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\documentclass[dvipdfmx]{book}
\newcommand{\VolumeName}{Volume 4: Axiom Developers Guide}
\input{bookheader.tex}
\mainmatter
\setcounter{chapter}{0} % Chapter 1
\begin{quote}
Confronting every new programmer learning a new language are
\begin{itemize}
\item The Cave of Artifacts
\item The Forest of Tooling
\item The Mountain of Language
\item The Cloud Castle of Mindset
\end{itemize}
-- Daniel Higginbotham in Clojure for the Brave and True
\end{quote}

\section{What is the purpose of the HACKPI domain?}

HACKPI is a hack provided for the benefit of the axiom interpreter.
As a mathematical type, it is the simple transcendental extension
\verb|Q(\pi)| of the rational numbers. This type allows interactive
users to use the name \verb|'%pi'| without a type both where a numerical
value is expected [ as in \verb|draw(sin x,x=-%pi..%pi)| ] or when the
exact symbolic value is meant.  The interpreter defaults a typeless
\verb|%pi| to HACKPI and then uses the various conversions to cast it
further as required by the context.

One could argue that it is unfair to single \verb|%pi| out from other
constants, but it occurs frequently enough in school examples
(specially for graphs) so it was worth a special hack. In a
non-interactive environment (library), HACKPI would not exist.

(Manuel Bronstein)

\section{How Axiom Builds}
\subsection{The environment variables}
Axiom uses a tree of Makefiles to build the system. Each Makefile
is created from the literate file (Makefile.pamphlet) and then executed.

In order to have a complete set of variables we create an ``environment''
that contains all of the shell variables (except the AXIOM variable).

These can be changed on the command line at the time of the top level
``make'' command. One common usage pattern is to override the NOISE
variable. This variable controls whether we see the full output or just
the echo of each individual step. Sometimes a build fails at a step and
we would like to know the details. By default they are written to 
\verb|$TMP/trace| but we can watch every detail with the command line:
\begin{verbatim}
  make NOISE=
\end{verbatim}
This overrides the output file and writes everything to the console.

Another common usage pattern is to override the tests that are run.
By default, all tests are run. This can be very time consuming. 
A particular subset can be run or, using the option ``notests'', none
will be run:
\begin{verbatim}
  make TESTSET=notests
\end{verbatim}

\begin{verbatim}
AWK=gawk 
BOOKS=/research/test/books 
BYE=bye 
CC=gcc 
CCF=-O2 -fno-strength-reduce -Wall -D_GNU_SOURCE -DLINUXplatform 
    -I/usr/X11/include 
COMMAND=/usr/local/axiom/mnt/ubuntu/bin/axiom 
DAASE=/research/test/src/share 
DESTDIR=/usr/local/axiom 
DOCUMENT=/research/test/mnt/ubuntu/bin/document 
GCLDIR=/research/test/lsp/gcl-2.6.8pre4 
GCLOPTS=--enable-vssize=65536*2 --enable-locbfd --disable-dynsysbfd 
        --disable-statsysbfd --enable-maxpage=512*1024 --disable-xgcl 
        --disable-tkconfig 
GCLVERSION=gcl-2.6.8pre4 
INC=/research/test/src/include 
INT=/research/test/int 
LDF= -L/usr/X11R6/lib -L/usr/lib -lXpm 
LISP=lsp 
LSP=/research/test/lsp 
MNT=/research/test/mnt 
NOISE=-o /research/test/obj/tmp/trace 
O=o 
OBJ=/research/test/obj 
PART=cprogs 
PATCH=patch 
PLF=LINUXplatform 
RANLIB=ranlib 
RUNTYPE=serial 
SPAD=/research/test/mnt/ 
SPADBIN=/research/test/mnt/ubuntu/bin 
SPD=/research/test 
SRC=/research/test/src 
SRCDIRS=interpdir sharedir algebradir etcdir clefdir docdir graphdir 
        smandir hyperdir browserdir inputdir 
SUBPART=everything 
SYS=ubuntu 
TANGLE=/research/test/mnt/ubuntu/bin/lib/notangle 
TAR=tar 
TESTSET=none 
TMP=/research/test/obj/tmp 
TOUCH=touch 
UNCOMPRESS=gunzip 
VERSION=Axiom (May 2010) 
WEAVE=/research/test/mnt/ubuntu/bin/lib/noweave 
XLIB=/usr/X11R6/lib 
ZIPS=/research/test/zips
\end{verbatim}

\section{The runtime structure of Axiom}
\begin{center}
\includegraphics[scale=0.5]{ps/v4architecture.eps}\\
{\bf Runtime Structure \cite{Bake14}}
\end{center}

\subsection{The build step}
This shows the steps taken to build Axiom in the sequence they happen.
Each level of indentation is another level of Makefile being executed.
\begin{verbatim}

