ;=tMD$ .@.=t\$ D$ 릋D$ ;= t"0% `% $&*** choose_27_8_16 needs size_watch=t !*** Mis-use of choose_27_8_16*** * came before choose_27_8_16 in proc_list_ !)proc_list_ has been automatically changed.The order is 8,16,27.The order is 27,8,16.$|$ |$|$|$ |$<$|$|$|$;= u( &=< |$|$t-|$~tD$D$Nj|$t$D$D$\$\$|$t-|$~tD$D$Nj|$Wt$@D$D$ \$\$D$|$  t$ \$ D$@$`6@6ËD$@D$ @=uD$ t|$ |$|$t-|$~tD$D$Nj|$uD$ D$D$ \$\$=|$D$t|$;=5y @; @ @9 @.@.=uV() L) X) |$=D$?\$D$|$7\$y @D$;=t|$ t% @ @9 @\$D$#9 @ @ @\$D$;=tMD$.@.=t\$D$릋D$;= t;) |$ t )  ) $'*** choose_30_47_21 needs size_watch=t !*** Mis-use of choose_30_47_21*** + came before choose_30_47_21 in proc_list_ !)proc_list_ has been automatically changed.The order is 47,21,30.The order is 30,47,21.<|$$|$ $|$8|$|$|$ |$|$|$|$4|$|$(|$,|$0;= u. =< |$|$t-|$~tD$D$Nj|$vt$_D$D$\$\$|$t-|$~tD$D$Nj|$t$D$D$ \$\$D$|$ t$ D$JD$$D$ JËD$$@=D$o@oD$D$JD$$D$ JËD$$@=tD$o@oD$\$ D$@D$`6@6ËD$@D$ @=uD$8t|$ |$|$t-|$~tD$D$Nj|$uD$ bD$D$ \$\$A=|$D$t|$;=5z @;@, @; @.@.=uV.  / / |$=D$?\$4D$4|$]\$4z @D$4v @  =tvv @ 13҉ t3CӉD$ @=u\$4v @D$4;=t|$8;= tC=< |$(|$(t-|$(~tD$(D$(Nj|$( 13҉ t3CӉD$,$\$$D$ |$$uD$ 8D$$@ \$ @D$ \$$\$$붉D$04D$(JD$,D$(JD$0 D$,@=uA D$0@=u%@, @; @\$4D$4H@; @, @\$4D$4#; @@, @\$4D$4;=tMD$4.@.=t\$4D$4릋D$4;= t;L/ |$8t |/  / <*** Mis-use of choose_70_65_8_47*** - came before choose_70_65_8_47 in proc_list_ !)proc_list_ has been automatically changed.70,8,47 has been inserted. |$|$|$ |$|$|$|$=< <$<$t*<$~t$$ˋ<$@ @| @ @< D$\$d'@'D$ '@'ËD$@=uTD$Jød'@'D$ '@'ËD$@=tD$t=|$ D$t|$;=5 @;D$.@.=uV`7 7 7 |$ =D$P\$D$ |$ \$ @D$;=t|$tËD$D$;=t.\$=tſ;=t-\$D$ƋD$;= t.|$t"7 7   <$|$=< |$|$t-|$~tD$D$Nj|$< $u @D$$ @=t @@; @D$k4$tF|$ u4t$t; @@ @D$@; @ @D$\$ @ @ *** Mis-use of choose_11_30/*** alg_length_reduction (11) or decoupling (30)+*** came before choose_11_30 in proc_list_ !)proc_list_ has been automatically updated.0|$,|$(|$$|$ |$|$|$|$|$ |$|$$;= c|$|$|$|$|$ =< |$D$D$ D$(1 @ Ë$Ë$H @H D$$|$t$Hu|$t|$ |$~u< D$@=u  D$ @=|$~ YD$(o@oD$(t$ D$o@oD$|$D$ËD$$@D$,\$(2@ø26@6ËD$,'@'D$t$uzD$ |$ ufD$ËD$$@D$,\$(2@ø26@6ËD$,'@'D$ D$D$$D$D$|$uD$|$ uD$ =|$D$t|$;=5 @;5 @;t5; @;uQ< = H= |$=D$Q\$D$!|$;=tK5 @;t&5 @;t5; @;u먋\$ @D$D$@=t/ D$@=t\$ @D$D$ @=t/ D$ @=t\$; @D$;=tMD$.@.=t\$D$릋D$;= t|= 01From now on lexicographic ordering of derivatives.5The current variable ordering is going to be reversed.|$|$$D$ \$t ;u ;=t@5B @;t-\$ D$ 볿;=;= uF t;=0tdC \$ D$ ;=0tC zz|$|$|$tPD$D$ @ z3 @3 @D$x@xD$D$뤋 z1D$ @ D$;=t-\$ D$ ƋD$ $D$%F|$$ D$|$t4D$$ =uFD$D$D$$|$ |$|$|$|$|$ |$$\$t9@9=|$ |$ @D$|$|$|$|$ u D$ D$ @ËD$D$D$D$ D$ =uD$CD$ D$ @ËD$t$D$뢉D$Ë$o @o \$ D$@|$  ug|$  uU\$  =t-\$ @=tt$|$$\$D$\$ |$ t/D$ L$3ۿ @ D$\$ \$ ŋ|$|$|$ |$ tAD$ D$ @ $\$m $\$ \$ 볋L$$@ ø6@@ .@.* Leading derivatives with non-zero symbol: ,|$(|$$|$ |$|$$|$|$ |$|$|$@á  =   @@@=t)D$t  @ -@-\I  @$ D$)@$ D$|$D$D$\$\$\$=t|$|$ |$ tW\$D$  =t D$ \$@=tD$ D$ 띋|$ <\$D$D$%|$|$ |$ D$ ;u @D$$S @$ =uGC@$ \$$@D$(@@$ ËD$(@=t\$D$$D$D$ D$ A\$ @|$t!  @ -@-,0No leading derivative has a non-zero coefficient.+Which term shall be substituted by the rule?Input its number + Enter: Input is not a number!This number is too big.#The new list of user defined rules: ,|$(|$$|$ |$|$$|$|$ |$|$|$ @ D$)@$ D$|$D$D$\$=|$|$ |$ tc\$D$  =t&D$ \$@=tD$ D$ 둋|$ uD$\$D$D$D$|$|$ |$ ,D$ D$$;u@D$S @$ =uxC@$ L$$\$ @ D$ u a@aD$(@@$ ËD$(@=tHt$$u\$D$D$'L$$\$5@\$D$D$ D$ |$uL t$;L \$ @M !