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+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input fixed.input}
+\author{The Axiom Team}
+\maketitle
+\begin{abstract}
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+\section{License}
+<<license>>=
+--Copyright The Numerical Algorithms Group Limited 1996.
+@
+<<*>>=
+<<license>>
+--fixed.input
+
+-------------------------------------------------------------------
+-- daly/10/06/92
+-------------------------------------------------------------------
+)clear all
+-- something is wrong with the integration of serices, when the
+-- coeffiennts are not constants. while the series is mathematically
+-- correct, this is not what one expects as the taylor series, which
+-- does not exist. 
+series(x/(x+log(x)))
+-- here integrate is treating log(x) as a constant, which is incorrect.
+integrate(%)
+
+-------------------------------------------------------------------
+-- bmt/10/19/92 integrate problem -- serious, wrong answer
+-------------------------------------------------------------------
+)clear all
+2*sin(t)*sqrt(1+cos(t))
+--correct answer is 2*sin(t)/sqrt(1+cos(t))
+integrate(%,t)
+
+-------------------------------------------------------------------
+-- bmt/10/20/92 multiple complex in type tower
+-------------------------------------------------------------------
+)clear all
+n:Complex ?
+-- an invalid type. can't have 2 complex constructors in a tower
+n:=x/y+%i
+
+-------------------------------------------------------------------------
+-- grabm/09/28/92 coercion bug
+-- comment: this coercion must carefully check whether there is a symbol
+--          among the kernels which is equal to an indeterminate of the
+--          polynomial ring, and not simply considering every expression
+--          as a constant
+-------------------------------------------------------------------------
+)clear all
+f:=(a-b-c-d)**2::EXPR INT
+f::DMP(['a,'b],EXPR INT)
+degree %
+
+-------------------------------------------------------------------
+-- bmt/10/26/92 wrong answer
+-- comment:I believe this problem simplifies to
+-- lfintegrate(sqrt(u**3+u**2),u) which returns the
+-- wrong answer due to some confusion in prootintegrate in INTPAF.
+-- I think the confusion happens with the use of radPoly and rootPoly.
+-- The answer is computed with respect to the result returned by rootPoly
+-- but the kernel which is substituted for "y" corresponds to radPoly.
+-------------------------------------------------------------------
+)clear all
+-- should be 2*sin(t)/sqrt(1+cos(t))
+integrate(sqrt(1+cos(x)),x)
+
+-------------------------------------------------------------------
+-- bmt/09/28/92 bug in ITRIGMNP
+-- gives:
+--    Cannot convert kernel to gaussian function
+-------------------------------------------------------------------
+)clear all
+integrate(exp(x**2),x)
+
+-------------------------------------------------------------------
+-- themos/11/06/92 integration bug
+--comment:
+--I suggest changing line 11 of limitedLogs in INTTR from:
+--      rows := max(degree den, 1 + "max"/[degree(u.contrib) for u in l1])
+--to:
+--      rows := max(degree den,
+--                  1 + reduce(max,[degree(u.contrib) for u in l1],0)$List(N))
+--Does you agree, thus max/ empty list would return 0.
+-------------------------------------------------------------------
+)clear all
+f:=log(1-(b*x/(a+c*x**2)))/x
+--               2
+--            c x  - b x + a
+--        log(--------------)
+--                  2
+--               c x  + a
+--   (1)  -------------------
+--                 x
+integrate(%,x)
+--                   2
+--                 %J c - %J b + a
+--             log(---------------)
+--           x          2
+--         ++         %J c + a
+--   (2)   |   -------------------- d%J
+--        ++            %J
+expand f
+--                 2               2
+--        - log(c x  + a) + log(c x  - b x + a)
+--   (3)  -------------------------------------
+--                          x
+integrate(%,x)
+--   >> Error detected within library code:
+--   No identity element for reduce of empty list using operation
+--   max
+
+-------------------------------------------------------------------
+-- bmt/11/17/92 interpreter resolve problem
+-------------------------------------------------------------------
+-- says cannot find an appropriate *
+)clear all
+%i/m*sqrt(m) 
+
+-------------------------------------------------------------------------
+-- tpd/09/22/92 this generates extra brackets
+-------------------------------------------------------------------------
+)clear all
+foo n == matrix[[i for i in 1..n] for j in 1..n]
+foo
+
+-------------------------------------------------------------------------
+--- tpd/09/24/92 matrix multiplication appears broken in the product.
+-- fails with
+--   >> System error:
+--   The function |deleteAssoc| is undefined.
+--   You are being returned to the top level of the interpreter.
+-- comment: (barry)
+-- This bug is due to the new autoloading scheme for the compiler, the
+-- function deleteAssoc needs to be moved into the in core system.
+-- As a work around, 
+-- if you compile a non existent file, the apropriate files will be loaded,
+-- e.g.:
+-- )co foo
+--  comment: this works in version 2
+-------------------------------------------------------------------------
+msq := Matrix SquareMatrix(2,POLY INT)
+m : msq := zero(2,2)
+m(1,1) := matrix([[1,2],[a,b]])
+m(1,2) := matrix([[a,b],[2,b]])
+m(2,2) := matrix([[1,2],[2,b]])
+m
+m*m
+m**2
+m**3
+(m*m)*m
+mm:=m*m
+mm*m
+
+-------------------------------------------------------------------------
+-- tpd/09/25/92 new equation.spad from johannes grabmeier
+-------------------------------------------------------------------------
+
+
+--Furthermore, we have function to put 0 or 1 on one side
+--and factor the left hand side, after the right hand side
+--is 0 and we have an IntegralDomain.