Makefile
  1 noweb
  2 copy SRC/scripts to AXIOM/bin
  3 extract Makefile.SYS from Makefile.pamphlet
  4 latex SRC/input/*.input.pamphlet
  5 extract SRC/algebra/Makefile.help from SRC/algebra/Makefile.pamphlet
    5a make SRC/algebra/Makefile.help parallelhelp
       5a1 extract syntax help from BOOKS/bookvol5
       5a2 extract help files from BOOKS/bookvol10.*
  6 extract BOOKS/Makefile from BOOKS/Makefile.pamphlet
    6a make BOOKS/Makefile
       6a1 copy SRC/scripts/tex/axiom.sty to AXIOM/doc
       6a2 create AXIOM/doc/*.pdf
           6a2a copy book*.pamphlet to AXIOM/doc
           6a2b extract latex for each book
           6a2c latex each book
           6a2d dvipdfm each dvi file
       6a3 make AXIOM/doc/toc.pdf
  7 extract AXIOM/doc/hypertex/Makefile1 from BOOKS/bookvol11
    7a make AXIOM/doc/hypertex/Makefile1
       7a1 extract all xhtml pages to AXIOM/doc/hypertex
       7a2 extract axiom1.bitmap from BOOK/bookvol11
       7a3 extract rcm3720.input from BOOK/bookvol11
       7a4 extract strang.input from BOOK/bookvol11
       7a5 extract signatures.txt from BOOK/bookvol11
       7a6 copy BOOKS/ps/doctitle.png to AXIOM/doc/hypertex
       7a7 copy BOOKS/ps/lightbayou.png to AXIOM/doc/hypertex
  8 make Makefile.SYS
    8a create the root directories
    8b create noweb if needed
    8c extract SRC/Makefile from SRC/Makefile.pamphlet
    8d make SRC/Makefile setup
       8d1 extract SRC/scripts/Makefile from SRC/scripts/Makefile.pamphlet
       8d2 make SRC/scripts/Makefile
          8d1a copy all scripts to AXIOM/bin
       8d3 extract SRC/lib/Makefile from SRC/lib/Makefile.pamphlet
       8d4 make SRC/lib/Makefile
          8d4a compile INT/lib/bsdsignal.c
          8d4b compile INT/lib/cursor.c
          8d4c compile INT/lib/edin.c
          8d4d compile INT/lib/fnct-key.c
          8d4e compile INT/lib/halloc.c
          8d4f compile INT/lib/openpty.c
          8d4g compile INT/lib/pixmap.c
          8d4h compile INT/lib/prt.c
          8d4i compile INT/lib/sockio-c.c
          8d4j compile INT/lib/spadcolors.c
          8d4k compile INT/lib/util.c
          8d4l compile INT/lib/wct.c
          8d4m compile INT/lib/xdither.c
          8d4n compile INT/lib/xshade.c
          8d4o compile INT/lib/xspadfill.c
          8d4p create libspad.a
          8d4q compile INT/lib/cfuns-c.c
          8d4r compile INT/lib/hash.c
          8d4s latex all files to INT/doc/src/lib
    8e extract LSP/Makefile from LSP/Makefile.pamphlet
    8f make LSP/Makefile gcldir
       8f1 untar ZIPS/gcl
       8f2 apply Axiom patches to gcl
       8f3 copy gcl_collectfn.lsp to OBJ/SYS/lsp
       8f4 copy sys-proclaim.lisp to OBJ/SYS/lsp
       8f5 make LSP/GCLVERSION/Makefile
       8f6 add BOOKS/tangle.lsp to gcl to create INT/SYS/lisp
    8g make SRC/Makefile
       8g1 make stanzas from SRCDIRS
          8g1a interpdir 
            8g1a1 copy bookvol5 to src/interp
            8g1a2 copy bookvol9 to src/interp
            8g1a3 copy bookvol10.5 to src/interp
            8g1a4 extract util.ht from BOOKS/bookvol7.1 to AXIOM/doc
            8g1a5 make SRC/interp/Makefile
              8g1a5a build SAVESYS=OBJ/SYS/bin/interpsys
                8g1a5a1 build DEPSYS=OBJ/SYS/bin/depsys
                8g1a5a2 compile all interp files
                8g1a5a3 call build-interpsys to make SAVESYS
                8g1a5a4 build warm.data
                8g1a5a5 build SAVESYS 
                8g1a5a6 copy SAVESYS to AXIOMSYS=AXIOM/bin/AXIOMsys
          8g1b sharedir 
            8g1b1 make share/Makefile
              8g1b1a copy SRC/share/algebra/command.list to AXIOM/lib
          8g1c algebradir 
            8g1c1 extract algebra/Makefile from SRC/algebra/Makefile.pamphlet
            8g1c2 copy bookvol10.2 to SRC/algebra
            8g1c3 copy bookvol10.3 to SRC/algebra
            8g1c4 copy bookvol10.4 to SRC/algebra
            8g1c5 copy bookvol10.5 to SRC/algebra
            8g1c6 extract 'findAlgebraFiles' 
                    from SRC/algebra/Makefile.pamphlet
            8g1c7 execute findAlgebraFiles and append output 
                    to SRC/algebra/Makefile
            8g1c8 make SRC/algebra/Makefile
              8g1c8a build INT/algebra nrlibs
              8g1c8b copy SRC/algebra/libdb.text to AXIOM/algebra
              8g1c8c construct AXIOM/bin/index.html
              8g1c8d copy SRC/share/algebra/gloss.text AXIOM/algebra
              8g1c8e copy SRC/share/algebra/glossdef.text AXIOM/algebra
              8g1c8f copy SRC/share/algebra/glosskey.text AXIOM/algebra
          8g1d etcdir 
            8g1d1 extract SRC/etc/Makefile from SRC/etc/Makefile.pamphlet
            8g1d2 make etc/Makefile
              8g1d2a copy SRC/doc/gloss.text INT/algebra
              8g1d2b copy SRC/doc/topics.data INT/algebra
              8g1d2c call make-databases
