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 Topics FrontPage FriCAS SandBoxFriCAS <-- You are here. SandBox SandBoxFriCAS <-- You are here. This page is set to execute \begin{axiom}... \end{axiom} commands using FriCAS. See also FriCASIntegration. fricas)version Value = "FriCAS 1.3.5 compiled at Thu Feb 21 18:22:14 UTC 2019" Any comments added here use this version of FriCAS. ... --meliusja, Tue, 08 Apr 2008 10:38:33 -0700 replyfricassolve(s=v*t+a*t^2/2,t) \begin{equation*} \label{eq1}\left[{{{2 \ t \ v}+{a \ {{t}^{2}}}-{2 \ s}}= 0}\right]?\end{equation*} Type: List(Equation(Fraction(Polynomial(Integer)))) ... --meliusja, Tue, 08 Apr 2008 10:39:28 -0700 replyfricasradicalSolve(s=v*t+a*t^2/2,t) \begin{equation*} \label{eq2}\left[{t ={{-{\sqrt{{4 \ {{v}^{2}}}+{8 \ a \ s}}}-{2 \ v}}\over{2 \ a}}}, \:{t ={{{\sqrt{{4 \ {{v}^{2}}}+{8 \ a \ s}}}-{2 \ v}}\over{2 \ a}}}\right]?\end{equation*} Type: List(Equation(Expression(Integer))) ... --meliusja, Tue, 08 Apr 2008 10:40:28 -0700 replyfricassolve(v*t+a*t^2/2-s=0,t) \begin{equation*} \label{eq3}\left[{{{2 \ t \ v}+{a \ {{t}^{2}}}-{2 \ s}}= 0}\right]?\end{equation*} Type: List(Equation(Fraction(Polynomial(Integer)))) ... --meliusja, Tue, 08 Apr 2008 10:41:13 -0700 replyfricassolve(v*t+a*t/2-s=0,t) \begin{equation*} \label{eq4}\left[{t ={{2 \ s}\over{{2 \ v}+ a}}}\right]?\end{equation*} Type: List(Equation(Fraction(Polynomial(Integer)))) ... --meliusja, Tue, 08 Apr 2008 10:42:37 -0700 replyfricassolve(v*t+a*t^2/2-s=0,t) \begin{equation*} \label{eq5}\left[{{{2 \ t \ v}+{a \ {{t}^{2}}}-{2 \ s}}= 0}\right]?\end{equation*} Type: List(Equation(Fraction(Polynomial(Integer)))) fricasradicalSolve(p^3 - p + 1/10=0,p) \begin{equation*} \label{eq6}\begin{array}{@{}l} \displaystyle \left[{p ={{{{\left(-{3 \ {\sqrt{- 3}}}+ 3 \right)}\ {{\root{3}\of{{-{3 \ {\sqrt{3}}}+{\sqrt{-{373}}}}\over{{60}\ {\sqrt{3}}}}}^{2}}}- 2}\over{{\left({3 \ {\sqrt{- 3}}}+ 3 \right)}\ {\root{3}\of{{-{3 \ {\sqrt{3}}}+{\sqrt{-{373}}}}\over{{60}\ {\sqrt{3}}}}}}}}, \: \right. \ \ \displaystyle \left.{p ={{{{\left(-{3 \ {\sqrt{- 3}}}- 3 \right)}\ {{\root{3}\of{{-{3 \ {\sqrt{3}}}+{\sqrt{-{373}}}}\over{{60}\ {\sqrt{3}}}}}^{2}}}+ 2}\over{{\left({3 \ {\sqrt{- 3}}}- 3 \right)}\ {\root{3}\of{{-{3 \ {\sqrt{3}}}+{\sqrt{-{373}}}}\over{{60}\ {\sqrt{3}}}}}}}}, \: \right. \ \ \displaystyle \left.{p ={{{3 \ {{\root{3}\of{{-{3 \ {\sqrt{3}}}+{\sqrt{-{37 3}}}}\over{{60}\ {\sqrt{3}}}}}^{2}}}+ 1}\over{3 \ {\root{3}\of{{-{3 \ {\sqrt{3}}}+{\sqrt{-{373}}}}\over{{60}\ {\sqrt{3}}}}}}}}\right] \end{array} \end{equation*} Type: List(Equation(Expression(Integer))) fricasR1 ==> Record(foo1: String, remLexs: List Integer) Type: Void fricasR2 ==> Record(foo2: String, remLexs: List Integer) Type: Void fricasr1: R1 := ["a", [1,2,3]] \begin{equation*} \label{eq7}\left[{foo 1 = \verb#"a"#}, \:{remLexs ={\left[ 1, \: 2, \: 3 \right]?}}\right]\end{equation*} Type: Record(foo1: String,remLexs: List(Integer)) fricasr2: R2 := ["b", [5]] \begin{equation*} \label{eq8}\left[{foo 2 = \verb#"b"#}, \:{remLexs ={\left[ 5 \right]?