+--Johannes
+
+eq1 := (-6*x**3+13*x**2+4)=(-x**4+12*x)
+--
+--
+--            3      2         4
+--   (1)  - 6x  + 13x  + 4= - x  + 12x
+--                                            Type: Equation Polynomial Integer
+--                                       Time: 1.61 (IN) + 1.47 (OT) = 3.08 sec
+eq2 := x**4+13*x**2-12*x = 6*x**3-4
+--
+--
+--         4      2          3
+--   (2)  x  + 13x  - 12x= 6x  - 4
+--                                            Type: Equation Polynomial Integer
+--                                       Time: 0.16 (IN) + 0.11 (OT) = 0.27 sec
+eq := eq1*y**2+eq2
+--
+--
+--             3      2      2    4      2            4        2     3
+--   (3)  (- 6x  + 13x  + 4)y  + x  + 13x  - 12x= (- x  + 12x)y  + 6x  - 4
+--                                            Type: Equation Polynomial Integer
+--               Time: 0.26 (IN) + 0.01 (EV) + 0.01 (OT) + 1.54 (GC) = 1.82 sec
+swap %
+--
+--
+--            4        2     3           3      2      2    4      2
+--   (4)  (- x  + 12x)y  + 6x  - 4= (- 6x  + 13x  + 4)y  + x  + 13x  - 12x
+--                                            Type: Equation Polynomial Integer
+--                                                   Time: 0.07 (OT) = 0.07 sec
+% + 4
+--
+--
+--            4        2     3       3      2      2    4      2
+--   (5)  (- x  + 12x)y  + 6x = (- 6x  + 13x  + 4)y  + x  + 13x  - 12x + 4
+--                                            Type: Equation Polynomial Integer
+--                                       Time: 0.69 (IN) + 0.01 (OT) = 0.70 sec
+%-6*x**3
+--
+--
+--            4        2       3      2      2    4     3      2
+--   (6)  (- x  + 12x)y = (- 6x  + 13x  + 4)y  + x  - 6x  + 13x  - 12x + 4
+--                                            Type: Equation Polynomial Integer
+--                                       Time: 0.19 (IN) + 0.01 (OT) = 0.20 sec
+leftZero %
+--
+--
+--             4     3      2            2    4     3      2
+--   (7)  0= (x  - 6x  + 13x  - 12x + 4)y  + x  - 6x  + 13x  - 12x + 4
+--                                            Type: Equation Polynomial Integer
+--                                       Time: 0.01 (IN) + 0.05 (OT) = 0.06 sec
+swap %
+--
+--
+--          4     3      2            2    4     3      2
+--   (8)  (x  - 6x  + 13x  - 12x + 4)y  + x  - 6x  + 13x  - 12x + 4= 0
+--                                            Type: Equation Polynomial Integer
+--                                                   Time: 0.01 (IN) = 0.01 sec
+factor lhs %
+--
+--
+--               2       2  2
+--   (9)  (x - 2) (x - 1) (y  + 1)
+--                                            Type: Factored Polynomial Integer
+--                           Time: 0.50 (IN) + 0.09 (EV) + 0.30 (OT) = 0.89 sec
+factorAndSplit eq
+--
+--
+--                             2
+--   (10)  [x - 2= 0,x - 1= 0,y  + 1= 0]
+--                                       Type: List Equation Polynomial Integer
+--                                       Time: 0.09 (EV) + 0.21 (OT) = 0.30 sec
+inv (eq :: EQ FRAC POLY INT)
+--
+--
+--                            1                                1
+--   (11)  - ------------------------------------= - ----------------------
+--              3      2      2    4      2            4        2     3
+--           (6x  - 13x  - 4)y  - x  - 13x  + 12x    (x  - 12x)y  - 6x  + 4
+--                                   Type: Equation Fraction Polynomial Integer
+--                                                   Time: 0.03 (IN) = 0.03 sec
+- %
+--
+--
+--                          1                              1
+--   (12)  ------------------------------------= ----------------------
+--            3      2      2    4      2          4        2     3
+--         (6x  - 13x  - 4)y  - x  - 13x  + 12x  (x  - 12x)y  - 6x  + 4
+--                                   Type: Equation Fraction Polynomial Integer
+--                                       Time: 0.01 (IN) + 0.01 (OT) = 0.02 sec
+
+
+-------------------------------------------------------------------
+-- bmt/09/29/92 coercion bug
+-- fails with:
+-- failed cannot be coerced to mode 
+-- (Record (: coef1 (Integer)) (: coef2 (Integer)))
+-- fixed on 09/29/92
+-------------------------------------------------------------------
+)clear all
+(p,q):UP(x,INT)
+p:=3*x**4+11*x**2-4
+q:=9*x**4+9*x**2-4
+myNextPrime: (INT,NNI) -> INT
+myNextPrime(x,n)==nextPrime(x)$PRIMES(INT)
+--  runs forever due to algebra bug in handling leading coefficients
+modularGcd([p,q])$InnerModularGcd(INT,UP(x,INT),67108859,myNextPrime)
+
+-------------------------------------------------------------------
+-- dewar/10/02/92
+-- actually, this was never wrong
+-------------------------------------------------------------------
+numeric(%e ** %pi)
+--gives
+   (61)
+--  23.1406926327 7926900563 5493084578 4861294789 0532254433 2912200665 
+-- 6744971913 2534837468 7163222983 2730035
+
+-------------------------------------------------------------------
+-- themos/10/07/92
+-------------------------------------------------------------------
+)clear all
+y:=operator 'y
+deqx:=differentiate(y(x),x,2)+differentiate(y(x),x)+y(x)
+solve(deqx,y,x)
+solve(deqx,y,x=0,[1])
+deqt:=differentiate(y(t),t,2)+differentiate(y(t),t)+y(t)
+solve(deqt,y,t)
+solve(deqt,y,t=0,[1]) -- bug
+deqz:=differentiate(y(z),z,2)+differentiate(y(z),z)+y(z)
+solve(deqz,y,z)
+solve(deqz,y,z=0,[1]) --bug
+
+-------------------------------------------------------------------
+-- themos/10/07/92 equation
+-------------------------------------------------------------------
+)clear all
+y:=operator 'y
+deq:=D(y(x),x)+x**2=(y x)/x-(y x)**2
+
+
+-------------------------------------------------------------------
+-- bmt/10/08/92 laplace
+-------------------------------------------------------------------
+)clear all
+laplace(exp(-x**3)*x**7,x,s) 
+
+-------------------------------------------------------------------
+-- bmt/09/30/92 nonlinear ODE 
+-------------------------------------------------------------------
+-- Any hope of solving the following which appeared in Barry
+-- Simon's computer algebra test suite:
+-- (Apparently it has Bernoulli form)
+y:=operator 'y
+x**2 * D(y x, x) + 2*x*(y x) - (y x)**3 = 0
+solve(%,y,x)
+
+
+-------------------------------------------------------------------
+-- sit/10/08/92 factor bug
+-------------------------------------------------------------------
+p:POLY FRAC INT := 3*(x+1)
+-- is wrong (missing factor of 3).
+factor(p)**2 
+
+-------------------------------------------------------------------
+-- henderson/10/08/92 
+-------------------------------------------------------------------
+)clear all
+--TextFile cliams that the operator readLineIfCan! returns "failed" when
+--the end of the file is reached.  In fact, I get a system error, which is
+--not very useful:
+--(16) ->readLineIfCan!(fp)
+--   >> System error:
+--   Unexpected end of #<input stream "/home/mcd/work/work">.
+--   You are being returned to the top level of the interpreter.
+fout:TextFile:=open("/tmp/foo","output")
+write!(fout,"foo")
+close!(fout)
+fin:TextFile:=open("/tmp/foo","input")
+readLineIfCan!(fin)
+readLineIfCan!(fin)
+close!(fin)
+)lis (system "rm /tmp/foo")
+
+-------------------------------------------------------------------
+-- bmt/10/08/92 factoring over SAEs
+-------------------------------------------------------------------
+)clear all
+a | a**2+1
+(x+a)*(x+a+1)
+factor %
+
+-------------------------------------------------------------------
+-- miller/10/09/92 local vars in types
+-------------------------------------------------------------------
+)clear all
+--Fixed (Victor's) bug with using local variables in type expressions
+--assigned to local variables.  It now gives a message and correctly uses
+--interpret-only mode.
+--The function changed was getAndEvalConstructorArgument from i-spec1.boot.
+
+-------------------------------------------------------------------
+-- themos/10/19/92
+-------------------------------------------------------------------
+)clear all
+-- Do this in a virgin system
+)set expose add constructor SquareMatrix
+S2:= SquareMatrix(2,FRAC POLY INT);
+V2: S2 := matrix([[v,-v],[-v,v]]);
+I2: S2 := 1;
+m:=5;
+l: List(S2) := append(cons(V2+h*I2,_
+            [(V2+2*h*I2) for i in 2 .. (m-1)]),_
+             [V2+h*I2])
+A: SquareMatrix(m, S2) := diagonalMatrix(l)
+-- did you get an error of deleteAssoc not defined ??