              8g1d2b copy INT/algebra/*.daase AXIOM/algebra
              8g1d2e compile asq.c
              8g1d2f copy OBJ/SYS/etc/asq AXIOM/bin
              8g1d2g copy SRC/etc/summary AXIOM/lib
              8g1d2h copy SRC/etc/copyright AXIOM/lib
          8g1e clefdir 
            8g1e1 extract SRC/clef/Makefile from SRC/clef/Makefile.pamphlet
            8g1e2 make clef/Makefile
              8g1e2a extract edible.c to OBJ/SYS/clef
              8g1e2b compile OBJ/SYS/clef/edible.c 
              8g1e2c link edible, fnct-key, edin, bsdsignal, prt, wct,
                          openpty, cursor into AXIOM/bin/clef
          8g1f docdir 
            8g1f1 extract SRC/doc/Makefile from SRC/doc/Makefile.pamphlet
            8g1f2 make SRC/doc/Makefile
              8g1f2a extract SRC/doc/axiom.bib to INT/doc
              8g1f2b extract SRC/doc/axiom.sty to AXIOM/bin/tex
              8g1f2c extract SRC/doc/refcard.dvi to AXIOM/doc
              8g1f2d extract SRC/doc/endpaper.dvi to AXIOM/doc
              8g1f2e copy SRC/doc/ps/* to AXIOM/doc/ps
              8g1f2f extract SRC/doc/rosetta.dvi to AXIOM/doc
       8g1f2g extract SRC/doc/booklet.c to INT
              8g1f2h compile booklet.c
              8g1f2i copy booklet to AXIOM/bin
          8g1g graphdir 
            8g1g1 extract SRC/graph/Makefile from BOOKS/bookvol8.pamphlet
            8g1g2 make graph/Makefile
              8g1g2a compile and link AXIOM/lib/viewman
              8g1g2b compile and link AXIOM/lib/view2d
              8g1g2c compile and link AXIOM/lib/view3d
              8g1g2d compile and link AXIOM/lib/viewalone
              8g1g2e extract AXIOM/graph/parabola.view from bookvol8
              8g1g2f extract psfiles from bookvol8 to AXIOM/lib/graph
          8g1h smandir 
            8g1h1 extract SRC/sman/Makefile from BOOKS/bookvol6.pamphlet
            8g1h2 make sman/Makefile
              8g1h2a extract INT/sman/session.c from bookvol6
              8g1h2b compile INT/sman/session.c to OBJ/SYS/sman/session.o
              8g1h2c link OBJ/SYS/sman/session.o to AXIOM/lib/session
              8g1h2d extract INT/sman/spadclient.c from bookvol6
              8g1h2e compile INT/sman/spadclient.c 
                       to OBJ/SYS/sman/spadclient.o
              8g1h2f link OBJ/SYS/sman/spadclient.o to AXIOM/lib/spadclient
              8g1h2g extract INT/sman/sman.c from bookvol6
              8g1h2h compile INT/sman/sman.c to OBJ/SYS/sman/sman.o
              8g1h2i link OBJ/SYS/sman/sman.o to AXIOM/lib/sman
              8g1h2j extract axiom shell script from bookvol6 to AXIOM/bin
              8g1h2k chmod axiom shell script to be executable
              8g1h2l create AXIOM/doc/bookvol6.dvi
          8g1i hyperdir 
            8g1i1 extract INT/hyper/Makefile from BOOKS/bookvol7.pamphlet
            8gli2 make INT/hyper/Makefile (to make hyperdoc)
              8g1i2a extract and compile AXIOM/lib/spadbuf
              8g1i2b extract and compile AXIOM/lib/ex2ht
              8g1i2c extract and compile AXIOM/bin/htadd
              8g1i2d extract and compile AXIOM/lib/hthits
              8g1i2e extract and compile AXIOM/bin/htsearch
              8g1i2f extract and compile AXIOM/lib/presea
              8g1i2g extract and compile AXIOM/bin/hypertex
              8g1i2h untar SPD/books/axbook.tgz to AXIOM/doc
              8g1i2j copy SPD/books/bigbayou.png to AXIOM/doc
              8g1i2k copy SPD/books/doctitle.png to AXIOM/doc
            8g1i3 extract INT/hyper/Makefile from BOOKS/bookvol7.1.pamphlet
            8g1i4 make INT/hyper/Makefile (to make hyperdoc pages)
              8g1i4a copy SPD/books/bookvol7.1 to AXIOM/doc
              8g1i4b htadd pages from AXIOM/doc/bookvol7.1
              8g1i4c copy SPD/books/bitmaps AXIOM/doc/bitmaps
              8g1i4d copy SPD/books/viewports AXIOM/doc/viewports
              8g1i4e untar AXIOM/doc/viewports .Z files
          8g1j browserdir 
            8g1j1 build of hyperdoc browser commented out
          8g1k inputdir 
            8g1k1 extract SRC/input/Makefile from SRC/input/Makefile.pamphlet
            8g1k2 make SRC/input/Makefile
              8g1k2a copy SRC/input/*.input INT/input
              8g1k2b lisp tangle input files from SRC/input/*.input.pamphlet
              8g1k2c extract INT/input/Makefile 
                       from SRC/input/Makefile.pamphlet
              8g1k2d make INT/input/Makefile TESTSET
                8g1k2d1 run regresstests
                8g1k2d2 run catstests
                8g1k2d3 run richtests
                8g1k2d4 run regression tests
                8g1k2d5 extract INT/input/Makefile.algebra 
                           from SRC/algebra/Makefile.pamphlet
                8g1k2d6 make INT/input/Makefile.algebra 
\end{verbatim}

