[Axiom-developer] 20080818.01.rhx.patch (demo Axiom type towers (Hemmecke))

[daly]

Demo the Axiom type towers. 
=======================================================================
diff --git a/changelog b/changelog
index f2bd54b..04a4914 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,5 @@
+20080818 tpd src/input/Makefile typetower.input added
+20080818 rhx src/input/typetower.input added
 20080817 tpd src/algebra/vector.spad doc ravel from CartesianTensor
 20080817 tpd src/algebra/vector.spad doc leviCivitaSymbol from CartesianTensor
 20080817 tpd src/algebra/vector.spad doc * from CartesianTensor
diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet
index a4d2218..e70b0d9 100644
--- a/src/input/Makefile.pamphlet
+++ b/src/input/Makefile.pamphlet
@@ -370,7 +370,7 @@ REGRES= algaggr.regress algbrbf.regress  algfacob.regress 
alist.regress  \
     tanhcoth.regress  tanatan.regress  tbagg.regress    test.regress \
     textfile.regress  torus.regress \
     triglim.regress   tsetcatvermeer.regress            tutchap1.regress \
-    void.regress      uniseg.regress
+    typetower.regress void.regress      uniseg.regress
 
 IN=     ${SRC}/input
 MID=   ${INT}/input
@@ -664,7 +664,8 @@ FILES= ${OUT}/algaggr.input  ${OUT}/algbrbf.input    
${OUT}/algfacob.input \
        ${OUT}/triglim.input  ${OUT}/tschirn.input ${OUT}/tsetcatbutcher.input \
        ${OUT}/tsetcatchemical.input ${OUT}/tsetcatvermeer.input \
        ${OUT}/tutchap1.input ${OUT}/tutchap2.input   ${OUT}/tutchap3.input \
-       ${OUT}/tutchap4.input ${OUT}/tutchap67.input  ${OUT}/typo.input     \
+       ${OUT}/tutchap4.input ${OUT}/tutchap67.input  ${OUT}/typetower.input \
+       ${OUT}/typo.input     \
        ${OUT}/uniseg.input   ${OUT}/up.input \
        ${OUT}/vector.input   ${OUT}/vectors.input    ${OUT}/viewdef.input \
        ${OUT}/void.input     ${OUT}/wiggle.input   \
@@ -1011,7 +1012,8 @@ DOCFILES= \
   ${DOC}/tsetcatvermeer.input.dvi    \
   ${DOC}/tutchap1.input.dvi    ${DOC}/tutchap2.input.dvi   \
   ${DOC}/tutchap3.input.dvi    ${DOC}/tutchap4.input.dvi   \
-  ${DOC}/tutchap67.input.dvi   ${DOC}/typo.input.dvi       \
+  ${DOC}/tutchap67.input.dvi   ${DOC}/typetower.input.dvi  \
+  ${DOC}/typo.input.dvi       \
   ${DOC}/uniseg.input.dvi      ${DOC}/up.input.dvi         \
   ${DOC}/vector.input.dvi      ${DOC}/vectors.input.dvi    \
   ${DOC}/viewdef.input.dvi     ${DOC}/void.input.dvi       \
diff --git a/src/input/typetower.input.pamphlet 
b/src/input/typetower.input.pamphlet
new file mode 100644
index 0000000..260d206
--- /dev/null
+++ b/src/input/typetower.input.pamphlet
@@ -0,0 +1,253 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input typetower.input}
+\author{Ralf Hemmecke}
+\maketitle
+\begin{abstract}
+This page demonstrates some features of Axiom.
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+@
+Let's begin with the construction of a polynomial ring $R$ in the
+indeterminate $z$ with coefficients from the ring $S$ of square
+matrices with entries that are polynomials $P=F[x]$ where $F$ is
+the Galois field with 3 elements.
+<<*>>=
+)spool typetower.output
+)set message test on
+)set message auto off
+)clear all
+ 
+--S 1 of 17
+F:=PrimeField 3
+--R 
+--R
+--R   (1)  PrimeField 3
+--R                                                                 Type: 
Domain
+--E 1
+
+--S 2 of 17
+P:=UnivariatePolynomial(x,F)
+--R 
+--R
+--R   (2)  UnivariatePolynomial(x,PrimeField 3)
+--R                                                                 Type: 
Domain
+--E 2
+
+--S 3 of 17
+S:=SquareMatrix(2,P)
+--R 
+--R
+--R   (3)  SquareMatrix(2,UnivariatePolynomial(x,PrimeField 3))
+--R                                                                 Type: 
Domain
+--E 3
+
+--S 4 of 17
+R:=UnivariatePolynomial(z,S)
+--R 
+--R
+--R   (4)
+--R   UnivariatePolynomial(z,SquareMatrix(2,UnivariatePolynomial(x,PrimeField 
3)))
+--R                                                                 Type: 
Domain
+--E 4
+
+@
+Ok, now we have the type $R$. Let's construct an element. We start
+with constructing some coefficients first.
+<<*>>=
+
+--S 5 of 17
+s1:S:=matrix [[2*x+1,x^2-1],[0,x-1]]
+--R 
+--R
+--R        +         2    +
+--R   (5)  |2x + 1  x  + 2|
+--R        |              |
+--R        +  0     x + 2 +
+--R                   Type: SquareMatrix(2,UnivariatePolynomial(x,PrimeField 
3))
+--E 5
+
+--S 6 of 17
+s2:=transpose s1
+--R 
+--R
+--R        +2x + 1    0  +
+--R   (6)  |             |
+--R        | 2           |
+--R        +x  + 2  x + 2+