@!D$~$t\$4M {L$3҉ t3CӉËD$@=t\$PM \$D$D$|$C@$ D$|$ udt$5@;uMtø@D$ D$\$D$ @ D$.tø@3ɋ\$A @ D$|$ uD$D$a@aD$D$a@a\$@D$ D$@ËD$ @\$c@ctø@\$ @ ø@ lM -@- @y @y  -@- @y @y ,;Total number of occurences of all unknowns in all equations:1Total number of equations in which unknowns occur:$|$|$|$ |$|$$|$|$ T=1|$|$D$D$D$<$|$|$D$D$@ D$ C@D$ D$3ۋD$ËD$o @o @ @@ ËD$ @ËD$@D$\$\$M\$D$T\$T\$T{ @{ T=1|$|$D$D$D$<$|$|$tKD$@@ ËD$=tD$o@oD$D$D$멋\$D$\$ D$ \$\$MD$ { @{ D$ ;=0tA@S To @o S D$ o @o $T|$$t @t D$|$tpD$a@a4 F=t9D$@=uFD$D$넋|$trD$@=uPT$ $$$@ $ D$\$|$D$a@aD$19@9=u4 D$=tD$\$\$D$ @ d|$tD $T$$$@ $|$ |$|$<$|$|$|$ |$|$;=8~;=TT8=tT\$D$T\$D$TT;=Tt-T\$TD$TƋ=4 |$|$UD$D$ &1=8D$=t|$<$<$t.$\$ =u$$Nj<$$\$D$$\$D$y|$<$<$t.$\$ =u$$Nj<$t.$\$D$$\$D$D$D$D$ @ ËD$D$D$ @ ËD$ËD$D$ |$|$ D$=t]É\$ \$D$D$=uD$&D$t$ D$ 뿃$SË$%|$$\$L$|$ ;= t @ @D$D$@ËD$@D$ @=tL$\$$ @ D$ @|$|$ |$|$$D$ =uD$|$D$D$<$|$ |$ D$ D$D$ =tL\$D$@ËD$\$4@4D$!\$D$\$4@4D$D$ D$ ED$D$ D$|$ |$|$$    \$D$ |$ t'D$$I|$ u8t$@;u'$D$D$<$ <$ <$ <$ 4$|$;u>$c\$ 4@4D$ $$A$1\$ 4@4D$ $|$t\$ D$D$ F|$|$$D$ @ D$|$tTD$D$$F=t\$ D$D$ FD$D$렋D$ @`H6@`H @|$|$ $\$D$ t2=W$3CD$6`H  =tS$\$ |$ D$ \$R\$4@4D$\$ \$ 뺋4$@;u!$\$D$4$ @;uS$\$ |$ bD$ \$\$ @ D$\$ \$ 뺋4$5@;<$~ $\$ID$ |$ D$ D$~uF$'@'ËD$\$D$D$ D$ v4$@;u<$\$=t3ۋ$CD$D$P @ =t9 @$ @@%ÍP @F =t @$ 3 $\$L$u ZF =t $8F\$D$ =tD$L$\$$돃 $|$ $\$L$T$ |$|$|$>l$ =t3 ; \$D$ =tjL$ \$$ @ É\$u3\$$@=tt\$L$D$0 @0 ËD$ =t It?D$1 @1 D$ D$1 @1 É\$D$ D$@=t \$D$@=t tL$ \$$ @ É\$\$D$ =t,\$$@=tXtQ\$L$D$0 @0 ËD$ =tt tt$|$$\$L$|$ |$|$=T$$<$$D$ =uWT$$L$D$2@2=t$\$ D$ $\$D$$$oL$\$D$ D$L$\$D$ËD$ËD$SP @ =t D$$F|$|$ $\$L$;= t @% @D$D$@ËD$@D$ @=tL$\$$ @ !D$ @=t tFÍ @;uÿ @;utÿ @% |$|$ $\$L$ @;u  @;u t@=t \$$@=t t3 @ D$ \$D$ @ D$ËD$ @=tt7\$ D$@=tL$\$$|$|$ $\$L$u3 @ D$ \$D$ @ D$ËD$ @=tt7\$ D$@=tL$\$$B |$|$<$ t3<$|$D$=~u @3@É\$$D$D$=u$?D$~u @3@t$D$막ø@ r @%r |$|$ <$D$\$=|$ <$L$9ËD$ @ ‰$T$ D$D$=u D$ D$ËD$ @ 4$$D$L$ |$ tYD$ ËD$ @ ËD$ \$D$\$\$L$ L$ 뛋D$%SË$u3B;t~t 3@FF|$ $\$D$<$t?$\$ =u%$\$D$$$붋<$4$;t<$~t$$^$@D$ $\$ËD$ D$$$ D$<$t%$\$D$$$ЋD$%|$$\$|$|$ = @ D$$D$ @ ËD$@D$4$;t-<$~u$ $\$8D$ 3ۋD$@=uD$ ;|$u$\$ \$ D$Ë$|$$A @A D$$ =t|$tt |$$ @ L$D$$ =u\$\$ʋ|$t>\$@@\$3FSPؿ@F\$$ @% @qH @qH6@qH @$\$L$D$  =t3CD$ 3D$ <$ $qH  =tV$$<$L$$\$TL$\$ @ D$ $$붋4$@;u%L$$\$D$ K4$5@; L$$\$D$ ~ <$~uA$'@'ËD$ D$ @=u @D$ q$\$Y @Y =B @D$ .4$@;$\$ =t!3ۋ$CD$ $\$Y @Y =t3۸D$ @D$ \$$Y @Y =t3۸D$ ^$ =t3L$$\$ @D$  @D$ D$ F|$$ @ D$;t%$ø@)F@=t$\$%|$|$$\$|$ |$|$ @ D$<$ 4$6@;$D$< @< D$L$3҉ t3CӉËD$y @y =t-$\$ $ $3ۋ$%D$<$ u4$@;u$$ $$<$$D$< @< D$L$3҉ t3CӉËD$y @y =t$\$D$$\$ D$ $$M|$u D$gL$3҉ t3CӉ3C@=t\$@D$\$\$|$ u D$ gL$ 3҉ t3CӉ3C@=t\$ @D$ \$ \$ |$t.L$\$ 6@D$ L$\$6@D$\$D$ %R @yHP @ yHU @(yHV @ |$$D$ 6@;ueD$$ËD$4@4Ë$4@4D$ yH  =<$ uZ4$6@;u>$MD$$ËD$4@4D$u$Ë$4@4D$E<$|$|$t2D$\$4@4D$\$\$‹D$ <$\$|$|$ @ $<$tU$a@aD$ \$@=u\$D$ D$F$$렋D$|$ \$<$|$D$=D$  u6 u"ËD$< @< ËD$ @ L$ $D$D$=u$gD$D$  u6 u"ËD$< @< ËD$ @ t$D$|$|$|$|$$\$D$ t@D$$MD$|$ uYt$@;uH$@\$ø@@D$- $\$@@D$\$D$ËD$| @| D$=;;D$@D$\$ @ \$ ً\$@@D$=) |$|$tKD$D$1=u1D$# @# 1D$D$멋L$\$ @$$|$$-@- u@;u a@aD$$@ËD$@Ftø@%FSP|$t2D$$ =uD$D$‹|$u3Vt$;t0|$~ut$u3@D$|$$\$|$ |$|$D$\$\$3Cb@bD$ |$u D$=|$~u\$\$L$L$D$D$\$D$@=5\$ $@$D$o@oD$븋$P@F =u9C@$ @@$x@%xʼn t @%@;tt Ëtt = =>= tu =u->= tu =tu uD% ǁ=t#3@сȉǁ>%J ÍF$\$=tPtH=t&$\$Ë$%$\$뢃$\$=tUtM2 @2 =t&$\$Ë$%$\$띃SPËD$2 @2 =tt  =tL$$S  @% F$\$ut.F.@.=t\$$뼃 $\$L$=utbȿ2 @2 =t,$\$2 @2 =tL$$\$  $\$L$u %1 @1 \$@F=t\$$ %\$$# @F# ËD$ %FSPËD$.@.=t%$1 @1 L$Ë$ @% D$ new function: new constant: $\$L$|$ ;= uM F =t9;=, t,, , \$  , \$$ @ D$ S D$ FS S |$ u"\$D$ S S ;=0tN|$ u#؅ FD$ % @%  D$ FD$ ÍFP;= t, , $@S FFS S F|$|$$\$L$D$ ËD$< @< / @F/ D$|$|$F|$tdD$D$Ë$Y @Y =t\$D$F=t\$ D$D$ FD$D$됋D$ %|$ \$ $|$|$D$|$t8D$ @ \$4@4D$\$\$뼋<$|$|$tdD$D$ \$=u\$ D$Y @Y =u\$D$ D$FD$D$됋D$%|$$=1|$ D$|$t@D$@@  1ËD$ 3@3D$ \$\$봋=1|$|$tMD$D$Ë$.@.=u\$ D$D$ D$D$맋|$ |$|$t%D$ @ D$D$ϋ 1\$ 83@38 1\$ ȿ3@31F u@;u @@ø@%S FÁt FWW@$D$<$ tD$$$3C\$\$֍ |$|$$|$|$|$ |$|$=-1 @1 D$$D$$$<$$1 @1 D$\$@=t$\$D$J\$D$@=t$\$ D$ $\$D$$$OD$ 2 D$D$2 D$D$2 ËD$ËD$ ÍF, |$|$$ uf$D$|$uGFD$D$ D$D$D$본 ()|$ |$$D$= <$ =z|$|$tMD$D$ Ë$.