}}\right]\end{equation*} Type: Record(foo2: String,remLexs: List(Integer)) fricasr1.remLexs \begin{equation*} \label{eq9}\left[ 1, \: 2, \: 3 \right]?\end{equation*} Type: List(Integer) fricasr2.remLexs \begin{equation*} \label{eq10}\left[ 5 \right]?\end{equation*} Type: List(Integer) ... --jib jabber, Fri, 07 Mar 2014 07:04:39 +0000 replyfricas)set output tex off fricas)set output algebra on guessPRec [1, 0, 1, 1, 4, 10, 35, 120, 455, 1792, 7413, 31780, 140833, 641928, 3000361, 14338702, 69902535, 346939792, 1750071307, 8958993507, 46484716684, 244187539270, 1297395375129, 6965930587924] (13) [ [ f(n): 2 2 (- n - 17 n - 72)f(n + 3) + (4 n + 30 n + 44)f(n + 2) + 2 2 (19 n + 113 n + 150)f(n + 1) + (14 n + 42 n + 28)f(n) = 0 , f(0) = 1, f(1) = 0, f(2) = 1] ] Type: List(Expression(Integer)) ... --panacea, Fri, 21 Nov 2014 18:25:05 +0000 replyfricasguessRec([1,1,0,1,- 1,2,- 1,5,- 4,29,- 13,854,- 685]) 2 (14) [[f(n): f(n + 2) + f(n + 1) - f(n) = 0, f(0) = 1, f(1) = 1]] Type: List(Expression(Integer)) integrate(exp(x-x^2), x) --mariuszi28cg, Mon, 25 Sep 2017 20:14:29 +0000 replyfricasintegrate(exp(x-x^2), x) 1 - 4 2 x - 1 +---+ %e erf(-------)\|%pi 2 (15) --------------------- 2 Type: Union(Expression(Integer),...) test 1+1 --aUser, Thu, 01 Feb 2018 14:32:47 +0000 replyfricas1+1 (16) 2 Type: PositiveInteger? ^ or **? --aUser, Thu, 01 Feb 2018 14:34:22 +0000 reply2^3-2**3 2^3 --aUser, Thu, 01 Feb 2018 14:34:46 +0000 reply2^3 ^ or **? --aUser, Thu, 01 Feb 2018 14:35:09 +0000 replyfricas2^3 (17) 8 Type: PositiveInteger? ^ or **? --aUser, Thu, 01 Feb 2018 14:35:49 +0000 replyfricas2**3 There are no library operations named ** Use HyperDoc Browse or issue )what op ** to learn if there is any operation containing " ** " in its name. Cannot find a definition or applicable library operation named ** with argument type(s) PositiveInteger PositiveInteger Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need. bug? --aUser, Thu, 01 Feb 2018 14:37:03 +0000 replyfricassqrt(2)^2-2 (18) 0 Type: AlgebraicNumber? bug? --aUser, Thu, 01 Feb 2018 14:38:36 +0000 replyfricasa:=sqrt(2) +-+ (19) \|2 Type: AlgebraicNumber? fricasb:=sqrt(a) +----+ | +-+ (20) \|\|2 Type: AlgebraicNumber? fricasb^2-a (21) 0 Type: AlgebraicNumber? bug? --aUser, Thu, 01 Feb 2018 14:41:20 +0000 replyfricasf(a) == { local b=sqrt(a); b^2-a} Line 1: f(a) == { local b=sqrt(a); b^2-a} ........A........B Error A: Missing mate. Error B: syntax error at top level Error B: Possibly missing a } 3 error(s) parsing ... --aUser, Thu, 01 Feb 2018 14:41:44 +0000 replyfricasf(a) == { local b:=sqrt(a); b^2-a} Line 1: f(a) == { local b:=sqrt(a); b^2-a} ..........A Error A: syntax error at top level Error A: Improper syntax. 2 error(s) parsing [q,r]? type --aUser, Thu, 01 Feb 2018 15:00:38 +0000 replyfricas[q,r] (22) [q, r] Type: List(OrderedVariableList([q,r])) ... --aUser, Thu, 01 Feb 2018 15:03:11 +0000 replyfricasf(a) == ( local b; b:=sqrt(a); b^2-a) Type: Void fricasf(2) fricasCompiling function f with type PositiveInteger -> AlgebraicNumber (24) 0 Type: AlgebraicNumber? fricasf(sqrt(2)) fricasCompiling function f with type AlgebraicNumber -> AlgebraicNumber (25) 0 Type: AlgebraicNumber? bug --aUser, Thu, 01 Feb 2018 15:03:46 +0000 replyfricasf(a) == ( local b; b:=sqrt(a); b^2-a) Compiled code for f has been cleared. 