\subsection{Where each output file is created}
Here we show which step in the above set of actions creates the file
that ends up in the final ship directory. We break it down by subdirectory
in the final image.
\subsubsection{AXIOM/algebra}
\begin{verbatim}
in AXIOM/algebra:
  *.o
  browse.daase
  category.daase
  compress.daase
  dependents.daase
  interp.daase
  operation.daase
  users.daase

\end{verbatim}
\subsubsection{AXIOM/autoload}
\begin{verbatim}
in AXIOM/autoload:
  ax.o
  bc-matrix.o
  br-con.o
  ht-util.o
  mark.fn
  mark.o
  nag-c02.o
  nag-c05.o
  nag-c06.o
  nag-d01.o
  nag-d02.o
  nag-d03.o
  nag-e01.o
  nag-e02.o
  nag-e04.o
  nag-f01.o
  nag-f02.o
  nag-f04.o
  nag-f07.o
  nag-s.o
  nspadaux.o
  pspad1.fn
  pspad1.o
  pspad2.fn
  pspad2.o
  topics.o
  wi1.fn
  wi1.o
  wi2.fn
  wi2.o

\end{verbatim}
\subsubsection{AXIOM/bin}
\begin{verbatim}
in AXIOM/bin:
  asq            8g1d2f  copy OBJ/SYS/etc/asq AXIOM/bin
  axiom          8g1h2k  chmod axiom shell script to be executable
  axiom.sty      6a1     copy SRC/scripts/tex/axiom.sty to AXIOM/doc
  AXIOMsys       8g1a5a6 copy SAVESYS to AXIOMSYS=AXIOM/bin/AXIOMsys
  booklet        8g1f2i  copy booklet to AXIOM/bin
  boxhead        2       copy SRC/scripts to AXIOM/bin
  boxtail        2       copy SRC/scripts to AXIOM/bin
  boxup          2       copy SRC/scripts to AXIOM/bin
  clef           8g1e2c  link edible, fnct-key, edin, bsdsignal, prt, wct,
                            openpty, cursor into AXIOM/bin/clef
  document       2       copy SRC/scripts to AXIOM/bin
  htadd          8g1i2c  extract and compile AXIOM/bin/htadd
  htsearch       8g1i2e  extract and compile AXIOM/bin/htsearch
  hypertex       8g1i2g  extract and compile AXIOM/bin/hypertex
  index.html     8g1c8c  construct AXIOM/bin/index.html
  lib            1       noweb
    btdefn
    cpif
    emptydefn
    finduses
    h2a
    htmldoc
    markup
    mnt
    nodefs
    noidx
    noindex
    noroff
    noroots
    notangle
    nountangle
    noweave
    noweb
    nt
    nuweb2noweb
    numtime
    pipedocs
    tmac.w
    toascii
    tohtml
    toroff
    totex
    unmarkup
  Makefile.pamphlet
  man
    man1
      cpif.1
      htmltoc.1
      nodefs.1
      noindex.1
      noroff.1
      noroots.1
      notangle.1
      nountangle.1
      noweave.1
      noweb.1
      nuweb2noweb.1
      sl2h.1
    man7
      nowebfilters.7
      nowebstyle.7
  showdvi            2       copy SRC/scripts to AXIOM/bin
? sman               8g1h2i  link OBJ/SYS/sman/sman.o to AXIOM/lib/sman
  SPADEDIT           2       copy SRC/scripts to AXIOM/bin
  tex                2       copy SRC/scripts to AXIOM/bin
    axiom.sty        8g1f2b  extract SRC/doc/axiom.sty to AXIOM/bin/tex
                     2       copy SRC/scripts to AXIOM/bin
    noweb.sty        1       noweb
    nwmac.tex        1       noweb
? viewalone          8g1g2d  compile and link AXIOM/lib/viewalone

\end{verbatim}
\subsubsection{AXIOM/doc}
\begin{verbatim}
AXIOM/doc:
  axbook
    *.xhtml
  axiom.sty
  bigbayou.png
  bitmaps
    *.bitmap
  bookvol0.out
  bookvol0.pdf
  bookvol0.toc
  bookvol10.1.out
  bookvol10.1.pdf
  bookvol10.1.toc
  bookvol10.2.out
  bookvol10.2.pdf
  bookvol10.2.toc
  bookvol10.3.out
  bookvol10.3.pdf
  bookvol10.3.toc
  bookvol10.4.out
  bookvol10.4.pdf
  bookvol10.4.toc
  bookvol10.5.out
  bookvol10.5.pdf
  bookvol10.5.toc
  bookvol10.out
  bookvol10.pdf
  bookvol10.toc
  bookvol11.out
  bookvol11.pdf
  bookvol11.toc
  bookvol12.out
  bookvol12.pdf
  bookvol12.toc
  bookvol1.out
  bookvol1.pdf
  bookvol1.toc
  bookvol2.out
  bookvol2.pdf
  bookvol2.toc
  bookvol3.out
  bookvol3.pdf
  bookvol3.toc
  bookvol4.out
  bookvol4.pdf
  bookvol4.toc
  bookvol5.out
  bookvol5.pdf
  bookvol5.toc
  bookvol6.out
  bookvol6.pdf
  bookvol6.toc
  bookvol7.out
  bookvol7.pdf
  bookvol7.toc
  bookvol7.1.out
  bookvol7.1.pamphlet
  bookvol7.1.pdf
  bookvol7.1.toc
  bookvol8.out
  bookvol8.pdf
  bookvol8.toc
  bookvol9.out
  bookvol9.pdf
  bookvol9.toc
  bookvolbib.pdf
  doctitle.png
  endpaper.dvi
  ht.db
  hypertex
    *.xhtml
  msgs
    s2-us.msgs
  ps
    *.ps
  refcard.dvi
  rosetta.dvi
  spadhelp
    *.help
  src
?    algebra
    algebra.Makefile.dvi
    books.Makefile.dvi
    clef
      axiom.sty
      edible.c.dvi
    clef.Makefile.dvi
    doc.Makefile.dvi
    etc.Makefile.dvi
?   hyper
    input
      *.input.dvi
    input.Makefile.dvi
?   interp
    interp.Makefile.dvi
    lib
      *.c.dvi
    lib.Makefile.dvi
    Makefile.dvi
    root.Makefile.dvi
    scripts.Makefile.dvi
    share.Makefile.dvi
?   sman
    src.Makefile.dvi
  toc.pdf
  util.ht
  viewports
    *.view
      data
      graph0
      image.bm
      image.xpm

\end{verbatim}
\subsubsection{AXIOM/graph}
\begin{verbatim}
AXIOM/graph
  parabola.view:     8g1g2e extract AXIOM/graph/parabola.view from bookvol8
    data
    graph0

\end{verbatim}
\subsubsection{AXIOM/input}
\begin{verbatim}
AXIOM/input:
  *.input files

\end{verbatim}
\subsubsection{AXIOM/lib}
\begin{verbatim}
AXIOM/lib:
  command.list       8g1b1a copy SRC/share/algebra/command.list to AXIOM/lib
  copyright          8g1d2h copy SRC/etc/copyright AXIOM/lib
  ex2ht              8g1i2b extract and compile AXIOM/lib/ex2ht
  graph              8g1g2f extract psfiles from bookvol8 to AXIOM/lib/graph
    colorpoly.ps
    colorwol.ps
    drawarc.ps
    drawcolor.ps
    drawIstr.ps
    drawline.ps
    drawlines.ps
    drawpoint.ps
    draw.ps
    drawrect.ps
    drawstr.ps
    drwfilled.ps
    end.ps
    fillarc.ps
    fillpoly.ps
    fillwol.ps
    header.ps
    setup.ps
  hthits         8g1i2d extract and compile AXIOM/lib/hthits
  presea         8g1i2f extract and compile AXIOM/lib/presea
  session        8g1h2c link OBJ/SYS/sman/session.o to AXIOM/lib/session
  spadbuf        8g1i2a extract and compile AXIOM/lib/spadbuf
  spadclient     8g1h2f link OBJ/SYS/sman/spadclient.o AXIOM/lib/spadclient
  SPADEDIT
  summary        8g1d2g copy SRC/etc/summary AXIOM/lib
  view2d         8g1g2b compile and link AXIOM/lib/view2d
  view3d         8g1g2c compile and link AXIOM/lib/view3d
  viewman        8g1g2a compile and link AXIOM/lib/viewman