@.=u\$D$ D$D$D$맸 $\$ @ D$D$ËD$B @B   = expr. with  terms in ,  |$ |$|$$|$|$|$D$<$$D$$$  @;S uJ@;u+ @ @D$;=0t0@=D$Ԏ D$ D$@@ D$=t  D$w @w D$@D$f|$uD$D$$ @$ <$t  D$o@oD$ D$ D$@D$D$|$uD$>D$ o@o\$@D$\$\$밉‹L$ @@D$\$@O@=tD$S D$FD$=uD$$ @$ <$t  \$D$@D$ = expr. with  terms in ,  |$ |$|$$|$|$|$D$<$$D$$$  @;. @ D$;=0t0@=D$\ D$p D$@@ D$=t|  D$w @w N\$t@\$ @@D$|$uD$D$$ @$ <$t  D$o@oD$9 D$ D$@D$D$|$uD$>D$ o@o\$@D$\$\$밉‹L$ @@D$\$@O@=tD$S D$FD$=uD$$ @$ <$t  \$D$@D$ =,  $|$|$$@$<$$D$$$S FD$D$|$t=t$;t, |$tD$D$$ @$ <$fȕ V F<$=t^;=0tQ$<$uA$3۸ @Fa @a $$빍F factors in  terms|$$;=0< t3@@;u@ @;u| t3@h =@u.$@-$z @z z @z D$0@=D$< <$ t3@4$@;u$ @ 4$ @;<$ t3@n$ =@u.$ @ 0$@$@P 2$@=t3@$@New level : Current level : Back to level : .|$$;=0;= <$u ș <$u     D$|$t+D$ D$D$ɸ*** Start of level %w%wStart of <$\$ F g @g ;= t^= ~PtKk@kܚ  p$8 @Ë$8 ;= t8 T$ a< F< Back to *** Back to level %w%w |$|$$N @N  3g @g ;= t1 $ z< < ;= t[= ~MtHk@k ( pD$8 @ËD$8 = |$|$D$3҉ t3CӉŹ 3҉ t3CӉÉ@=tzD$=t#D$ @ !D$ @ D$ D$ D$ equations read from disk :  number of equations : number of lin. equations : total no of terms total no of factors  equations in  variable equations with derivative:s:(*#al), no equations function with  arguments :  :  constant : no functions or constants<|$8|$4|$0|$,|$(|$$|$ |$|$|$|$|$ |$$\$;=05 |$(|$8|$0|$4|$ |$|$D$ D$,D$D$=@  =u&H  D$h ;=u D$h D$@ t D$  $3҉ t3CӉ;=uuD$ <$|$$|$$t?D$$S @ =tD$ o@oD$ D$$D$$ D$  D$ <$|$$D$|$$uD$8D$$@ \$@D$\$$\$$붉D$ <$|$$D$|$$uD$8D$$&@ \$@D$\$$\$$붉D$,\$  =u0Ԟ D$ D$,<$D$ D$0<$|$$|$$D$$D$ @ D$ =t33҉ t3CӉD$D$|$ u |$ t\$D$  =t\$0D$D$00\$D$ @=t|$|$ D$D$0D$$D$$\$0$ @ $|$0-L$03҉ t3CӉD$(D$( 3CD$(@=t   D$  |$ t   ;=+tD$0 @ D$0;=+D$0D$\$0\$0D$3҉ t3CӉ$ D$< t$uL T D$D$|$0|$D$0D$,|$ZFD$,@=uD$,D$ D$$D$@  ËD$$@ø@ËD$,@D$,D$\ D$@ D$I @ D$=}D$$D$o@o\$$@=u d D$$o@oD$$뵋D$o@oËD$,@D$,D$@  u l ;=8tTD$)@ =t8D$)@ 89@9=t t ;=u'D$S @ =t |  D$D$= |$0D$ o@oD$  |$|$$|$$t2D$$ t\$4D$4D$$D$$‹|$4D$40 @0 D$41 @1 D$8D$8@=UD$(\$0|$4taD$41 @1 \$8 =t;D$(o@oD$(D$4\$0D$0\$4\$43ۋD$(@=3ۋD$8@=D$( 3CD$(@=t   D$8ğ 3CD$8@=tԟ  JD$( 3CD$(@=t   D$,1D$03 @3 D$0|$0t\|$, uD$,D$,o@oD$,D$0D$0D$0=t 똋|$4tD$41 @1 3HD$8 < contradiction8|$4|$0|$,|$(|$$|$ |$|$|$|$|$ |$$\$;=( t  |$$|$|$(|$|$0|$4D$,D$D$ D$|$ uD$8D$ &@ \$@D$\$ \$ 붉D$ <$|$D$ |$uD$ 8D$@ \$ @D$ \$\$붉D$ $3҉ t3CӉD$ 13҉ t3CӉD$e-‹L$\$ D$"ËD$ á D$4;=zu 8%<$gD$,D$<$|$ |$ D$ D$ @ D$=t33҉ t3CӉD$D$|$,u |$t\$D$, =t\$D$D$0\$D$,@=t|$|$,D$D$D$ D$ \$$ @ $L$3҉ t3CӉ\$,\$0D$0\$4D$0D$4\$0|$|$ |$ t2D$  t\$(D$(D$ D$ ‹|$(D$(0 @0 D$(1 @1 D$,D$,@=D$$|$(tID$(1 @1 \$, =t#D$$o@oD$$\$(\$(뫋\$,D$$\$0D$0|$(tD$(1 @1 3HD$,>\$4D$0D$48%8$|$ |$D$D$|$tTD$/@ \$@=tD$/@ D$D$D$렸D$ <$|$|$tND$/@ \$ =tD$\$ D$ D$D$릋|$ t$D$ ËD$\$D$\$ $ @ $D$@=tD$@D$=t<$ D$=What is the maximal number of terms of equations to be shown? equations :  equations with derivatives: no equations|$|$$|$|$|$ |$<$;=@=4$;@ =|$ @!@!D$ |$=<$|$|$tQD$D$@ \$ @=u\$D$D$D$D$룋D$D$<$|$ ;=+D$ @ D$|$D$3҉ t3CӉ D$ D$t @t D$D$MD$t @t ̲ Non-vanishing expressions: .Lists with at least one non-vanishing sub-list /(ie. a sub-list of which no element vanishes.): {,}$|$ |$|$|$ |$|$$|$|$4$;8 $D$ |$ D$ D$L @L =t,D$\$D$"t\$@\$D$D$ D$ oD$B @B |$|$ |$ t%D$ a @a D$ D$ ϋ4$;X  $D$ |$ D$ D$ĵ |$|$|$ND$D$ĵ D$D$|$|$|$D$D$ L @L =t,D$ \$D$"t\$ @\$D$D$D$oD$B @B |$tV|$t̵ |$|$|$t%D$a @a D$D$ϸԵ D$D$Ե D$ D$ M$FDo you want to see the coefficients of all derivatives in all equations:in factorized form which may take relatively much time? y/n: 0|$$\$|$ |$,|$|$|$(|$|$ |$$\$|$t2D$ t\$D$D$D$‹=|$ , @D$0 @0 D$$0 @0 $4  !@!D$(=yt=nu=nu \$$D$$t|$ =|$e|$ uD$D$΋D$D$\$\$L$ <$|$|$D$D$@@ \$.