+
+-- load definition of deleteAssoc
+)li (load "/spad/obj/rios/interp/c-util")
+
+A: SquareMatrix(m, S2) := diagonalMatrix(l)
+--now , can print.
+
+-------------------------------------------------------------------
+-- bmt/10/19/92 squareFree bug
+-------------------------------------------------------------------
+-- has an extra factor of 2.
+squareFree((2*x*y+1)*(x*y+1)**2)
+
+-------------------------------------------------------------------
+-- bmt/10/19/92 division by zero error
+-------------------------------------------------------------------
+limit(atan(1/sin(x)),x=0)
+
+-------------------------------------------------------------------
+-- bmt/10/19/92 limit problem
+-------------------------------------------------------------------
+)clear all
+-- the left and right hand limits don't agree
+limit(atan(-sin(x)/(cos(x)+e)),x=acos(-e))
+-- the left and right hand limits don't agree
+limit(atan(1/(cos(x)+e)),x=acos(-e))
+
+-------------------------------------------------------------------
+-- grabmeier/10/19/92 unable to compute the determinant of an 8x8
+-------------------------------------------------------------------
+)clear all
+D := MATRIX FRAC(POLY INT)
+
+
+d : (INT, Boolean) -> POLY INT
+d(i,ss)  ==
+  ex := i rem 4
+  if ex < 0 then ex := ex+4
+  ss =>
+    ex = 0 => 's
+    ex = 1 => 'sd
+    ex = 2 => 'sdd
+    'sddd
+  ex = 0 => 1
+  ex = 1 => 'd
+  ex = 2 => 'dd
+  'ddd
+
+
+
+-- 1,d,dd,ddd,s,sd,sdd,sddd
+
+mTV4 : D := new(8,8,0)
+
+for i in 1..8 repeat
+  for j in 1..8 repeat
+    mTV4(i,j) :=
+      i <= 4 =>
+        j <= 4 => d(i+j-2, false)
+        d(-i+j,true)
+      j <= 4 => d(i+j-2,true)
+      d(-i+j,false)
+mTV4
+gdd4 := determinant mTV4
+
+-------------------------------------------------------------------
+-- sutor/09/28/92 operator bug?
+-------------------------------------------------------------------
+)clear all
+L n ==
+  n = 0 => 1
+  n = 1 => x
+  (2*n-1)/n * x * L(n-1) - (n-1)/n * L(n-2)
+dx:=operator("D")::OP(POLY FRAC INT)
+evaluate(dx,p+-> differentiate(p,'x))
+E n == (1-x**2)*dx**2-2*x*dx+n*(n+1)
+L 15
+-- used to fail on this line
+E 15
+
+-------------------------------------------------------------------
+-- bmt/10/12/92 EFSTRUC recursion problem
+-------------------------------------------------------------------
+)clear all
+bug:=(1+x**(1/4))**(1/3)/(x**(1/2))
+-- gives a value stack overflow
+integrate(bug,x) -- seems to go into infinite recursion in rischNormalize
+
+-------------------------------------------------------------------
+-- james/10/28/92 coerce bug
+-------------------------------------------------------------------
+)clear all
+-- improper character in Roman numeral: G
+g::ROMAN
+
+-------------------------------------------------------------------
+-- grabm/10/28/92 runs forever
+-------------------------------------------------------------------
+)clear all
+-- this is quick
+factor 1068303355883998767544567663620885466990173600
+-- but this runs forever
+sqrt 1068303355883998767544567663620885466990173600
+
+
+-------------------------------------------------------------------
+-- themos/11/05/92 fortran output bug
+-------------------------------------------------------------------
+)clear all
+--      REAL T7,T6,T5,T4,T3,T2,T1
+--      T1=x*x
+--      T2=2.*y**5
+--      T3=4.*T1
+--      T4=y**3
+--      T5=2.*T7
+--      T6=x**3
+--      T7=x**4
+--      R46=((T2+(T3+8.*x+8.)*T4+(T5+8.*T6+(-40.*T1))*y)*SIN(x)+(-T2+(-T3+
+--     &16.*x)*T4+(-T5+16.*T6)*y)*COS(x))/(y**8+4.*T1*y**6+6.*T7*y**4+4.*x
+--     &**6*y*y+x**8)
+-- T7 is referenced before it is defined
+a1:=sin(x)/(x**2+y**2);a2:=D(a1,[x,y]);a2+D(a2,x)
+
+-------------------------------------------------------------------
+-- grabm/11/05/92 
+-- comment:
+--I believe the "bug" is that the interpreter is converting gb which is
+--a list of dmp style polynomials to a list of POLY type polynomials.
+--Unfortunately the result of such a conversion is no longer a groebner basis.
+--I don't see any reasonable way to prevent this, and would therefore have to
+--classify this as "user error", when working with grobner bases, the user
+--needs to force all his polynomials to be the appropriate type (thus having
+--the correct ordering).
+-------------------------------------------------------------------
+-- expressing a symmetric functions in terms of elementary symm. functions
+-- working in POLY INT is o.k.
+R := FRAC INT
+--   (1)  Fraction Integer
+lpx : List POLY R := [x1+x2+x3, x1*x2+x1*x3+x2*x3,x1*x2*x3]
+--   (2)  [x3 + x2 + x1,(x2 + x1)x3 + x1 x2,x1 x2 x3]
+lip := [lpx.1-e3, lpx.2-e2, lpx.3-e1]
+--   (3)  [x3 + x2 + x1 - e3,(x2 + x1)x3 + x1 x2 - e2,x1 x2 x3 - e1]
+gbp := groebner lip
+--   (4)
+--                         2                   2
+--   [x3 + x2 + x1 - e3, x2  + (x1 - e3)x2 + x1  - e3 x1 + e2,
+--      3        2
+--    x1  - e3 x1  + e2 x1 - e1]
+normalForm(x1**2+x2**2+x3**2,gbp)
+--          2
+--   (5)  e3  - 2e2
+-- working in DMP is o.k.
+dmp := DMP([x1,x2,x3,e1,e2,e3],INT)
+--   (6)  DistributedMultivariatePolynomial([x1,x2,x3,e1,e2,e3],Integer)
+li : List dmp := [lpx.1-e1, lpx.2-e2, lpx.3-e3]
+--   (7)  [x1 + x2 + x3 - e1,x1 x2 + x1 x3 + x2 x3 - e2,x1 x2 x3 - e3]
+gb := groebner li
+--   (8)
+--                         2                     2
+--   [x1 + x2 + x3 - e1, x2  + x2 x3 - x2 e1 + x3  - x3 e1 + e2,
+--      3     2
+--    x3  - x3 e1 + x3 e2 - e3]
+p:dmp:=(x1**2+x2**2+x3**2)
+--          2     2     2
+--   (9)  x1  + x2  + x3
+normalForm(p,gb)
+--           2
+--   (10)  e1  - 2e2
+-- but we have some problems (coercion? ordering?) here:
+normalForm(x1**2+x2**2+x3**2,gb)
+--            2                      2              2
+--   (11)  2x2  + (2x1 - 2e1)x2 + 2x1  - 2e1 x1 + e1
+
+
+
+
+-------------------------------------------------------------------
+-- bmt/11/17/92 Ifintegrate
+-------------------------------------------------------------------
+--It seems that one cannot simply parametrize reducible curves.