+--R                   Type: SquareMatrix(2,UnivariatePolynomial(x,PrimeField 
3))
+--E 6
+
+@
+And now we build the polynomial
+<<*>>=
+
+--S 7 of 17
+r:R:=z^2+s1*z+s2
+--R 
+--R
+--R         2   +         2    +    +2x + 1    0  +
+--R   (7)  z  + |2x + 1  x  + 2|z + |             |
+--R             |              |    | 2           |
+--R             +  0     x + 2 +    +x  + 2  x + 2+
+--RType: 
UnivariatePolynomial(z,SquareMatrix(2,UnivariatePolynomial(x,PrimeField 3)))
+--E 7
+
+@
+Of course, since we work in characteristic 3, the following sum must be zero
+<<*>>=
+--S 8 of 17
+r+2*r
+--R 
+--R
+--R   (8)  0
+--RType: 
UnivariatePolynomial(z,SquareMatrix(2,UnivariatePolynomial(x,PrimeField 3)))
+--E 8
+
+@
+Note that this is not the integer 0, but it is still a polynomial of type $R$.
+Asking for the degree of $r$ is no problem, because $R$ is a univariate 
+polynomial ring.
+<<*>>=
+
+--S 9 of 17
+degree r
+--R 
+--R
+--R   (9)  2
+--R                                                        Type: 
PositiveInteger
+--E 9
+
+@
+So let's see what happens if we multiply $r$ by itself.
+<<*>>=
+
+--S 10 of 17
+r2:=r*r
+--R 
+--R
+--R   (10)
+--R                               + 2             +     + 4              +
+--R      4   +         2    + 3   |x  + 2x    0   | 2   |x  + 2x     0   |
+--R     z  + |x + 2  2x  + 1|z  + |               |z  + |                |z
+--R          |              |     |  2       2    |     |          4     |
+--R          +  0    2x + 1 +     +2x  + 1  x  + 2+     +   0     x  + 2x+
+--R   + 
+--R     + 2                    +
+--R     |x  + x + 1      0     |
+--R     |                      |
+--R     |             2        |
+--R     +    0       x  + x + 1+
+--RType: 
UnivariatePolynomial(z,SquareMatrix(2,UnivariatePolynomial(x,PrimeField 3)))
+--E 10
+
+@
+Well, of course there is a common factor of $r_2$ and $r$. Can Axiom find it?
+<<*>>=
+
+--S 11 of 17
+gcd(r2,r)
+--R 
+--R   There are 4 exposed and 3 unexposed library operations named gcd 
+--R      having 2 argument(s) but none was determined to be applicable. 
+--R      Use HyperDoc Browse, or issue
+--R                               )display op gcd
+--R      to learn more about the available operations. Perhaps 
+--R      package-calling the operation or using coercions on the arguments
+--R      will allow you to apply the operation.
+--R 
+--RDaly Bug
+--R   Cannot find a definition or applicable library operation named gcd 
+--R      with argument type(s) 
+--RUnivariatePolynomial(z,SquareMatrix(2,UnivariatePolynomial(x,PrimeField 3)))
+--RUnivariatePolynomial(z,SquareMatrix(2,UnivariatePolynomial(x,PrimeField 3)))
+--R      
+--R      Perhaps you should use "@" to indicate the required return type, 
+--R      or "$" to specify which version of the function you need.
+--E 11
+
+@
+Ooops. What does that say?
+\begin{verbatim}
+  Cannot find a definition or applicable library operation named gcd
+\end{verbatim}
+Ah, of course, the coefficient ring of $R$ is the matrix ring $S$ and this
+is unfortunately not a field (not even an integral domain) so we cannot
+simply apply Euclid's algorithm. Axiom simply stops by telling you 
+that there is no applicable operation.
+
+Of course, Axiom can compute a gcd of univariate polynmials.
+<<*>>=
+
+--S 12 of 17
+p1:=s1(1,1)
+--R 
+--R
+--R   (11)  2x + 1
+--R                                   Type: UnivariatePolynomial(x,PrimeField 
3)
+--E 12
+
+--S 13 of 17
+ps:=s1(1,2)
+--R 
+--R
+--R          2
+--R   (12)  x  + 2
+--R                                   Type: UnivariatePolynomial(x,PrimeField 
3)
+--E 13
+
+--S 14 of 17
+gcd(p1,p2)
+--R 
+--R
+--R   (13)  1
+--R          Type: UnivariatePolynomial(p2,UnivariatePolynomial(x,PrimeField 
3))
+--E 14
+
+@
+Ok, let us do that again
+<<*>>=
+
+--S 15 of 17
+q1:UP(x,INT):=2*x+1
+--R 
+--R
+--R   (14)  2x + 1
+--R                                        Type: 
UnivariatePolynomial(x,Integer)
+--E 15
+
+--S 16 of 17
+q2:UP(x,INT):=x^2+2
+--R 
+--R
+--R          2
+--R   (15)  x  + 2
+--R                                        Type: 
UnivariatePolynomial(x,Integer)
+--E 16
+
+--S 17 of 17
+gcd(q1,q2)
+--R 
+--R
+--R   (16)  1
+--R                                        Type: 
UnivariatePolynomial(x,Integer)
+--E 17
+
+@
+Nice! Depending on where I compute these polynomials either they have
+a common factor or are coprime. Well, it all depends on the underlying
+ring, of course.
+<<*>>=
+
+)spool 
+)lisp (bye)
+ 
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} nothing
+\end{thebibliography}
+\end{document}