@.= @D$ D$,|$,xD$,\$ =:|$ |$|$t=\$,D$ =uD$D$뷋t$,;t!\$,@D$(D$,D$(|$$|$uzC@D$ tø@3A\$( @ e @e \$@ËD$,\$ D$ C@D$ tø@3A\$( @ e @e \$@\$ËD$t$r|$u*D$,\$\$ D$ <\$D$ËD$t$D$,D$,xD$D$|$ D$ D$,\$ \$ ;t@D$(L$(|$$D$(i@i @y @y  i@i˿y @y \$,@i@i @y @y D$( D$, 0 Functions : with  terms variables : |$ |$$\$=t $ @  <$|$D$ |$uD$ fD$ u, @;u @ 3@\$ @D$ \$\$ |$t  D$% @% F$\$L$T$ ;=0tH` @` D$d @d \$ D$e @e \$$h @h F|$ |$|$$ t(=t@=t3D3@<@;u;3҉ t3CӉ@% @;uCQD$$3ËD$@%@;t 6@;u5@;~u)@=t3@$@D$D$ D$$\$@=uL\$D$@\$6@6ËD$ '@'D$ D$o@oD$뉋D$ h @;uXD$\$|$t4D$ËD$'@'D$\$\$D$3@|$$=u 3f u t 3@;D$$ËD$@%F $\$D$=u3y u t3@\@ @@ D$\$@=tD$%$@ @@ ËD$@ F|$$@ @@ D$4$u3$@ @@ ËD$@% uV;uIu:u% t tÍF @%  u ttÿ;t ÍFP u? u1Ë$%<$-@F-$<$ u@<$ u/<$ u$$빍F<$ u, $$6@$$|$$ u" u t=t 3@e3\D$$ËD$@Ë$@%Í|$$u3 @ D$4$u3$ @ ËD$@%|$<$ t=t3@3@;t4 @;t'6@;t@;t @;uR\$$|$u$3D$_$@$\$\$3@ |$$\$ u" u t39@9=tB$\$~D$$\$hËD$ @%$\$3@|$|$|$|$<$|$ D$=D$z @z \$É\$ $D$D$=u$9D$D$z @z \$t$ D$ 뭿 @ D$=t]É\$\$D$D$=uD$&D$t$D$뿃|$|$|$|$ |$|$$=|$|$ |$\$D$$$<$$D$\$@=t$\$D$J\$D$@=t$\$ D$ $\$D$$$TD$D$D$ ËD$ËD$%|$|$|$|$ |$|$$=|$|$ |$\$D$$$<$$D$\$@=t$\$D$J\$D$@=t$\$ D$ $\$D$$$TD$D$D$ ËD$ËD$%|$$|$|$|$ |$|$=\$D$$$<$$D$\$I @I =t$\$D$J\$D$I @I =t$\$ D$ $\$D$$$TD$D$D$ ËD$ËD$|$ |$|$|$<$|$ D$9D$@ \$L$ $D$D$=u$9D$D$@ \$t$ D$ 뭿{ @{ D$=t]É\$\$D$D$=uD$&D$t$D$뿃|$|$ |$|$$=;D$\$D$ $$<$t]$\$ @ =t$\$D$$\$ D$ $$똋D$.D$D$ !ËD$%i F|$$=t@@D$$ËD$4@%4|$ |$<$|$|$1/ @/ D$ |$ D$ D$D$=1<$<$tK$@\$.@.=uD$o@oD$$$몋\$D$\$D$\$ \$ LD$|$<$|$ D$=D$q @q =uD$D$r @r É\$ $D$D$=u$KD$D$q @q =uD$D$r @r t$ D$ 뛃0|$,|$(|$$|$ |$|$|$|$|$ $\$D$ @ D$ |$|$<$|$u D$D$|$$|$(= |$ u D$ D$,ËD$ ËD$,D$(D$$D$ D$ =uD$$): ?or type in one of the following properties that is to be changed(e.g. vars ): : +Shall the flag for all equations be set (Y) +or be removed ? (N) =Shall the property list for all equations be set to nil (Y/N) current value for new value (e.g. (x y z) ENTER): The new value of Input not recognized.|$|$|$ |$$D$ @ $0 @0 $d     =|$$ @!@!D$=, ` !@!D$<$|$ |$ D$ |$yu\$ @ \$@D$ D$ 몋 D$=e !@!D$=yuE<$|$ |$ D$ \$x@xD$ D$ ‹<$|$ |$ D$ D$ D$$ \$D$  !@!D$\$D$x@x D$$ \$D$ D$ D$ &|$M @u 0 |$=0Please give the name of the file in double quotes without `;' : ls %w%w%w: $Indicate the file you want to load by"entering the corresponding number: This is not a number >0 and <= ! Try again: ,|$ D$\$|$$|$(|$ |$|$|$<$|$t=|$$ @  !@!D$( D$ ø*  =u%\$ ?  =_\$( p@ @ D$ D$ D$  $= u\$D$D$D$ "=t \$ pD$<$u D$ =u\$D$D$D$ }|$D$|$|$ |$ tJD$o@oD$ D$ D$ D$  H !@!D$ |$ ~u]D$ @=uGL$3҉ t3CӉËD$ @=t_p L$3҉ t3CӉ !@!D$ 3\$ D$D$("tL$(jk@kD$(=e |$ $D$( @ |$=e |$|$ |$ t%D$  D$ D$ ϋ|$|$ |$ tED$  u!@@x@xD$ D$ 믋|$$T= @ S @ S @ @ ;=`t@ @;=`t `}TU @U D$(Th @h D$(,%w%d%w.%d.%w%w.%d.%d.%w. |$|$$ D$|$D$D$<$tc~uB @=tL$$h p{L$$t pg$ pU~u9 @=t\$ p\$ p ø p$\$\$ $ Í0Please give the name of the file in double quotes without `;' : off echo$ backup_:='$end$ |$|$|$|$D$ \$|$<$tua=<$ @ F !@!D$<$="tL$jk@kD$r@D$sD$= |$|$= tD$g@h@h( 8 F @ D$\$D$ T @T ËD$ H P D$t|$= D$}g@@F <$jk@k$ @ $ $*Ë$$døcË$$F @ $eøiË$F @ Í;= ui=t~uOt~u4=uÁt-@%4@%4<$\$L$ |$|$T$D$|$D$ uc$<$tC$S FÉ\$t\$D$D$$$벋D$D$f @ \$D$\$ D$ @ D$\$ @ \$S S D$"No statistical history is recorded.&To record one enter: as {size_watch,t};size_hist%w.%w size_hist:='$end$|$<$;= u, 0 =|$\ @d jt p$v@v < o @o  $~@~|$=0Enter a list of equations, like e2,e5,e35; from: #To write all equations just enter ; These are no equations: Try again.0Shall the name of the equation be written? (y/n) 0Please give the name of the file in double quotes without `;' : load crack$ lisp(nfct_:=)$ lisp(nequ_:=off batch_mode$list_of_variables:=list_of_functions:=depend ,$:=list_of_equations:=list_of_inequalities:="solution_:=crack(list_of_equations,% list_of_inequalities," list_of_functions,# list_of_variables)$ := sub(second first solution_,)end$1These data were produced with the following input:lisp( old_history := 'L|$D|$@|$<|$8|$4|$0|$,|$(|$$|$ |$|$ $\$|$|$|$|$=|$H @ $B @B  @D$DD$ =u <$|$\$\$ 9t  uhD$@|$@t@D$@D$<$=t\$D$<D$D$@D$@봋$D$ @ D$ D$ |$ t& D$  |$ ( !