\end{verbatim}
\subsubsection{AXIOM/src}
\begin{verbatim}
AXIOM/src:
? algebra

\end{verbatim}
\subsubsection{AXIOM/timestamp}
\begin{verbatim}
AXIOM/timestamp
\end{verbatim}

\section{How Axiom Works}
\subsection{Input and Type Selection}
First we change the default setting for autoload messages to
turn off the noise of file loading from the library:
\begin{verbatim}
(1) -> )set mes auto off
\end{verbatim}

Next we tell the interpreter to show us the modemaps used to
classify input and select types. This is known as ``bottomup''
messages. We can watch the interpreter ponder the input.
\begin{verbatim}
(1) -> )set mes bot on
\end{verbatim}

Now we give it something nontrivial to ponder.
\begin{verbatim}
(1) -> f:=1/(a*x+b)
\end{verbatim}

After parsing the input Axiom begins to figure out the type
of the expression. In this case it starts with the multiply 
operator in the denominator.

Axiom has determined that ``a'' is of type VARIABLE and ``x''
is of type VARIABLE. It is looking for function of the form
\begin{verbatim}
    VARIABLE * VARIABLE
\end{verbatim}
so it looks in the domain of the left argument ``a'' which is
VARIABLE and does not find the required function. Similarly
it looks in the domain of the right argument ``x'' which is
VARIABLE and, not surprisingly, does not find the required function.

It tried to promote each VARIABLE to SYMBOL and looks for a way
to mulitply VARIABLES and SYMBOLS or SYMBOLS and SYMBOLS. Neither
succeeds.
\begin{verbatim}

 Function Selection for *
      Arguments: (VARIABLE a,VARIABLE x) 
   -> no appropriate * found in Variable a 
   -> no appropriate * found in Variable x 
   -> no appropriate * found in Symbol 
   -> no appropriate * found in Variable a 
   -> no appropriate * found in Variable x 
   -> no appropriate * found in Symbol 

 Modemaps from Associated Packages 
   no modemaps
\end{verbatim}

Since it cannot find a specific modemap that uses the exact types
it now expands the search to look for the general modemaps. It
searches these modemaps in order to try to find one that fits.
\begin{verbatim}

 Remaining General Modemaps 
   [1] (D,D1) -> D from D
            if D has XFALG(D2,D1) and D2 has ORDSET and D1 has RING
\end{verbatim}

The first match will fail because Symbol does not have RING. We can
determine this by asking the interpreter:
\begin{verbatim}
SYMBOL has RING

   (1)  false
                                                       Type: Boolean
\end{verbatim}

The following modemaps will fail for various similar reasons:
\begin{verbatim}
   [2] (D1,D) -> D from D
            if D has XFALG(D1,D2) and D1 has ORDSET and D2 has RING
   [3] (Integer,D) -> D from D
            if D has VECTCAT D2 and D2 has TYPE and D2 has ABELGRP
   [4] (D1,D) -> D from D
            if D has VECTCAT D1 and D1 has TYPE and D1 has MONOID
   [5] (D,D1) -> D from D
            if D has VECTCAT D1 and D1 has TYPE and D1 has MONOID
   [6] (D,D1) -> D1 from D
            if D has SMATCAT(D2,D3,D4,D1) and D3 has RING and D4 has 
            DIRPCAT(D2,D3) and D1 has DIRPCAT(D2,D3)
   [7] (D1,D) -> D1 from D
            if D has SMATCAT(D2,D3,D1,D4) and D3 has RING and D1 has 
            DIRPCAT(D2,D3) and D4 has DIRPCAT(D2,D3)
   [8] (D,D) -> D from D if D has SGROUP
   [9] (D,D1) -> D from D if D has RMODULE D1 and D1 has RNG
   [10] (D,D) -> D from D if D has MONAD
   [11] (D,D) -> D from D
            if D has MATCAT(D1,D2,D3) and D1 has RING and D2 has FLAGG 
            D1 and D3 has FLAGG D1
   [12] (D1,D) -> D from D
            if D has MATCAT(D1,D2,D3) and D1 has RING and D2 has FLAGG 
            D1 and D3 has FLAGG D1
   [13] (D,D1) -> D from D
            if D has MATCAT(D1,D2,D3) and D1 has RING and D2 has FLAGG 
            D1 and D3 has FLAGG D1
   [14] (Integer,D) -> D from D
            if D has MATCAT(D2,D3,D4) and D2 has RING and D3 has FLAGG 
            D2 and D4 has FLAGG D2
   [15] (D,D1) -> D1 from D
            if D has MATCAT(D2,D3,D1) and D2 has RING and D3 has FLAGG 
            D2 and D1 has FLAGG D2
   [16] (D1,D) -> D1 from D
            if D has MATCAT(D2,D1,D3) and D2 has RING and D1 has FLAGG 
            D2 and D3 has FLAGG D2
   [17] ((D5 -> D6),(D4 -> D5)) -> (D4 -> D6) from MappingPackage3(D4,
            D5,D6)
            if D4 has SETCAT and D5 has SETCAT and D6 has SETCAT
   [18] (D1,D) -> D from D if D has LMODULE D1 and D1 has RNG
   [19] (PolynomialIdeals(D1,D2,D3,D4),PolynomialIdeals(D1,D2,D3,D4))
             -> PolynomialIdeals(D1,D2,D3,D4)
            from PolynomialIdeals(D1,D2,D3,D4)
            if D1 has FIELD and D2 has OAMONS and D3 has ORDSET and D4 
            has POLYCAT(D1,D2,D3)
   [20] (D1,D) -> D from D
            if D has GRMOD(D1,D2) and D1 has COMRING and D2 has ABELMON
            
   [21] (D,D1) -> D from D
            if D has GRMOD(D1,D2) and D1 has COMRING and D2 has ABELMON
            