1 old definition(s) deleted for function or rule f Type: Void fricasf(2) fricasCompiling function f with type PositiveInteger -> AlgebraicNumber (27) 0 Type: AlgebraicNumber? fricasf(sqrt(2)) fricasCompiling function f with type AlgebraicNumber -> AlgebraicNumber (28) 0 Type: AlgebraicNumber? fricasf(z) fricasCompiling function f with type Variable(z) -> Expression(Integer) (29) 0 Type: Expression(Integer) fricasf(%i) fricasCompiling function f with type Complex(Integer) -> Expression( Complex(Integer)) (30) 0 Type: Expression(Complex(Integer)) )display properties --aUser, Thu, 01 Feb 2018 15:08:53 +0000 replyfricasf(a) == ( local b; b:=sqrt(a); b^2-a) Compiled code for f has been cleared. 1 old definition(s) deleted for function or rule f Type: Void fricasf(2) fricasCompiling function f with type PositiveInteger -> AlgebraicNumber (32) 0 Type: AlgebraicNumber? fricasc:=f(sqrt(2)) fricasCompiling function f with type AlgebraicNumber -> AlgebraicNumber (33) 0 Type: AlgebraicNumber? fricas)display properties c Properties of c : Value (has type AlgebraicNumber): 0 ... --aUser, Thu, 01 Feb 2018 15:21:08 +0000 replyfricasf(a) == ( local b; b:=sqrt(a); b^2 = a) Compiled code for f has been cleared. 1 old definition(s) deleted for function or rule f Type: Void ... --aUser, Thu, 01 Feb 2018 15:21:23 +0000 reply \begin{AxiomCode?} f(a) == ( local b; b:=sqrt(a); b^2-a) f(2) \end{AxiomCode?} ... --aUser, Thu, 01 Feb 2018 15:22:39 +0000 replyfricasf(a) == ( local b; b:=sqrt(a); b^2 = a) 1 old definition(s) deleted for function or rule f Type: Void fricasf(2) fricasCompiling function f with type PositiveInteger -> Equation( AlgebraicNumber) (36) 2 = 2 Type: Equation(AlgebraicNumber?) fricasf(%i) fricasCompiling function f with type Complex(Integer) -> Equation( Expression(Complex(Integer))) (37) %i = %i Type: Equation(Expression(Complex(Integer))) fricasf(sqrt(2)) fricasCompiling function f with type AlgebraicNumber -> Equation( AlgebraicNumber) +-+ +-+ (38) \|2 = \|2 Type: Equation(AlgebraicNumber?) fricasf(2^(1/2)) +-+ +-+ (39) \|2 = \|2 Type: Equation(AlgebraicNumber?) fricasf(2^(1/3)) 3+-+ 3+-+ (40) \|2 = \|2 Type: Equation(AlgebraicNumber?) )set output --aUser, Thu, 01 Feb 2018 15:29:41 +0000 replyfricasf(a) == ( local b; b:=sqrt(a); b^2-a) Compiled code for f has been cleared. 1 old definition(s) deleted for function or rule f Type: Void fricasf(sqrt(2)) fricasCompiling function f with type AlgebraicNumber -> AlgebraicNumber (42) 0 Type: AlgebraicNumber? fricas)set output fortran --------------------------- The fortran Option ---------------------------- Description: create output in FORTRAN format )set output fortran is used to tell FriCAS to turn FORTRAN-style output printing on and off, and where to place the output. By default, the destination for the output is the screen but printing is turned off. Also See: )set fortran Syntax: )set output fortran