@!D$D=yt=yt=nt=nu=yt=yuD$t`  !@!D$Dv@vg@h@h    |$t!    |$|$@|$@tCD$@ @ D$ =t\$4@4D$D$@D$@뱸 \$@-@-x @y @y |$|$@|$@tCD$@@@ D$ =t\$4@4D$D$@D$@뱸 \$@-@-x @y @y |$|$@|$@D$@D$<S F=t0 D$<S D$<FD$|$t3D$D$8< D$8 D$D$D D$@D$@=|$|$|$@|$@D$@D$< @y @y L -@-˿y @y C@D$< tø@-@- @y @y D$@D$@WT \$@-@-x @y @y T |$<|$|$|$@u D$@C@ tø@D$D$: Choose another name.  is already in use.Give a list of variables depends on, for example x,y,z; For constant input a `;' |$|$|$$D$ =|$ @  @ D$D$t    D$  !@!D$\$ =t o@o C$D$=t/, D$H D$|$!S D$FS S ` D$  D$ @D$  ug @;u D$  u\$S S $=tËD$S S $D$# @# $1D$# @# 1|$=\$$%#Which equation is to be simplified? #This is not the name of an equation! replaces  : |$ $\$L$|$|$=|$< @D !@!D$ $=up C@D$  D$$D$ @$3<$=t==l$T$ D$@D$D$ D$ $D$@$ËD$=t9D$ D$@D$b @b |$=\$$% Is there a further( new function in the changed/new PDE thatis to be calculated (y/n)? BIf you want to replace a pde then type its name, e.g. e_23 .5If you want to add a pde then type `new_pde' . Input of a value for the new pde..9You can use names of other pds, e.g. 3*e_12 - df(e_13,x); &Terminate the expression with ; or $ : Do you want the equation to be)- simplified (e.g. e**log(x) -> x) without+ dropping non-zero factors and denominators2 (e.g. to introduce integrating factors) (1)3- simplified completely (2)  is added replaces A pde ' does not exist! (Back to previous menu)0|$,|$(|$ $\$L$|$|$|$|$ |$$=|$h @p |$$t    !@!D$ =yt=yu>D$ @ D$\$;t\$ D$ D$$t|$ nt|$ n= ( !@!D$ =j @t$=d |$ j @u  D$    @a@aD$<$|$(|$(tbD$(D$,C@ tø@ËD$,ËD$' @' D$\$(\$( , \   !@!D$=u D$D$t|$ j @t$D$ @$;=8tv|$ |$(|$(tbD$(D$,ËD$q @q =t&8D$,13 @3 8D$(D$(3<$=|$===l$T$ D$@D$|$ j @u |$ tL$ ( @x@xD$|$ j @u   D$ $D$@$&$ D$ 0 |$=\$$0%Pick  from this list:Pick from this list;>a sublist and input it in the same form. Enter ; to choose all.Wrong number picked. is not allowed.|$ $\$=|$  @|$t&  D$F  4  $B @B L  F|$uT  @D$ |$tb3҉ t3CӉËD$F =u  D$ m|$ u<$|$ X$D$ @ =t7$D$ @   D$ |$=D$ )How many equations do you want to select? (number ) : Please select  equations from: . pde: "Error!!! Please select a pde from: |$|$|$ |$$\$=*D$=|$" @|$u1" " !@!D$" D$" 3CD$@=t # # $D$D$ t\$D$@=|$  D$o@oD$3CD$@=tD$# $# !@!D$ $D$ =ug0# $3CD$@=tD$# $# !@!D$ |$ $D$ $\$D$ D$|$=D$%Í |$$\$|$uZD$D$|$t0D$Ë$ @ D$D$ċD$D$듃 ÍP<$u@$@S FS F$S $뮃 |$|$$ؿ-@- @ D$|$u~D$-@- @ D$|$t;D$Ë$ @ -@-D$D$빋D$D$o  @% |$$\$= |$ =e |$ $ |$ @u @D @ tU=V95W5VމwU@wƉ5U$Xø @Z=VD$_95[b5VU|$=e  ;\ D$ D$]@]=tt %^%_ |$$\$tU=V95W5VމwU@wƉ5U\$$Xø1 @Z=VD$_95[G5VU\D$]@]=tD$ %^%_$|$ |$|$|$|$ $\$L$=O|$ Q@O=T|$T @@D$tU=V95W5VމwU@wƉ5U$@XD$D$@XD$ D$@X‹L$ \$ @Z=VD$ _95[o5VU\D$@|$=O|$=TD$ ]@]=t@ D$ $%^%_(|$$|$ |$|$|$$\$L$T$ =O|$ Q@O=T|$T @tU=V95W5VމwU@wƉ5U$XD$D$XD$ D$XD$$D$ XŋT$$L$ \$C @"Z=VD$_95[U5VU\|$=O|$=TD$]@]=t D$(%^%_$|$|$|$|$ |$|$$D$ =O|$ Q@O=T|$T @tU=V95WJ5VމwU@wƉ5U|$|$<$|$u D$C@ tø@D$D$D$D$=uD$=D$C@ tø@t$D$먉ø@D$1@D$|$|$ =4 |$u wD$tø@D$ D$D$D$=uD$2D$tø@t$ D$ 볉ø@\$D$Xø @Z=VD$ _95[U5VU\|$=O|$=TD$ ]@]=t D$ $%^%_|$ |$<$L$ىø @$tU=V95W5VމwU@wƉ5U$@D$D$@ËD$Xø @Z=VD$ _95[A5VU\D$ ]@]=t D$ %^%_(SPLIT_SIMPLIFY DOES NOT HAVE 3 ARGUMENTS.|$$|$|$|$ |$3҉ t3CӉ=t0 $@D$$-@-D$$-@-\$tU=V95W5VމwU@wƉ5UD$XD$\$D$X\$' @Z=VD$ _95[Q5VU\D$ ]@]=t\$ t@%^%_|$|$|$$\$=O|$  Q@O=T|$T @tU=V95W5VމwU@wƉ5U$@D$D$@ËD$Xø @Z=VD$_95[`5VU\|$ =O|$=TD$]@]=tD$@%^%_|$ |$$=O|$ Q@O=T|$ T @tU=V95W5VމwU@wƉ5U$Xø@Z=VD$_95[U5VU\|$=O|$ =TD$]@]=t D$%^%_Stop of a subroutine.Less than 100000 free cells.