   [22] (D1,D2) -> D from D
            if D has FMCAT(D1,D2) and D1 has RING and D2 has SETCAT
   [23] (D1,D2) -> D from D
            if D has FAMONC(D2,D1) and D2 has SETCAT and D1 has CABMON
            
   [24] (Equation D1,D1) -> Equation D1 from Equation D1
            if D1 has SGROUP and D1 has TYPE
   [25] (D1,Equation D1) -> Equation D1 from Equation D1
            if D1 has SGROUP and D1 has TYPE
   [26] (D,D1) -> D from D
            if D has DIRPCAT(D2,D1) and D1 has TYPE and D1 has MONOID
         
   [27] (D1,D) -> D from D
            if D has DIRPCAT(D2,D1) and D1 has TYPE and D1 has MONOID
         
   [28] (DenavitHartenbergMatrix D2,Point D2) -> Point D2
            from DenavitHartenbergMatrix D2
            if D2 has Join(Field,TranscendentalFunctionCategory)
   [29] (PositiveInteger,Color) -> Color from Color
   [30] (DoubleFloat,Color) -> Color from Color
   [31] (CartesianTensor(D1,D2,D3),CartesianTensor(D1,D2,D3)) -> 
            CartesianTensor(D1,D2,D3)
            from CartesianTensor(D1,D2,D3)
            if D1: INT and D2: NNI and D3 has COMRING
   [32] (PositiveInteger,D) -> D from D if D has ABELSG
   [33] (NonNegativeInteger,D) -> D from D if D has ABELMON
   [34] (Integer,D) -> D from D if D has ABELGRP

\end{verbatim}

Eventually the interpreter decides that it can coerce Symbol to
Polynomial(Integer). We can do this in the interpreter also:
\begin{verbatim}
a::Symbol::POLY(INT)

   (1)  a
                                                     Type: Polynomial Integer
\end{verbatim}

And the interpreter can find multiply in POLY(INT):
\begin{verbatim}
 
 [1]  signature:   (POLY INT,POLY INT) -> POLY INT
      implemented: slot $$$ from POLY INT
 [2]  signature:   (POLY INT,POLY INT) -> POLY INT
      implemented: slot $$$ from POLY INT
\end{verbatim}

We can see this signature exists by asking the interpreter to show
us the domain POLY(INT) (truncated here for brevity):
\begin{verbatim}

)show POLY(INT)
 Polynomial Integer is a domain constructor.
 Abbreviation for Polynomial is POLY 
 This constructor is exposed in this frame.
 Issue )edit src/algebra/POLY.spad to see algebra source code for POLY 

------------------------------- Operations --------------------------------

 ?*? : (Fraction Integer,%) -> %       ?*? : (Integer,%) -> %
 ?*? : (PositiveInteger,%) -> %        ?*? : (%,Fraction Integer) -> %
 ?*? : (%,Integer) -> %                ?*? : (%,%) -> %

\end{verbatim}

Having found multipy the interpreter now starts a search for the operation
\begin{verbatim}
     (POLY(INT)) + (VARIABLE)
\end{verbatim}

It cannot find this modemap 
\begin{verbatim}

 Function Selection for +
      Arguments: (POLY INT,VARIABLE b) 
   -> no appropriate + found in Polynomial Integer 
   -> no appropriate + found in Variable b 
   -> no appropriate + found in Variable b 
 
\end{verbatim}
so it promotes VARIABLE to POLY(INT) and finds the operation:
\begin{verbatim}
   (POLY(INT)) + (POLY(INT))
\end{verbatim}

\begin{verbatim}
 [1]  signature:   (POLY INT,POLY INT) -> POLY INT
      implemented: slot $$$ from POLY INT
 
\end{verbatim}

Next it tackles the division operation where the numerator is
PI (PositiveInteger) and the denominator is POLY(INT). It tries 
to find
\begin{verbatim}
    (PI) / (POLY(INT))
\end{verbatim}

in PositiveInteger, Polynomial Integer and Integer. All attempts fail.
\begin{verbatim}

 Function Selection for /
      Arguments: (PI,POLY INT) 
   -> no appropriate / found in PositiveInteger 
   -> no appropriate / found in Polynomial Integer 
   -> no appropriate / found in Integer 
   -> no appropriate / found in PositiveInteger 
   -> no appropriate / found in Polynomial Integer 
   -> no appropriate / found in Integer 

 Modemaps from Associated Packages 
   no modemaps

\end{verbatim}

So now it turns to the general modemaps:
\begin{verbatim}
 Remaining General Modemaps 
   [1] (D,D1) -> D from D if D has VSPACE D1 and D1 has FIELD
   [2] (D,D1) -> D from D
            if D has RMATCAT(D2,D3,D1,D4,D5) and D1 has RING and D4 has
            DIRPCAT(D3,D1) and D5 has DIRPCAT(D2,D1) and D1 has FIELD
         
   [3] (D1,D1) -> D from D if D has QFCAT D1 and D1 has INTDOM
   [4] (D,D1) -> D from D
            if D has MATCAT(D1,D2,D3) and D1 has RING and D2 has FLAGG 
            D1 and D3 has FLAGG D1 and D1 has FIELD
   [5] (D,D1) -> D from D
            if D has LIECAT D1 and D1 has COMRING and D1 has FIELD
   [6] (D,D) -> D from D if D has GROUP
   [7] (SparseMultivariatePolynomial(D2,Kernel D),
            SparseMultivariatePolynomial(D2,Kernel D)) -> D
            from D if D2 has INTDOM and D2 has ORDSET and D has FS D2
         
   [8] (Float,Integer) -> Float from Float
   [9] (D,D) -> D from D if D has FIELD
   [10] (D,D) -> D from D
            if D = EQ D1 and D1 has FIELD and D1 has TYPE or D = EQ D1 
            and D1 has GROUP and D1 has TYPE
   [11] (DoubleFloat,Integer) -> DoubleFloat from DoubleFloat
   [12] (D,D1) -> D from D
            if D has AMR(D1,D2) and D1 has RING and D2 has OAMON and D1
            has FIELD
 