+--Any attempt at parametrization can only follow one branch.
+--For curves of the form y**2=a**2*x**2+b*x+c, these are only reducible
+--if a**2*x**2+b*x+c is a perfect square. Basically I think we need to
+--choose 1 branch in this case. Under some circumstances we could
+--try to proceed down both branches and then unify the result, but for
+--the time being, perhaps we should just perform the substitution
+--y=sqrt(a**2*x**2+b*x+c) whenever that is a perfect square.
+--(then we don't even need to do any back subsitution or change to dx).
+--
+--For another example, perhaps due to the same essential difficulty try
+integrate(normalize(sqrt(1+cos(x)),x),x)
+-- this gets a non-invertible error
+
+--I have fixed integrate(sqrt(1+cos(x)),x) and
+--integrate(normalize(sqrt(1+cos(x)),x),x)
+--fix was to function prootintegrate in INTPAF in intaf.spad
+
+
+-------------------------------------------------------------------
+-- james/02/11/93 integrate
+-- used to give error:
+--   No identity element for reduce of empty list using operation
+--   max
+-------------------------------------------------------------------
+integrate(((-x-1)*log((x**2+x))**2+2*log(x))/(x+1),x)
+
+-------------------------------------------------------------------
+-- dewar/02/15/93
+-------------------------------------------------------------------
+)clear all
+--i::Polynomial(Integer)
+--%::Union(Symbol,Polynomial Integer)
+--list %
+--
+--this is a bug in old style union handling, this style of unions
+--will become obsolete, so the suggested fix is to use tagged unions instead:
+i::POLY INT
+%::Union(s:Symbol, p:POLY INT)
+list % -- works
+
+-------------------------------------------------------------------
+-- grabm/03/04/93
+-- parser bug? not for system commands
+-------------------------------------------------------------------
+)clear all
+-- gives error: Sorry, but no option begins with upd;
+)load /spad/mnt/rios/algebra/EQ )upd; a:=1
+-- works
+)load /spad/mnt/rios/algebra/EQ )upd ; a:=1
+
+-------------------------------------------------------------------
+-- bronstein/03/08/93
+-------------------------------------------------------------------
+)clear all
+I:=operator 'I
+J:=operator 'J
+eq := mu * D(I x,x) = - (K + S) * I(x) + S*J(x)
+solve(eq,I,x)
+--> wrong particular solution
+
+-------------------------------------------------------------------
+-- dewar/03/16/93
+-- bug writing backslash to files
+--comment:
+--This is not a bug, what is written to the file is the lisp representation
+--of the string "\\test" (since '\' is lisp escape character).
+--The file is read and written using lisp primitives and thus the escape
+--characters need to be doubled. This is consistent, since if you read the
+--item back in from the file, the correct string is reconstituted.
+--
+--If you want to create a text file which simply contains lines of text
+--instead of lines of lisp strings, use:
+--ofile : TextFile := open("test","output")  ...
+--then you will get a file of literal strings with no string delimiters and
+--no escape characters.
+-------------------------------------------------------------------
+)clear all
+ofile: File String := open("/tmp/test","output")
+-- this writes "\\\\test" but should write "\\test"
+write!(ofile,"\\test"::String)
+close! ofile
+
+-------------------------------------------------------------------
+-- themos/04/07/93
+-- this sometimes fails
+-- comment:
+--thanks for tracing down the problem. Basically the lines for monicizing
+--the result need to be done before we try to 
+--retract back from the algebraic extension.
+-------------------------------------------------------------------
+)clear all
+pol:DMP([x,y,z],PF(2)):=x**2*y**2+x**2*y*z+x**2*z**2+x*y*z**2+y**3*z+y*z**3
+factor pol
+
+-------------------------------------------------------------------
+-- williamson/04/21/93
+-- comment:
+--I ran into some problems when factoring polynomials over number fields.
+--Evidently the interpretor doesn't choose the correct function 'factor'.
+--It uses a function from MULTFACT, when it should be using a function from
+--SAEFACT. 
+-------------------------------------------------------------------
+)clear all
+up := UP('w,FRAC INT)
+p : up := w**4 + w**3 + w**2 + w + 1
+sae := SAE(FRAC INT,up,p)
+q : UP('x,sae) := x**5 - 1
+factor q -- used to report: x**5-1 
+saefact := SAEFACT(up,sae,UP('x,sae))
+factor(q)$saefact
+
+-------------------------------------------------------------------
+-- grabm/04/30/93
+-- comment:
+--This bug has been fixed at yorktown, in radix.spad, change line 163, the
+--third line of intgroup to be:
+--      empty? rest li => intToExpr first(li)
+--and then recompile RADIX
+-------------------------------------------------------------------
+)clear all
+-- results in 10 and not A
+10::RadixExpansion(16)
+
+-------------------------------------------------------------------
+-- bronstei@inf.ethz.ch (manuel bronstein)
+-------------------------------------------------------------------
+)clear all
+r:=rule 'x == 1
+-- used to return x, now returns 1
+r x
+
+-------------------------------------------------------------------
+-- ADK@scri.fsu.edu (tony kennedy)
+-------------------------------------------------------------------
+)clear all
+factor(-12)
+%**2
+
+-------------------------------------------------------------------
+-- bronstei@inf.ethz.ch (manuel bronstein)
+-- >> Error: cannot retract nonconstant polynomial
+-------------------------------------------------------------------
+)clear all
+complexNumeric(log(sqrt(-3)))
+
+-------------------------------------------------------------------
+-- quitte@knuth.univ-poitiers.fr/8/15/93 (Claude Quitte)
+-------------------------------------------------------------------
+)clear all
+--)abbreviation package TEST Test
+--
+--Test() : with
+--
+--   leftProductBy : Integer -> (Integer -> Integer)
+--   rightProductBy : Integer -> Mapping(Integer, Integer)
+--
+-- == add
+--
+--   leftProductBy(n) == n * #1
+--
+--   rightProductBy(n : Integer) : Mapping(Integer, Integer) == #1 * n
+--------------------------------------------------------------------------
+--
+--Why is it impossible to specify the target/cible types for the
+--functions leftProductBy and rightProductBy in the body of the package ???
+--
+--The code for leftProductBy is correct but the code for rightProductBy
+--troubles the compiler !!!! It seems to compile a LOCAL function with
+--ANOTHER signature.
+--
+--------------------------------------------------------------------------
+--   initializing NRLIB TEST for Test
+--   compiling into NRLIB TEST
+--   compiling exported leftProductBy : Integer -> Integer -> Integer
+--Time: 0.35 SEC.
+--
+--   compiling local rightProductBy : Integer -> Integer
+--****** comp fails at level 2 with expression: ******
+--error in function rightProductBy
+--
+--****** level 2  ******
+--$x:= #1
+--$m:= $EmptyMode
+--$f:=
+--((((|n| # #) (|rightProductBy| # # #) (|n| #) (|rightProductBy| # # #) ...)))