%Number of garbage collections exceeds . Heidadeife 4Memory seems to run out. Less than 100000 free cells! Qo@oQO@=u(;T @=t~;=0ta5  @=t 5 !5 T5 5 & @%& ;0t, @=t6  |$|$|$ |$$|$|$=O|$ Q@O=R|$ =S|$=T|$T @;=m u)n @@-@-D$t;=h @$.@.=uw; t @$.@.=uU @h@hH@@-@- @@D$ttU=V95W5VމwU@wƉ5U$Xø @ø@Z=VD$_95[@5VU|$tM @h@hH@h@h-@- @@|$t!n @h@h-@-|$ =R|$=S\|$=O|$=TD$]@]=uH|$ uL|$ u:t$;t(t$;u$@ D$ %^%_ |$|$|$ |$$|$|$=O|$ Q@O=R|$ =S|$=T|$T @;=m t)n @h@h-@-D$t;=h @$.@.=uw; t @$.@.=uU @h@hH@@-@- @@D$ttU=V95W5VމwU@wƉ5U$Xø @Z=VD$_95[P5VU|$tM @h@hH@h@h-@- @@|$t!n @@-@-|$ =R|$=S\|$=O|$=TD$]@]=uH|$ u\|$ uJt$;t8t$;u 3A$@ø@ D$ %^%_ @=H@ |$|$|$|$|$$\$=O|$  Q@O=R|$=S|$=T|$T @tU=V95W5VމwU@wƉ5U$XD$D$X‹L$=H@ø-@Z=VD$_95[5VU|$=R|$=S\|$ =O|$=TD$]@]=t 3C3@tø@ D$ %^%_Equation factorized. It is a consequence of .8|$$|$ $|$(|$|$|$|$|$4|$|$|$ |$,\$0;=%t3CD$0@D$( JD$(|$(M|$4=|$,-D$(@ D$=t.~ j@=OD$(C@ 3Ctø@m @m D$;u3C@=|$|$,D$a@aD$ u t\$\$뉋1@D$D$3C@=u ;D$D$t|$uD$(C@ D$D$D$|$u@D$D$$\$$D$@=u,\$D$c@cD$D$$o@oD$$뺋|$tD$D$D$D$@ @@ D$|$0|$ |$ |$D$ @ \$ =\$(D$  =uD$ C@ D$ =u7D$ C@ @D$ =tD$ D$,\$D$ D$ |$83CD$\$4D$4\$\$ |$,D$(C@ @D$$C@D$, @\$$6@\$, @D$(\$0@D$0ËD$( @ 7|$uJ|$4tt$4;u-D$4D$(@x@x|$u#L$4D$(@x@x=|$$|$$t1D$$ËD$( @ D$$D$$ÿ;=BtD$(C@ D$D$(@@ D$$D$( @ D$ 3|$0=t|$ =l$$L$4D$(\$ @ ;=Bt{D$( @ D$$D$(C@ @D$ D$@\$ 6@\$$ @@D$(\$0 @ D$(\$0 @ |$4u |$,te;=0tX`? D$(p? |$,t ? D$,? D$(D$(|$,t$D$0|$4t$8|$$|$|$|$ <$ 4$;<$ uw<$ u`$$b@b\$D$$$U<$ u<$ t\$$D$<$|$|$ u@|$ u.|$ uD$D$봃|$u\$$D$r3C$D$3CD$ËD$c@cD$ u&\$D$ËD$ D$\$$D$D$ |$|$|$|$ <$|$|$|$ @ $D$  L$\$|$|$ u9|$ u'\$D$ =u\$D$D$\$뎋D$\$D$\$\$]D$|$ u|$ t&|$\$$D$D$D$=uD$D$D$|$D$D$ uh uZ3C$D$ 3CD$ËD$ c@cD$u\$D$D$\$$D$D$]\$$D$D$ @ =u4 @ @ @ @ @ @F ø4|$(|$ |$|$|$|$\$ L$$|$|$,|$0|$$t3@ø@ $ø5@@D$\$ \$ |$|$(D$=D$ u @;t D$ D$L$\$ D$" @" D$D$o@oD$\$,D$D$,D$D$$|$|$ |$ t7D$ ~t\$$D$$D$ D$ 뽹L$$3҉ t3CӉ3C@=t\$$ @D$$\$$\$$\$0D$$D$0L$(L$D$D$=bD$ u @;t D$ D$L$\$ D$" @" D$D$o@oD$\$,D$D$,D$D$$|$|$ |$ t7D$ ~t\$$D$$D$ D$ 뽹L$$3҉ t3CӉ3C@=t\$$ @D$$\$$\$$\$0D$$D$0t$(D$(\$0@@D$0\$,@\$0@4%0|$,|$$$\$L$T$ l$=|$|$|$|$ |$(@D$$3ۋD$$@=t1\$$D$\$D$D$$@D$$뷋\$D$ \$D$\$D$$\$D$$@=L$$\$D$" @" D$(\$D$$D$L$$ø5@\$( @L$ ø@D$ D$$o@oD$$]D$ @D$,\$@\$,@0%4|$0$\$L$T$ |$|$|$|$|$ |$$|$(|$,;ul$l$DD$D$\$D$D$<$u[\$D$D$_ËD$ " @" D$ø@\$@<$u=3A\$D$ " @" D$ø@\$@f<$u=3A\$D$ " @" D$ø@\$@#<$u3A\$D$ " @" D$\$D$ " @" D$ \$@@D$0L$ \$@\$0@<$3A\$D$ " @" D$\$D$ " @" D$ \$ @@D$0L$ \$@\$0@<$3A\$D$ " @" D$\$D$ " @" D$ \$D$ " @" D$$L$ \$ @L$$ø@@D$0T$$L$ \$@\$0@Y<$3A\$D$ " @" D$\$D$ " @" D$ \$D$ " @" D$$L$ \$ @L$$ø@@D$0T$$L$ \$@\$0@<$3A\$D$ " @" D$\$D$ " @" D$ \$D$ " @" D$$\$D$ " @" D$(L$ \$ @T$(L$$ø@@D$0l$(T$$L$ \$@"\$0@<$D$D$D$\$\$L$,|$t^D$\$L$ËD$ " @" D$\$\$\$,D$,D$o@oD$떋\$,@D$0\$,@\$0@3A\$D$ " @" D$\$D$ " @" D$ \$D$ " @" D$$\$D$ " @" D$(L$ \$ @D$0L$(\$$ @\$0@@D$0l$(T$$L$ \$@"\$0@4Old: New: |$$\$L$D$ D$|$D$D$;= t&X  @D$  @D$ L$\$@D$ ;= tX D$ L$ @D$x@xE@D$x@xG@D$x@x@D$ @ D$D$$|$$D$ \$|$tED$D$$=t\$ D$D$ D$D$믋D$ % $\$L$ tRD$ =t4\$$ =ttL$\$$뒃 New functions list: |$<$\$ L$1á83 @3 8|$ |$|$D$$@@ 13 @3 @@$x@x)@$ 13 @3 )@$x@xD$D$kL$1D$ @ ;= tZ 1|$ |$$\$  =t$%|$|$ <$ u>$\$D$=t\$ 4@4D$ $$뷋D$ |$ |$$\$ { =t $%|$|$ <$ u>$\$D$=t\$ 4@4D$ $$뷋D$  @ ø@%F         F  V   F R R ;= u( @@-@-;=u!L @@-@-;=u!@@-@-;= u' @@F-@-;=t!@h@h-@-;=V u!W @@-@-;= u" @@F-@- m F   ;=m u(n @@-@-;= u! @@-@-Í  =uP; t' @h@Fh-@-! @@-@-   =uU;t,L @Fh@h-@-)L @@-@-   =uO;t&@h@h-@-!@@-@-   =uU; t, @Fh@h-@-) @@-@-   =uO;t&@h@h-@-!