\end{verbatim}
it eventually promotes PI to FRAC(POLY(INT)) and
POLY(INT) to FRAC(POLY(INT)) and finds the match:
\begin{verbatim}
    (FRAC(POLY(INT))) / (FRAC(POLY(INT)))
\end{verbatim}

We can ask the intepreter to show us this operation (again, the
output is truncated for brevity):
\begin{verbatim}
 )show FRAC(POLY(INT))
 Fraction Polynomial Integer is a domain constructor.
 Abbreviation for Fraction is FRAC 
 This constructor is exposed in this frame.
 Issue )edit src/algebra/FRAC.spad to see algebra source code for FRAC 

------------------------------- Operations --------------------------------

 ?*? : (Fraction Integer,%) -> %       ?*? : (Integer,%) -> %
 ?*? : (PositiveInteger,%) -> %        ?*? : (%,Fraction Integer) -> %
 ?*? : (%,%) -> %                      ?**? : (%,Integer) -> %
 ?**? : (%,PositiveInteger) -> %       ?+? : (%,%) -> %
 ?-? : (%,%) -> %                      -? : % -> %
 ?/? : (%,%) -> %                      ?<? : (%,%) -> Boolean
\end{verbatim}

\begin{verbatim}
 [1]  signature:   (FRAC POLY INT,FRAC POLY INT) -> FRAC POLY INT
      implemented: slot $$$ from FRAC POLY INT
 
\end{verbatim}

At this point the interpreter has succeeded in finding a type for
the expression and eventually returns the result badged with the
appropriate type:
\begin{verbatim}

           1
   (1)  -------
        a x + b
                                            Type: Fraction Polynomial Integer

\end{verbatim}
\subsection{A simple integral}

Now we will show an integration with successive levels of expansion
of explanation. We will use the expression above:
\begin{verbatim}
(1) -> f:=1/(a*x+b)

           1
   (1)  -------
        a x + b
                                            Type: Fraction Polynomial Integer
(2) -> integrate(f,x)

        log(a x + b)
   (2)  ------------
              a
                                          Type: Union(Expression Integer,...)

\end{verbatim}
\subsection{A simple integral, expansion 1 interpreter}

\begin{verbatim}
(2) -> integrate(f,x)
\end{verbatim}

Here we assume the previous discussion of modemap handling for the
expression f and we only look at the modemap handling for the integrate
function. We are looking for a modemap of the form:

\begin{verbatim}
      integrate(FRAC(POLY(INT)),VARIABLE x)
\end{verbatim}

So first we look in the domains of the arguments, that is, in
Fraction Polynomial Integer, and Variable. Neither one succeeds:
\begin{verbatim}

 Function Selection for integrate
      Arguments: (FRAC POLY INT,VARIABLE x) 
   -> no appropriate integrate found in Fraction Polynomial Integer 
   -> no appropriate integrate found in Variable x 
   -> no appropriate integrate found in Fraction Polynomial Integer 
   -> no appropriate integrate found in Variable x 

 Modemaps from Associated Packages 
   no modemaps

\end{verbatim}

Next we look at the general modemaps to find one that might work:
\begin{verbatim}
 Remaining General Modemaps 
   [1] (D,D1) -> D from D
            if D1 = SYMBOL and D has UTSCAT D2 and D2 has RING and D2 
            has ACFS INT and D2 has PRIMCAT and D2 has TRANFUN and D2 
            has ALGEBRA FRAC INT or D1 = SYMBOL and D has UTSCAT D2 and
            D2 has RING and D2 has variables: D2 -> List D1 and D2 has 
            integrate: (D2,D1) -> D2 and D2 has ALGEBRA FRAC INT
   [2] (D,D1) -> D from D
            if D1 = SYMBOL and D has UPXSCAT D2 and D2 has RING and D2 
            has ACFS INT and D2 has PRIMCAT and D2 has TRANFUN and D2 
            has ALGEBRA FRAC INT or D1 = SYMBOL and D has UPXSCAT D2 
            and D2 has RING and D2 has variables: D2 -> List D1 and D2 
            has integrate: (D2,D1) -> D2 and D2 has ALGEBRA FRAC INT
         
   [3] (D,D1) -> D from D
            if D1 = SYMBOL and D has ULSCAT D2 and D2 has RING and D2 
            has ACFS INT and D2 has PRIMCAT and D2 has TRANFUN and D2 
            has ALGEBRA FRAC INT or D1 = SYMBOL and D has ULSCAT D2 and
            D2 has RING and D2 has variables: D2 -> List D1 and D2 has 
            integrate: (D2,D1) -> D2 and D2 has ALGEBRA FRAC INT
   [4] (Polynomial D2,Symbol) -> Polynomial D2 from Polynomial D2
            if D2 has ALGEBRA FRAC INT and D2 has RING
   [5] (D,D1) -> D from D
            if D has MTSCAT(D2,D1) and D2 has RING and D1 has ORDSET 
            and D2 has ALGEBRA FRAC INT
   [6] (Fraction Polynomial D4,Symbol) -> Union(Expression D4,List 
            Expression D4)
            from IntegrationResultRFToFunction D4
            if D4 has CHARZ and D4 has Join(GcdDomain,RetractableTo 
            Integer,OrderedSet,LinearlyExplicitRingOver Integer)
   [7] (Expression Float,List Segment OrderedCompletion Float) -> 
            Result
            from AnnaNumericalIntegrationPackage
   [8] (Expression Float,Segment OrderedCompletion Float) -> Result
            from AnnaNumericalIntegrationPackage
   [9] (GeneralUnivariatePowerSeries(D2,D3,D4),Variable D3) -> 
            GeneralUnivariatePowerSeries(D2,D3,D4)
            from GeneralUnivariatePowerSeries(D2,D3,D4)
            if D3: SYMBOL and D2 has ALGEBRA FRAC INT and D2 has RING 
            and D4: D2
   [10] (D2,Symbol) -> Union(D2,List D2) from FunctionSpaceIntegration(
            D4,D2)
            if D4 has Join(EuclideanDomain,OrderedSet,
            CharacteristicZero,RetractableTo Integer,
            LinearlyExplicitRingOver Integer) and D2 has Join(
            TranscendentalFunctionCategory,PrimitiveFunctionCategory,
            AlgebraicallyClosedFunctionSpace D4)
   [11] (Fraction Polynomial D4,SegmentBinding OrderedCompletion 
            Fraction Polynomial D4) -> Union(f1: OrderedCompletion Expression
            D4,f2: List OrderedCompletion Expression D4,fail: failed,
            pole: potentialPole)
            from RationalFunctionDefiniteIntegration D4
            if D4 has Join(EuclideanDomain,OrderedSet,
            CharacteristicZero,RetractableTo Integer,
            LinearlyExplicitRingOver Integer)
   [12] (Fraction Polynomial D4,SegmentBinding OrderedCompletion 
            Expression D4) -> Union(f1: OrderedCompletion Expression D4,f2: 
            List OrderedCompletion Expression D4,fail: failed,pole: 
            potentialPole)
            from RationalFunctionDefiniteIntegration D4
            if D4 has Join(EuclideanDomain,OrderedSet,
            CharacteristicZero,RetractableTo Integer,
            LinearlyExplicitRingOver Integer)
   [13] (D2,SegmentBinding OrderedCompletion D2) -> Union(f1: 
            OrderedCompletion D2,f2: List OrderedCompletion D2,fail: failed,
            pole: potentialPole)
            from ElementaryFunctionDefiniteIntegration(D4,D2)
            if D2 has Join(TranscendentalFunctionCategory,
            PrimitiveFunctionCategory,AlgebraicallyClosedFunctionSpace 
            D4) and D4 has Join(EuclideanDomain,OrderedSet,
            CharacteristicZero,RetractableTo Integer,
            LinearlyExplicitRingOver Integer)