+--
+--   >> Apparent user error:
+--   no mode found for
+--   #1
+--   You are being returned to the top level of the interpreter.
+------------------------------------------------------------------------
+-- answer (barry):
+--
+--Your problem in TEST about specifying target types which are Mapping's
+--was a bug in the compiler and has been fixed for Axiom release 2.0.
+--
+
+-------------------------------------------------------------------
+-- bronstei@inf.ethz.ch/10/6/93 (Manuel Bronstein)
+-------------------------------------------------------------------
+)clear all
+--Here is algfunc.spad with an old bug fixed: eventh-roots of negative floats
+--coerced to EXPR FLOAT (e.g. sqrt(-1::EXPR FLOAT)) used to produced an error
+--message since the sqrt from FLOAT was called. 
+--I have now fixed iroot from AF(R,)
+--to call the sqrt from R only if either R is algebraically closed, or the
+--root is odd, or the argument is positive. Here's the new behaviour:
+sqrt(-1::EXPR FLOAT)
+sqrt(2::EXPR FLOAT)
+nthRoot(-2::EXPR FLOAT, 3)
+nthRoot(-2::EXPR FLOAT, 4)
+--As a side-effect, this fixes the problem with numeric
+-- (which was instantiating sqrt(-1)$EXPR(FLOAT)).
+
+-------------------------------------------------------------------
+-- bronstei@inf.ethz.ch/10/6/93 (Manuel Bronstein)
+-------------------------------------------------------------------
+)clear all
+--Here is an updated efstruc.spad where ker2trigs now knows about abs.
+real abs(4 + %i * 5) -- does the right thing.
+
+-------------------------------------------------------------------
+-- bronstei@inf.ethz.ch/10/5/93 (Manuel Bronstein)
+-------------------------------------------------------------------
+)clear all
+--Here is again an enhanced elemntry.spad. What happens now is that
+exp(q * %i * %pi)
+--gets automatically simplifed whenever q is a rational number whose
+--denominator is one of  {1,2,3,4,6}.
+exp(5/3*%i*%pi)
+
+-------------------------------------------------------------------
+-- bronstei@inf.ethz.ch/10/4/93 (Manuel Bronstein)
+-- luczak@nag.com (Richard Luczak)
+-------------------------------------------------------------------
+)clear all
+exp(log(-1))
+sum((-1)**k * (k+m),k=0..n)
+
+-------------------------------------------------------------------
+-- bronstei@inf.ethz.ch/10/4/93 (Manuel Bronstein)
+-------------------------------------------------------------------
+)clear all
+--simplifies (abs now checks quotients instead of just retraction to R).
+abs((1/2)::EXPR(INT))
+
+-------------------------------------------------------------------
+-- bronstei@inf.ethz.ch/10/4/93 (Manuel Bronstein)
+-------------------------------------------------------------------
+)clear all
+integrate(1/(x**2 + %i*a),x)
+
+-------------------------------------------------------------------
+-- bmt@spadserv.watson.ibm.com/9/28/93 (Barry Trager)
+-------------------------------------------------------------------
+)clear all
+limit(1/2**n,n=%plusInfinity)
+
+-------------------------------------------------------------------
+-- bronstei@inf.ethz.ch/9/22/93 (Manuel Bronstein)
+-------------------------------------------------------------------
+)clear all
+--Here is an update to efstruc.spad that handles complex constants much better.
+--Negative square roots and negative logs are now recognized as complex and
+--treated properly by real?, real, imag, and complexForm:
+x := sqrt(-3) + sqrt 2 + sqrt(- exp a) + log(-a**2-1)
+real? x
+real x
+imag x
+--As a result, integrals involving sqrt(-2) etc... are now treated correctly
+--(this was the case only for sqrt(-1) with the older version).
+
+-------------------------------------------------------------------
+-- bronstei@inf.ethz.ch/9/22/93 (Manuel Bronstein)
+-------------------------------------------------------------------
+)clear all
+-->> haha := rule x*x == z
+-->> haha(a*a + b*b + c**2)        -->  3z
+-->> haha(a*a + b*b + c**2 + d*d)  -->  z
+-->>
+-->> the bug is that the last line returns z instead of 4z.
+--
+--Sorry guys, this is not a bug: haha is so general a rule that it matches
+--the integer 4 (as 2 squared), so the rewrite chain for the last example is:
+--
+--a*a + b*b + c**2 + d*d  --->  z + z + z + z  = 4 * z ---> z * z ---> z
+--
+--Here is a console showing what exactly happens:
+haha := rule x*x == z
+haha 4
+haha 3
+haha(4*z)
+haha(3*z)
+--To see the whole rewrite chain:
+a*a + b*b + c**2 + d*d
+applyRules([haha], %, 1)$APPRULE(INT,INT,EXPR INT)
+applyRules([haha], %, 1)$APPRULE(INT,INT,EXPR INT)
+applyRules([haha], %, 1)$APPRULE(INT,INT,EXPR INT)
+--I think it's actually the proper behavior for the pattern matcher. This
+--example shows that rules can bite, something Mma users are quite aware about!
+
+-------------------------------------------------------------------
+-- dewar/9/16/93 (mike dewar)
+-------------------------------------------------------------------
+--There's an odd discontinuity about the behaviour of the following
+--function:
+
+harm(1) == 1
+harm(n) == harm(n-1) + 1/n
+harm : Integer -> Fraction Integer
+harm(1023) -- takes a little while, as expected
+harm(1024) -- takes forever?
+
+-------------------------------------------------------------------
+-- williams@inf.ethz.ch/9/1/93 (Clifton Williamson)
+-------------------------------------------------------------------
+--> we should probably support integrate(%,x) on series types.
+--> x could be either the variable of expansion or a parameter variable.
+--
+--We've discussed this before with regard to the function 'differentiate'.
+--The technical problem is that "generalized" power series may have
+--"coefficients" involving the series variable:
+series(x**x,x=0)
+--In this case, as currently implemented, 'differentiate' returns incorrect
+--answers:
+)set mess test off
+differentiate %
+)set mess test on
+--The solution we discussed (perhaps "resolution" is a better word) was to
+--create a separate series type for generalized series.  This would be a
+--carbon copy of UPXS, except that differentiate(series,variable(series)) and
+--integrate(series,variable(series)) would return an error message.  The
+--error message could also suggest that the user first apply 'approximate',
+--then compute a derivative or integral.  We would also have a coercion from
+--UPXS to this type.  The function 'series' would return a UPXS, when the
+--coefficients do not involve the series variable, and a "generalized series",
+--when the coefficients involve the series variable.
+--
+--If this sounds cool to you, I'll go ahead with it when I have time.  (I'm
+--trying to meet a 16 September deadline for a MEGA '94 submission.  I'm still
+--getting my results together, so time is tight.  Let me know if there is a
+--time deadline for the AXIOM code!)
+--
+--> Also we need better tools for taking finite truncations of series objects.
+--> In particular since we now mostly use series(expr,var) to create a series,
+--> UPXS if over a coefficient domain which supports the symbol of expansion,
+--> should be able to truncate a series to an EXPR.