@@-@-  V  =uU;V t,W @Fh@h-@-)W @@-@-   =uO; t& @h@h-@-! @@-@- R  =uU;t,H@Fh@h-@-)H@@-@-R  R m  =uO;m t&n @h@h-@-!n @@-@-   =uU; t, @Fh@h-@-) @@-@-  FP @ = tu  @ 7 @7 =t#$@-@%-F$ @ @tu   @ 7 @7 =t#$@-@%-F$-@%-|$<$D$$2D$@=t54$;t%D$o@oD$$$봍F4$;t4$džSP =tz\$$ =uI4$m @;t>4$t24$ t&4$;t4$ @;t t2A partial substitution has been repeated too often.It will now be made rigorously.$A step to find overdet. sub-systems (A derivative replacement (An algebraic length reduction (!A length reducing simplification (A Groebner Basis computation (A step () was preventedto avoid a cycle.|$$|$|$ |$=|$m @;D$ |$tB\$$ @ =tD$ o@oD$ D$D$벻D$ @=t'g g tD$ D$ |$\$$ =tw$\$ @\$@@=t&D$ o@oD$ \$\$S|$tB\$$ @ =u#D$ o@oD$ \$\$벋|$tiL$ ˸@@D$ 3҉ t3CӉËD$@=tD$j\$\$=|$3ۋD$ @=tW\$D$ @ =t1D$D$\$\$D$ @D$ 둃|$ ;=04$ @;uh Q4$ uDh >4$ udh +4$uh 4$;uh h $h h 3ۋD$ @=ttD$|$u$9 @9 D$ off echo$ backup_:='(-% A list of dependencies, like ((f x y) (g x)) % A list of unsolved equations% A list of assignments'% A list of free or unresolved functions%% A list of non-vanishing expressions.4% A list of or-lists. Each or-list has elements that 5% are factor-list, such that for each or-list at least5% from one factor-list all elements must be non-zero. )$end$$|$D$ \$L$T$l$|$ <$|$=t=|$ fjk@k D$ oøsËD$  D$ D$ 4@4D$ v@v m m |$|$|$tDD$S FÉ$t\$$D$D$D$밸,m FD$ `m hm D$  `m m D$ `m m D$ `m Fm D$ `m Fn @n |n D$ `m Fn n D$ ~@~D$ $[]|$ |$|$$D$<$|$ |$ tjD$ D$D$o@oD$q D$q FD$@D$ D$ 늃Sub: in: gives a singular result. Substitutions give singularities.$\$L$|$ |$<$|$$D$ $$|$t|$|$L$ @ D$|$s=hpr D$ @|r D$@r |$ur D$ *** sol1 ***: Remaining equations: *** sol2 ***: free param in sol1: free param in sol2: *Initial preparation of unsolved eqn in sol1*Initial preparation of unsolved eqn in sol2sol1 substitutions in sol2gb=@2The remaining equations of solution sol2 are in the1ideal of the remaining equations of solution sol1.Equation &of solution sol2 is not in the ideal of(the remaining equations of solution sol1.&--> sol1 is not a special case of sol2.[]Sub: new value=singular result1which vanishes modulo the remaining eqn.s of sol1.8which does not vanish modulo the remaining eqn.s of sol1.-Even substitutions in the numerator is giving singularities like log(0).(Substitutions in numerators give all zero regular_sb: singular_sb: #Substitutions lead to singularities.Solution  has to be transformed.9The following are expressions which vanish due to sol1 and>which lead to singularities when used for substitutions in sol2@The following are all free parameters in sol2 for which there aresubstitutions in sol1 singular_eli: regular_eli: Param in sol1 and not in sol2: /ERROR, there must be an assignment of a in sol2! Assignment in * of a variable that is a free parameter in  : gauge_of_s2=Because / is either in flin_ or appears linearly in sol2,it gets a higher priority.next relation to be used: 0= try_to_sub=&After dropping single variable factors  factor(s) remain&New relation used for substitution: sb=try to sub next: num_sb=Trafo was singular.7In order to avoid a singularity when doing substitutions>the supposed to be more general solution was transformed using:The new gauge_of_s2: remain_c2=7After completing the trafo the new list of parameters ofsol2 is: sol1 has free parameters: @Something seems wrong in merge_sol(): after the transformation ofCsol2, all free parameters of sol1 should be free parameters of sol2.6All free parameters of sol1 are free parameters of sol2"The following subst. was singular: Remaining substitution: Substitution of  by: In the following S1 stands for and S2 stands for  . ASolution S1 fulfills all conditions of solution S2 when conditionsBare made denominator free. But, after rewriting solution S2 so that@all free parameters of solution S1 are also free parameters of S2Cthen the new solution S2 now requires the vanishing of an expression@in these free parameters which is not allowed by S1. Therefore S1is not a special case of S2.remain_c2_cp after subst = % is not less restrictive than solution( and fulfills all conditions of solution  .