where arg can be one of on turn FORTRAN printing on off turn FORTRAN printing off (default state) console send FORTRAN output to screen (default state) fp<.fe> send FORTRAN output to file with file prefix fp and file extension .fe. If not given, .fe defaults to .sfort. If you wish to send the output to a file, you must issue this command twice: once with on and once with the file name. For example, to send FORTRAN output to the file polymer.sfort, issue the two commands )set output fortran on )set output fortran polymer The output is placed in the directory from which you invoked FriCAS or the one you set with the )cd system command. The current setting is: Off:CONSOLE f(sqrt(2)) (43) 0 Type: AlgebraicNumber? fricas)set output tex ----------------------------- The tex Option ------------------------------ Description: create output in TeX style )set output tex is used to tell FriCAS to turn TeX-style output printing on and off, and where to place the output. By default, the destination for the output is the screen but printing is turned off. Syntax: )set output tex where arg can be one of on turn TeX printing on off turn TeX printing off (default state) console send TeX output to screen (default state) fp<.fe> send TeX output to file with file prefix fp and file extension .fe. If not given, .fe defaults to .stex. If you wish to send the output to a file, you must issue this command twice: once with on and once with the file name. For example, to send TeX output to the file polymer.stex, issue the two commands )set output tex on )set output tex polymer The output is placed in the directory from which you invoked FriCAS or the one you set with the )cd system command. The current setting is: Off:CONSOLE f(sqrt(2)) (44) 0 Type: AlgebraicNumber? fricas)set output script ---------------------------- The script Option ---------------------------- Description: display output in SCRIPT formula format )set output script is used to tell FriCAS to turn IBM Script formula-style output printing on and off, and where to place the output. By default, the destination for the output is the screen but printing is turned off. Syntax: )set output script where arg can be one of on turn IBM Script formula printing on off turn IBM Script formula printing off (default state) console send IBM Script formula output to screen (default state) fp<.fe> send IBM Script formula output to file with file prefix fp and file extension .fe. If not given, .fe defaults to .sform. If you wish to send the output to a file, you must issue this command twice: once with on and once with the file name. For example, to send IBM Script formula output to the file polymer.sform, issue the two commands )set output script on )set output script polymer The output is placed in the directory from which you invoked FriCAS or the one you set with the )cd system command. The current setting is: Off:CONSOLE f(sqrt(2)) (45) 0 Type: AlgebraicNumber? ... --aUser, Thu, 01 Feb 2018 15:30:46 +0000 replyfricasf(a) == ( local b; b:=sqrt(a); b^2-a) Compiled code for f has been cleared. 