+--
+--I thought I had you on this one!  The signature is there (in pscat.spad):
+--
+--    if Coef has coerce: Symbol -> Coef then
+--      if Coef has "**":(Coef,Expon) -> Coef then
+--        approximate: ($,Expon) -> Coef
+--          ++ \spad{approximate(f)} returns a truncated power series with the
+--          ++ series variable viewed as an element of the coefficient domain.
+--
+--It certainly works for Laurent series:
+laurent(cos(a+x)/x,x=0)
+approximate(%,3)
+--But, unfortunately, I never implemented it for Puiseux series:
+puiseux(cos(a+x)/x,x=0)
+approximate(%,3)
+series(cos(x**(2/3) + a),x=0)
+approximate(%,2)
+approximate(%% 1,7/5)
+
+
+-------------------------------------------------------------------
+-- bronstei@inf.ethz.ch/10/6/93 (manuel bronstein)
+-------------------------------------------------------------------
+)clear all
+)set message type off
+
+-- Here is algfunc.spad with an old bug fixed: eventh-roots of negative floats
+-- coerced to EXPR FLOAT (e.g. sqrt(-1::EXPR FLOAT)) used to produced an error
+-- message since the sqrt from FLOAT was called. 
+-- I have now fixed iroot from AF(R,)
+-- to call the sqrt from R only if either R is algebraically closed, or the
+-- root is odd, or the argument is positive. Here's the new behaviour:
+sqrt(-1::EXPR FLOAT)
+sqrt(2::EXPR FLOAT)
+nthRoot(-2::EXPR FLOAT, 3)
+nthRoot(-2::EXPR FLOAT, 4)
+--As a side-effect, this fixes the problem with numeric
+-- (which was instantiating sqrt(-1)$EXPR(FLOAT)).
+
+-------------------------------------------------------------------
+-- bronstei@inf.ethz.ch/9/22/93 (manuel bronstein)
+-------------------------------------------------------------------
+)clear all
+--Here is a (rather major) bug fix to ffactor in FSUPFACT.NRLIB. It causes
+--a large family of integrals to return 0, because ffactor(?**2+expr) returned
+--?**2 when expr involved a parameter. This is fixed now.
+
+-------------------------------------------------------------------
+-- jhd@maths.bath.ac.uk/8/15/93 James Davenport
+-------------------------------------------------------------------
+)clear all
+integrate(1/(x*(log(x)**2+a**2-1)),x)
+--   (1)
+--   VCONCAT
+--                             log(x)
+--                       atan(---------)
+--                             +------+
+--                             | 2
+--           2                \|a  - 1
+--      ((- a  + 1<0) -> ---------------)
+--                           +------+
+--                           | 2
+--                          \|a  - 1
+--  ,
+--      PAREN
+--               2
+--           (- a  + 1>0)
+--         ->
+--                                      +--------+
+--                          2    2      |   2           2
+--               log((log(x)  - a  + 1)\|- a  + 1  + (2a  - 2)log(x))
+--             +
+--                           2    2
+--               - log(log(x)  + a  - 1)
+--          /
+--               +--------+
+--               |   2
+--             2\|- a  + 1
+--  ,
+--                            1
+--      ( Or (a - 1=0) -> - ------)
+--           (a + 1=0)      log(x)
+--                                          Type: ConditionalExpression Integer
+--
+--is still fairly ugly.
+
+-------------------------------------------------------------------
+-- bronstei@inf.ethz.ch/8/9/93 (manuel bronstein)
+-------------------------------------------------------------------
+)clear all
+--Here is efstruc.spad with a change to normalize so that
+normalize(2**(1/2) + 2**(1/4)) now returns  2**(1/4) ** 2 + 2**(1/4).
+
+
+-------------------------------------------------------------------
+-- satoshi@yktvmv.vnet.ibm.com/8/7/93 satoshi hamaguchi
+-------------------------------------------------------------------
+)clear all
+-- In axiom, both
+  integrate(%e**x,x=0..1)
+--and
+  integrate(log(x),x=1..2)
+--yield a warning  potentialPole    .
+--If I use "noPole", they give right answers. Isn't this a bug?
+--... Satoshi
+
+-------------------------------------------------------------------
+-- bronstei@inf.ethz.ch/8/5/93 (manuel bronstein)
+-------------------------------------------------------------------
+)clear all
+simplify(2**(1/3)*2**(1/2)) -- will return 2**(5/6).
+
+-------------------------------------------------------------------
+-- bronstei@inf.ethz.ch/8/4/93 (manuel bronstein)
+-------------------------------------------------------------------
+)clear all
+integrate(1/sqrt(1+cos(x)), x)
+
+-------------------------------------------------------------------
+-- bronstei@inf.ethz.ch/8/4/93 (manuel bronstein)
+-------------------------------------------------------------------
+)clear all
+normalize atan(cos(x)/sin(x))  -- used to be division by 0 error
+
+-------------------------------------------------------------------
+-- bronstei@inf.ethz.ch/8/4/93 (manuel bronstein)
+-------------------------------------------------------------------
+)clear all
+a := 2**(1/6)
+[a**n for n in 2..13]
+
+-------------------------------------------------------------------
+-- bronstei@inf.ethz.ch/8/4/93 (manuel bronstein)
+-------------------------------------------------------------------
+)clear all
+int:=sqrt(a*(1-u**2)/(1+u**2))/u
+integrate(eval(int,a=1),u) -- works
+integrate(eval(int,a=sqrt(-1)),u) -- seems to run forever
+integrate(eval(int,a=1)*(-1)**(1/4),u) -- dies after a long time
+ -- with an elt index error.
+
+-------------------------------------------------------------------
+-- bronstei@inf.ethz.ch/8/4/93 (manuel bronstein)
+-------------------------------------------------------------------
+)clear all
+sqrt((1-x**2)*(1-k**2*x**2))
+integrate(x/%,x)
+
+-------------------------------------------------------------------
+-- bronstei@inf.ethz.ch/7/26/93 (manuel bronstein)
+-------------------------------------------------------------------
+)clear all
+rad:=last zerosOf((2+y)**8-3,y)
+k:=first kernels %
+eval(rad,k,rad)
+
+-------------------------------------------------------------------
+-- nagttt@vax.ox.ac.uk/7/15/93 themos tsikas
+-------------------------------------------------------------------
+)clear all
+f := (x - y) / (x + y)
+eval(f,x=1/x)
+
+-------------------------------------------------------------------
+-- nagttt@vax.ox.ac.uk/7/15/93 themos tsikas
+-------------------------------------------------------------------
+)clear all
+digits 200
+--I claim that these two should give the same result (they do in Reduce):
+--(precision was 200)
+a:=4*sin(2*%pi/9)*sin(5*%pi/9)/sqrt(3)
+b:=1/(2*sin(%pi/9))
+a::EXPR FLOAT
+b::EXPR FLOAT                                 
+
+-------------------------------------------------------------------
+-- bmt@spadserv.watson.ibm.com/9/15/93 (Barry Trager)
+-------------------------------------------------------------------
+)clear all
+limit(tanh(x),x=%plusInfinity)
+
+-------------------------------------------------------------------
+-- bronstei@inf.ethz.ch/6/14/93 (manuel bronstein)
+-------------------------------------------------------------------
+)clear all
+a := x::EXPR INT
+b := x::EXPR COMPLEX INT
+zeroOf(a**4+1,x)
+zeroOf(b**4+1,x)
+
+-------------------------------------------------------------------
+-- bronstei@inf.ethz.ch/6/11/93 (manuel bronstein)
+-------------------------------------------------------------------
+)clear all
+normalize(0**a) -- now returns 0  (was crashing before)
+
+-------------------------------------------------------------------
+-- bronstei@inf.ethz.ch/6/11/93 (manuel bronstein)
+-------------------------------------------------------------------
+)clear all
+-  new export  complexForm(f)  which returns the rectangular complex form of f:
+log(a+%i * b)
+complexForm %
+complexIntegrate(1/(x-%i),x)
+complexForm %
+
+-------------------------------------------------------------------
+-- jcf@posso.ibp.fr
+-------------------------------------------------------------------
+--  normalForm est faux pour les polynomes a coef dans un anneau:
+D:=DMP([a,b],INT)
+l:List D:=[2*a-1,b-a]
+g:=groebner l
+p:D:=a
+normalForm(p,g)
+-- was  1
+--   au lieu de 1/2 !!