@But it was not possible for the program to re-formulate solution , to include both solutions in a single set of/assignments without vanishing denominators. :-( h0=dropped_assign_in_s2=new_assign_in_s2=h1=a=h2=(A seemingly successful transformation of 0went singular when performing the transformation finally on the whole solution.Strange: merging  and  without dropping inequalities!Probably  had already been merged with  or similar before. includes by dropping inequality inequalities$$$$$|$x$\$L$T$ l$|$\|$4|$0|$l$|$8|$h|$$|$`|$d|$|$t|$ $|$L|$P|$<|$($|$H$|$|$$$|$,|$D|$@|$X|$T|$p$$$|$|$$$3s $t$;t& t D$P @P D$ @ (t D$t$ ;t& t D$ P @P D$  @ $$$$ @ |$t$u s$uy, $D D$P t D$ $ ;=0w D$ $Ё $3҉ t3CӉ=u    $$$t.$@$$ËD$sol_list%w%w+Do you want to see the list of names of the single solution?  solutions? (y/n) :Is this the list to work on (Y) 2or shall all solution files of this session in the :current directory be collected and used? (N): The following list is used: |$D$=<$ @$ j4 pD$ =u @ D$ @ ;=tC 3҉ t3CӉ@=t @  3҉ t3CӉD$@ |$u t D$ !@!=yu   !@!=nu9 @ \ <$= 7Do you want to merge solutions computed in this session,'i.e. since loading CRACK the last time? Comparing  with: Strange:  is contained in *but both have same number of free unknowns!1One of them has probably undergone earlier merging0|$,|$(|$$|$<$|$|$ |$|$|$|$|$ ;=ju @ .   =u @ @ =<$|$|$qt$;\D$D$ \$\$=e |$  $ @ |$ =e =T|$;=0t& D$  |$|$|$;D$D$\$\$=e |$  $ @ |$ =e =T|$;=0t  D$t$;t$;|$$|$(D$JD$$D$(|$$ tD$(D$$D$$3C\$(\$(ӉD$$\$3҉ t3CӉËD$$@D$$|$(|$,D$JD$(D$,|$( tD$,D$(D$(3C\$,\$,ӉD$(\$3҉ t3CӉËD$(@ËD$$@=t)tT$L$\$D$ @ D$9|$$|$(D$JD$$D$(|$$ tD$(D$$D$$3C\$(\$(ӉD$$\$3҉ t3CӉËD$$@D$$|$(|$,D$JD$(D$,|$( tD$,D$(D$(3C\$,\$,ӉD$(\$3҉ t3CӉËD$(@ËD$$@=t)tT$L$ \$D$ @ D$tT$L$\$D$ @ É\$u$tT$L$ \$D$ @ D$|$tR D$ $ D$< l \$ D$ =t\$D$D$[\$D$ =B\$4@0ÍFsol_list%w%w off echo$ sol_list:=nil sol_list:='$end$<$ j p$v@v ;=u  (  8 @ $~@~sol_list%w%w%s.%wmv %s %sls %s* off echo$ sol_list:='$end$|$|$|$ |$|$<$;=t|$ |$ j pD$ \$$ pD$D$L$\$0 p$=\$@ p@ @ D$  D$D$ }L D$ =u3@ D$@=t!D$o@oD$3@ zD$v@vT d  t | D$~@~$<$==TD$ @ T4@4D$v@vT d  t | D$~@~L$\$0 p$=t 3@ <$uθ8Name of the session in double quotes (e.g. "bu263393-"): 5< @!@F!j<$=.html-s%w%w%d%w2Solution ���� ��� to problem �������&

Solution 


&Remaining equations | Expressions | Parameters | Inequalities | Relevance | Back to problem  
 %

Equations

'The following unsolved equations remain:
	Equations:
'

Expressions

7The solution is given through the following expressions:&

Parameters

9Apart from the condition that they must not vanish to give6a non-trivial solution and a non-singular solution with=non-vanishing denominators, the following parameters are free:
  free unknowns: (
Inequalities
@In the following not identically vanishing expressions are shown.CNext come so-called OR-lists of FACTOR-lists in the following sense.KEach FACTOR-list represents the factors of an expression and at least one ofIthese expressions must not vanish in each OR-list. In other words, in eachMOR-list at least one FACTOR-list must not vanish, i.e. none of the expressions!in the FACTOR-list may vanish.
OR-list::
Relevance for the application:
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