1 old definition(s) deleted for function or rule f Type: Void fricasf(sqrt(2)) fricasCompiling function f with type AlgebraicNumber -> AlgebraicNumber (47) 0 Type: AlgebraicNumber? fricas)set output fortran on f(sqrt(2)) R48=0 (48) 0 Type: AlgebraicNumber? fricas)set output tex on f(sqrt(2)) R49=0 (49) 0 \begin{equation} \label{eq11}0\end{equation} Type: AlgebraicNumber? fricas)set output script on f(sqrt(2)) .eq set blank @ :df. 0 :edf. R50=0 (50) 0 \begin{equation} \label{eq12}0\end{equation} Type: AlgebraicNumber? ... --aUser, Thu, 01 Feb 2018 15:31:59 +0000 replyfricasf(a) == ( local b; b:=sqrt(a); b^2-a) Compiled code for f has been cleared. 1 old definition(s) deleted for function or rule f Type: Void fricasf(sqrt(2)) fricasCompiling function f with type AlgebraicNumber -> AlgebraicNumber .eq set blank @ :df. 0 :edf. R52=0 (52) 0 \begin{equation} \label{eq13}0\end{equation} Type: AlgebraicNumber? fricas)set output fortran console f(sqrt(2)) .eq set blank @ :df. 0 :edf. R53=0 (53) 0 \begin{equation} \label{eq14}0\end{equation} Type: AlgebraicNumber? fricas)set output tex console f(sqrt(2)) .eq set blank @ :df. 0 :edf. R54=0 (54) 0 \begin{equation} \label{eq15}0\end{equation} Type: AlgebraicNumber? fricas)set output script console f(sqrt(2)) .eq set blank @ :df. 0 :edf. R55=0 (55) 0 \begin{equation} \label{eq16}0\end{equation} Type: AlgebraicNumber? ... --aUser, Thu, 01 Feb 2018 15:33:05 +0000 replyfricasf(a) == ( local b; b:=sqrt(a); b^2-a) Compiled code for f has been cleared. 1 old definition(s) deleted for function or rule f Type: Void fricasf(sqrt(2)) fricasCompiling function f with type AlgebraicNumber -> AlgebraicNumber .eq set blank @ :df. 0 :edf. R57=0 (57) 0 \begin{equation} \label{eq17}0\end{equation} Type: AlgebraicNumber? fricas)set output fortran on fricas)set output fortran console f(sqrt(2)) .eq set blank @ :df. 0 :edf. R58=0 (58) 0 \begin{equation} \label{eq18}0\end{equation} Type: AlgebraicNumber? fricas)set output tex on fricas)set output tex console f(sqrt(2)) .eq set blank @ :df. 0 :edf. R59=0 (59) 0 \begin{equation} \label{eq19}0\end{equation} Type: AlgebraicNumber? fricas)set output script on fricas)set output script console f(sqrt(2)) .eq set blank @ :df. 0 :edf. R60=0 (60) 0 \begin{equation} \label{eq20}0\end{equation} Type: AlgebraicNumber? ... --aUser, Thu, 01 Feb 2018 15:35:28 +0000 replyfricasf(a) == ( local b; b:=sqrt(a); b^2-a) Compiled code for f has been cleared. 1 old definition(s) deleted for function or rule f Type: Void fricasf(sqrt(2)) fricasCompiling function f with type AlgebraicNumber -> AlgebraicNumber .eq set blank @ :df. 0 :edf. R62=0 (62) 0 \begin{equation} \label{eq21}0\end{equation} Type: AlgebraicNumber? fricas)set output fortran on f(sqrt(2)) .eq set blank @ :df. 0 :edf. R63=0 (63) 0 \begin{equation} \label{eq22}0\end{equation} Type: AlgebraicNumber? fricas)set output fortran off fricas)set output tex on f(sqrt(2)) .eq set blank @ :df. 0 :edf. (64) 0 \begin{equation} \label{eq23}0\end{equation} Type: AlgebraicNumber? fricas)set output tex off fricas)set output script on f(sqrt(2)) .eq set blank @ :df. 0 :edf. (65) 0 Type: AlgebraicNumber? fricas)set output script off test --jr, Sat, 03 Feb 2018 13:20:20 +0000 reply((1+2)/3)::SquareMatrix?