+
+-------------------------------------------------------------------
+-- nagttt@vax.ox.ac.uk/6/9/93 themos tsikas
+-------------------------------------------------------------------
+)clear all
+squareFreePart(50)
+
+-------------------------------------------------------------------
+-- mcd@maths.bath.ac.uk/6/4/92 mike dewar
+-------------------------------------------------------------------
+)clear all
+f:=complexIntegrate(1/((x-%i)*(x-2*%i)),x)
+g:=subst(f,x=1)
+imag g
+
+-------------------------------------------------------------------
+-- daly
+-------------------------------------------------------------------
+)clear all
+-- not wrong but the order is obscure...
+primes(1,10)
+
+-------------------------------------------------------------------
+-- nagttt@vax.ox.ac.uk/1/11/93 themos tsikas
+-------------------------------------------------------------------
+)clear all
+m:=matrix[[1,a,0],[b,0,a],[0,b,c]]
+ll:=radicalEigenvectors m
+-- ends in error:
+-- Internal system problem in function appInfix : fewer than 2 arguments to
+-- an infix function
+--
+-- but this works
+ll 1
+ll 2
+ll 3
+
+
+-------------------------------------------------------------------
+-- bronstei@inf.ethz.ch/11/24/93 Manuel Bronstein
+-------------------------------------------------------------------
+--)clear all
+--Here's a bug that appears with the )math option only.
+--
+--)tr MATRIX )ops determinant )math
+--
+--   Parameterized constructors traced:
+--      MATRIX
+-- this integrate is commented out since it runs forever (at least many hours)
+--integrate(sqrt(1/x+1)*sqrt(x)-sqrt(x+1),x)
+--1<enter Matrix.determinant,26 :
+--
+-- Internal system problem in function appInfix :
+-- fewer than 2 arguments to an infix function
+
+-------------------------------------------------------------------
+-- copper@yktvmv/12/1/93 Don Coppersmith
+-------------------------------------------------------------------
+)clear all
+--Attached is the list of integers n such that 2**512-n is prime
+--and n is between 0 and 5000:
+--
+--   4893,4653,4475,4005,3893,3669,3459,3143,2967,
+--   2807,2529,1827,1695,975,875,629,569
+
+--It was gotten from Axiom by issuing the commands
+qrimes : Stream Integer := generate(nextPrime,2**512-5000)
+rrimes := [ 2**512-p for p in qrimes while p < 2**512 ]
+srimes := complete rrimes
+[srimes.i for i in [1..18]]
+[srimes.i for i in [10..18]]
+
+-------------------------------------------------------------------
+-- miker@num-alg-grp.co.uk/12/13/93 (Mike Richardson)
+-------------------------------------------------------------------
+)clear all
+--
+--This is fixed now. You get an Expression Float.
+--
+--> 
+--> I suspect that this is an example of the same thing - but it looks
+--> worse:
+--> 
+--> (145) ->X := log(0.7*%i*x)
+--> 
+-->    >> Error detected within library code:
+-->    negative root
+--> 
+--> Mike
+--> 
+
+
+-------------------------------------------------------------------
+--  sutor@yktpub.watson.ibm.com/12/15/93 (Robert S. Sutor)
+-------------------------------------------------------------------
+)clear all
+--enter Integer in input box
+--choose domains
+--choose Description
+--click on Convertible(String)
+--
+--page cannot be formatted
+
+-------------------------------------------------------------------
+-- bmt@posso.dm.unipi.it12/15/93 (barry trager)
+-------------------------------------------------------------------
+)clear all
+log2() --> Float
+exp1() --> DoubleFLoat ???
+
+
+-------------------------------------------------------------------
+--  quitte@knuth.univ-poitiers.fr/12/14/93 Claude Quitte
+-------------------------------------------------------------------
+)clear all
+--There is a bug in the function delete_! for the type List ; this function,
+--has two arguments, a list L and an integer n : delete_!(L, n) returns
+--a list L with the n nth-item deleted, MODIFYING its argument L.
+--
+--But for n = 1, the argument L is not MODIFIED !!! (exactly for n =
+--minIndex(L)). An extract of the package ListAggregate (abbreviation LSAGG,
+--file aggcat.spad).
+--
+--
+----------------------------------------------------------------------
+--ListAggregate(S:Object): Category == Join(StreamAggregate S,
+--   FiniteLinearAggregate S, ExtensibleLinearAggregate S) with
+--      list: S -> $
+--	++ list(x) returns the list of one element x.
+-- add
+--   ..........
+--
+--   delete_!(x:$, i:I) ==
+--     i < (m := minIndex x) => error "index out of range"
+--     i = m => rest x   			****** <- here is the error
+--     y := rest(x, (i - 1 - m)::N)
+--     setrest_!(y, rest(y, 2))
+--     x
+--
+--A little session :
+--
+L : List(String) := ["There is", "it seems", "a real bug", "here"]
+--   (1)  ["There is","it seems","a real bug","here"]
+--                                                            Type: List String
+--
+L1 := delete!(L, 1)
+--   (2)  ["it seems","a real bug","here"]
+--                                                            Type: List String
+L
+--   (3)  ["There is","it seems","a real bug","here"]
+--                                                            Type: List String
+--
+L2 := delete!(L, 2)
+--   (4)  ["There is","a real bug","here"]
+--                                                            Type: List String
+L
+--   (5)  ["There is","a real bug","here"]
+--
+--Is there a way to fix this bug quickly ?? (I am ready to modify the source
+--Axiom aggcat.spad).
+--The problem with ! functions lies in the documentation not in the code. We
+--need to explain more clearly that ! functions are ALLOWED to update their
+--arguments, not REQUIRED to do so. 
+--Whenever using ! functions you should always
+--use the returned result, not try to use the updated argument. This is how
+--the code is designed. 
+--For example if you try to do insert! into an empty list,
+--the argument is not updatable and thus a new list is created. The reason for
+--having ! operations is for efficiency to do less copying, not to guarantee
+--that given arguments will always be updated. So the correct use is to only
+--depend on the results returned by updating functions.