(3,Integer) ... --jr, Sat, 03 Feb 2018 13:21:15 +0000 replyfricas((1+2)/3)::SquareMatrix(3,Integer) +1 0 0+ | | (66) |0 1 0| | | +0 0 1+ Type: SquareMatrix?(3,Integer) ... --jr, Sat, 03 Feb 2018 13:21:37 +0000 replyfricas((1+2)/3)::SquareMatrix(3,Integer) +1 0 0+ | | (67) |0 1 0| | | +0 0 1+ Type: SquareMatrix?(3,Integer) integrate(exp(x-x^2), x) --test int, Sun, 10 Jun 2018 12:51:40 +0000 replyfricasintegrate((x+sin(x))^(1/2), x) >> Error detected within library code: integrate: implementation incomplete (constant residues) integrate(exp(x-x^2), x) --test int, Sun, 10 Jun 2018 12:52:12 +0000 reply\begin{fricas} integrate((x+sin(x))^(1/2), x) \end{fricas} trying something difficult; see comments --h-x, Sat, 09 Nov 2019 02:31:34 +0000 replyfricas""; "Notice how the CAS computes the second (displayed) integral, while saying 'proved non-elementary' about the first one,"; "despite the fact that the two are equal for any x in [-1..-1/sqrt(2)]."; "My guess is that the system does not apply transformations that force additional restrictions on the function domain..."; "Notice also how transforming the integrand into a form that the CAS can find an elementary antiderivative for requires"; "quite some discipline in keeping track of the integrand domain, as well as using some not-so-obvious transfromations..."; t:=sqrt(1-x*x); nume:=(acos(x)*t)^(-1); deno:=log(2*acos(x)/%pi); transformedDeno:=log(2+asin(2*x*t)/%pi); failedIntegral:=integrate(nume/transformedDeno,x); integrate(nume/deno, x) 2 acos(x) (68) - log(log(---------)) %pi Type: Union(Expression(Integer),...) Still no CAS knows if this one is elementary... Probably it is not, but a fun fact... --h-x, Sat, 09 Nov 2019 02:35:01 +0000 replyfricasintegrate((x+sin(x))^(1/3),x) >> Error detected within library code: integrate: implementation incomplete (constant residues) Axiom is lying here, the example comes from Bronstein's tutorial, pp. 29-31 --h-x, Sun, 10 Nov 2019 09:26:19 +0000 replyfricast:=x+log(x); nume:=(x^2+x+1)*sqrt(t)+(3*x+1)*log(x)+3*x*x+x; deno:=x*t*sqrt(t)+x*x*t; check:=integrate(nume/deno,x); differentiate(2*sqrt(t)+2*log(x+sqrt(t)), x) 2 +----------+ 2 (x + 2 x + 1)\|log(x) + x + (3 x + 1)log(x) + 3 x + x (69) -------------------------------------------------------- 2 +----------+ 2 3 (x log(x) + x )\|log(x) + x + x log(x) + x Type: Expression(Integer) No lie, compare: fricasnume - numer(differentiate(2*sqrt(t)+2*log(x+sqrt(t)), x)) +----------+ (70) - x\|log(x) + x Type: Expression(Integer) Still, shame that FriCAS can not do corrected example... Axiom fails again, see Bronstein's Tutorial pp. 32-33 for details on this example --h-x, Sun, 10 Nov 2019 13:21:41 +0000 replyfricasnume:= 3*(x+exp(x))^(1/3)+(2*x*x+3*x)*exp(x)+5*x*x; deno:= x*(x+exp(x))^(1/3); "integrate(nume/deno, x)" (q:=nume/deno, differentiate(3*x*(x+exp(x))^(2/3)+3*log(x), x)) (71) +-------+ 3| x 2 x 2 3 \|%e + x + (2 x + 3 x)%e + 5 x [-------------------------------------, +-------+ 