+
+
+-------------------------------------------------------------------
+--  quitte@knuth.univ-poitiers.fr/1/13/94 Claude Quitte
+-------------------------------------------------------------------
+)clear all
+K := Fraction(Integer)   
+PolK := UP('X, K) 
+X : PolK := monomial(1, 1) 
+n : PositiveInteger := 15 
+E := SimpleAlgebraicExtension(K, PolK, X**n + X**(n-3) -1) 
+y : E := X::E
+minimalPolynomial(y)$E
+--
+--Internal Error
+--The function minimalPolynomial with signature SimpleAlgebraicExtension(
+--Fraction Integer,UnivariatePolynomial(X,Fraction Integer),X**15+X**12-1)
+---> UnivariatePolynomial(X,Fraction Integer) is missing from domain
+--SimpleAlgebraicExtension(Fraction (Integer))
+--(UnivariatePolynomial X (Fraction (Integer)))((15 1 . 1) (12 1 . 1) (0 -1 . 1))
+
+-------------------------------------------------------------------
+-- luczak@nag.com/02/20/94 (Richard Luczak)
+-------------------------------------------------------------------
+)clear all
+--
+--I'm trying to define the following rule
+--
+tr := rule cos(x)**(n | integer? n and even? n)==(1-sin(x)**2)**(n/2)
+--
+--Axiom returns
+--
+--   Cannot find a definition or library operation named even? with argument
+--   types
+--                             Expression Integer
+--
+--   We will attempt to step through and interpret the code.
+--
+--                                  n
+--                                  -
+--              n             2     2
+--   (2)  cos(x)  == (- sin(x)  + 1)
+--                        Type: RewriteRule(Integer,Integer,Expression Integer)
+--
+--
+--and henceforth the rule does not work as desired. Variations on the same bug
+--occur involving positive?. Of course, I get what I want using integer?(n/2)
+--but ... Is this a bug?
+--
+-- from barry:
+--
+--Should use integer? n and even? integer n  instead.
+--
+--I toyed with the idea of implementing:
+--    even? x == integer? x and even? integer x
+--    odd? x == integer? x and odd? integer x
+--in EXPR, but this would mean that   (x +-> even? x or odd? x)  is not always
+--true anymore and I'm not sure it's a good idea. If you guys don't think it's
+--a problem, it takes 2 minutes to add it to EXPR R when R has RETRACT INT
+--(don't use integer? but retractIfCan@Union(Integer, "failed")).
+
+-------------------------------------------------------------------
+-- copper@yktvmv.vnet.ibm.com/02/22/94 don coppersmith
+-------------------------------------------------------------------
+)clear all
+--Questions that came up using Axiom on Aixproj:
+--
+--How does one find a Pade approximation?  I tried  pade(2,2,y)
+--pade(y,2,2)  Pade(2,2,y)  PadeApproximation(2,2,y)
+--(where y is a series in x) to no avail.
+--
+--Complaint: The error message
+--
+--    Could not find library function "pade" with arguments
+--         (integer, integer, series).
+--
+--is misleading.  If it can't find any "pade", then say
+--
+--    Could not find library function "pade".
+--
+--If there is such a function, but I've got the arguments wrong,
+--why not say
+--
+--    The library function "pade" takes arguments
+--          (series, integer, integer)
+--
+--or at least make that information available somehow.
+--
+-- from bob sutor:
+--
+--Don,
+--  Try using )what op pade  to see what is available. This will give
+--you all operations with "pade" in their names. You can also search for
+--*pade* via the HyperDoc Browse utility. If you don't use Browse, following
+--up with, say, )display op pade, will give you a more information, though
+--it is badly formatted (I will look at that).
+--
+--Aixproj is back-level, so I'm not sure exactly what is up their
+--relative to the current system.
+--
+--I agree we could use better messages regarding unknown functions
+--and I will look at that.
+--  Bob
+--
+-- from bob sutor:
+--
+--The pade operation requires a taylor series as its third object while
+--"series" returns a UnivariatePuiseuxSeries. If you set
+--  y := taylor(1+x)**(1/5)
+--then things work. You can also do
+--  pade(1,1,y :: UTS(EXPR INT, x, 0) )
+--but this is more work if you really have a Taylor series.
+--
+--This is unintuitive and there should be a "pade" for Puiseux series that
+--fails if the series is not appropriate. Barry? I will now look at adding
+--a bit more coercion support to the interpreter to allow Puiseux series
+--to possibly retract to Laurent series and then possibly to Taylor series.
+--
+--Don,
+--  I added the retractions I mentioned before to the development system.
+--When the system is unsuccessful in matching a UnivariatePuiseuxSeries
+--to an operation, it will first try to retract it to a
+--UnivariateLaurentSeries. If that fails still, it will try to retract to
+--a UnivariateTaylorSeries.
+--
+--  Be aware that this sort of thing happens when you are evaluating
+--an expression. If you are defining a function, you should add
+--explicit coercions or create objects of the correct type for
+--efficiency. You function may still work, but it may not run in
+--a compiled form.
+
+-------------------------------------------------------------------
+-- barry@num-alg-grp.co.uk/02/24/94 Barry Trager
+-------------------------------------------------------------------
+)clear all
+sqrt(2)*2.0 -- fails
+sqrt(2)::EXPR INT * 2.0 -- works
+
+-- tpd: this now fails with sample : () -> Reference Integer is missing
+-- from domain: Integer
+-- Internal Error
+
+
+-------------------------------------------------------------------
+-- bronstei@inf.ethz.ch/02/28/94 Manuel Bronstein
+-------------------------------------------------------------------
+)clear all
+f := exp(exp(x)*exp(1/exp(x))) / exp exp x
+--)tr LIMITPS )math
+--)tr DFINTTLS )math
+--)tr DFINTTLS_;findLimit
+-- the following exits back to the shell after limit returns "failed":
+computeInt(first tower f, f, 0, %plusInfinity, false)$DFINTTLS(INT, EXPR INT)
+
+-------------------------------------------------------------------
+-- jhd@maths.bath.ac.uk/03/02/94 James Davenport
+-------------------------------------------------------------------
+)clear all
+--I've fixed coercions of List and Vector to DirectProduct.
+--You can now do things like
+[1,2,3] :: DirectProduct(3, Fraction Integer)
+--
+--I also fixed coercions from Vector to List, which seems to have
+--been removed via a library change.
+
+-------------------------------------------------------------------
+-- jhd@maths.bath.ac.uk/03/09/94 James Davenport
+-------------------------------------------------------------------
+)clear all
+-- used to give a lisp error
+x**10+1::Polynomial PrimeField 2
+
+-------------------------------------------------------------------
+-- themos@num-alg-grp.co.uk/03/11/94 Themos Tsikas
+-------------------------------------------------------------------
+)clear all
+f(x)==x**2
+draw(f,-1..1)
+
+-------------------------------------------------------------------
+-- themos@num-alg-grp.co.uk/03/16/94 Themos Tsikas
+-------------------------------------------------------------------
+)clear all
+--The problem was that nobody was around to catch the coerceOrCroaker.
+--Just comment out the first line of intCodeGenCoerce1 in i-code.boot
+--(the one with the explicit THROW). Bothe the riemannSphereDraw and JHD's
+--original example will then work. -- bob sutor
+)read conformal
+riemannSphereDraw(1..2,0..%pi,4,4,"polar")
+
+
+
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} nothing
+\end{thebibliography}
+\end{document}