3| x x\|%e + x +-------+ 3| x 2 x 2 3 \|%e + x + (2 x + 3 x)%e + 5 x -------------------------------------] +-------+ 3| x x\|%e + x Type: Tuple(Expression(Integer)) testing --h-x, Sun, 02 Feb 2020 04:20:03 +0000 replyfricas)clear value t integrate(sqrt(A-2*t*t/((1-t*t)^2))*2/(1+t*t), t) (72) [ +---------+ \|- 4 A - 2 * log +------------------------+ | 4 2 3 +---------+ |A t + (- 2 A - 2)t + A 4 (2 t - 2 t)\|- 4 A - 2 |------------------------ + A t | 4 2 \| t - 2 t + 1 + 2 (- 6 A - 4)t + A / 4 2 t + 2 t + 1 + +------------------------+ | 4 2 2 |A t + (- 2 A - 2)t + A (t - 1) |------------------------ | 4 2 +-+ \| t - 2 t + 1 - 2 \|2 atan(-----------------------------------) +-+ t\|2 / 4 , +-------+ +-------+ t\|4 A + 2 - \|4 A + 2 atan(-----------------------------------) +------------------------+ | 4 2 2 |A t + (- 2 A - 2)t + A (t - 1) |------------------------ | 4 2 \| t - 2 t + 1 + +------------------------+ | 4 2 2 |A t + (- 2 A - 2)t + A (t - 1) |------------------------ | 4 2 +-+ \| t - 2 t + 1 - \|2 atan(-----------------------------------) +-+ t\|2 / 2 ] Type: Union(List(Expression(Integer)),...)  Some or all expressions may not have rendered properly, because Latex returned the following error:This is pdfTeXk, Version 3.141592-1.40.3 (Web2C 7.5.6) \write18 enabled. %&-line parsing enabled. entering extended mode (./5900402145956781665-16.0px.tex LaTeX2e <2005/12/01> Babel and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, arabic, farsi, croatian, ukrainian, russian, bulgarian, czech, slov ak, danish, dutch, finnish, basque, french, german, ngerman, ibycus, greek, mon ogreek, ancientgreek, hungarian, italian, latin, mongolian, norsk, icelandic, i nterlingua, turkish, coptic, romanian, welsh, serbian, slovenian, estonian, esp eranto, uppersorbian, indonesian, polish, portuguese, spanish, catalan, galicia n, swedish, ukenglish, pinyin, loaded. 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[2] [3] [4] [5] [6] (/usr/share/texmf-texlive/tex/latex/base/t1cmtt.fd) [7] [8] [9] [10] LaTeX Error: Environment AxiomCode undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.153 \begin{AxiomCode} Missing$ inserted. $l.154 f(a) == ( local b; b:=sqrt(a); b^ 2-a) LaTeX Error: \begin{document} ended by \end{AxiomCode}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.156 \end{AxiomCode} \newpage Missing$ inserted. $l.156 \end{AxiomCode} \newpage [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] LaTeX Error: Environment fricas undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.183 \begin{fricas} Missing$ inserted. $l.184 integrate((x+sin(x))^ (1/2), x) LaTeX Error: \begin{document} ended by \end{fricas}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.185 \end{fricas} Missing$ inserted. \$ l.185 \end{fricas} [25] (./5900402145956781665-16.0px.aux) ) (see the transcript file for additional information) Output written on 5900402145956781665-16.0px.dvi (25 pages, 8104 bytes). Transcript written on 5900402145956781665-16.0px.log. `

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