\end{verbatim}
Modemap [6] wins because we can construct the first argument
by matching 
\begin{verbatim}
  Fraction Polynomial Integer
\end{verbatim}
to
\begin{verbatim}
  Fraction Polynomial D4
\end{verbatim}
so we can infer that D4 == Integer
\begin{verbatim}

   [6] (Fraction Polynomial D4,Symbol) -> Union(Expression D4,List 
            Expression D4)
            from IntegrationResultRFToFunction D4
            if D4 has CHARZ and D4 has Join(GcdDomain,RetractableTo 
            Integer,OrderedSet,LinearlyExplicitRingOver Integer)

\end{verbatim}

Given that match we have two requirements on Integer, both of which
we can check with the interpreter:

\begin{verbatim}
INT has CHARZ

   (3)  true
                                                    Type: Boolean
(4) -> INT has Join(GcdDomain,RetractableTo Integer,OrderedSet,_
                     LinearlyExplicitRingOver Integer)

   (4)  true
                                                    Type: Boolean
\end{verbatim}

So we have a match
\begin{verbatim}
 
 [1]  signature:   (FRAC POLY INT,SYMBOL) -> Union(EXPR INT,LIST EXPR INT)
      implemented: slot (Union (Expression (Integer))
                         (List (Expression (Integer))))
                         (Fraction (Polynomial (Integer)))(Symbol) 
                   from IRRF2F INT
 [2]  signature:   (EXPR INT,SYMBOL) -> Union(EXPR INT,LIST EXPR INT)
      implemented: slot (Union (Expression (Integer)) 
                         (List (Expression (Integer))))
                         (Expression (Integer))(Symbol) 
                   from FSINT(INT,EXPR INT)
 
\end{verbatim}

Now we invoke 
\begin{verbatim}
    integrate(FRAC(POLY(INT)),SYMBOL) -> Union(EXPR INT,LIST EXPR INT)
      from IRRF2F(INT)

    integrate(1/(a*x+b),x)

\end{verbatim}
can print the result:
\begin{verbatim}

        log(a x + b)
   (2)  ------------
              a
                                          Type: Union(Expression Integer,...)

\end{verbatim}
\subsection{A simple integral, expansion 2 integrate}
Now that we know how the interpreter has matched the input and called
the function we need to follow the first level call into the function.

Axiom provides a trace tool that will allow us to walk into the function
invocation and watch what happens. We will follow this same invocation
path many times, each time we will descend another layer, repeating the
information as we do.

For now, we look at the domain IRRF2F from irexpand.spad.
The categorical definition of this domain reads (we remove parts
of the definition for brevity):
\begin{verbatim}
IntegrationResultRFToFunction(R): Exports == Implementation where
  R: Join(GcdDomain, RetractableTo Integer, OrderedSet,
           LinearlyExplicitRingOver Integer)

  RF  ==> Fraction Polynomial R
  F   ==> Expression R
  IR  ==> IntegrationResult RF
  OF  ==> OutputForm

  Exports ==> with
    expand          : IR -> List F
       ++ expand(i) returns the list of possible real functions
       ++ corresponding to i.
    if R has CharacteristicZero then
      integrate       : (RF, Symbol) -> Union(F, List F)
        ++ integrate(f, x) returns the integral of \spad{f(x)dx}
        ++ where x is viewed as a real variable..

  Implementation ==> add
    import IntegrationTools(R, F)
    import TrigonometricManipulations(R, F)
    import IntegrationResultToFunction(R, F)

    toEF: IR -> IntegrationResult F

    toEF i          == map(#1::F, i)$IntegrationResultFunctions2(RF, F)
    expand i        == expand toEF i
    complexExpand i == complexExpand toEF i

    if R has CharacteristicZero then
      import RationalFunctionIntegration(R)

      if R has imaginary: () -> R then 
        integrate(f, x) == complexIntegrate(f, x)
      else
        integrate(f, x) ==
          l := [mkPrim(real g, x) for g in expand internalIntegrate(f, x)]
          empty? rest l => first l
          l

@
\end{verbatim}

We can see that this domain constructor takes one argument which, in
this case, is Integer. We've already determined that Integer has the
required Joins:
\begin{verbatim}
(4) -> INT has Join(GcdDomain,RetractableTo Integer,OrderedSet,_
                     LinearlyExplicitRingOver Integer)

   (4)  true
                                                    Type: Boolean
\end{verbatim}
and we can see that:
\begin{verbatim}
(5) -> INT has CharacteristicZero

   (5)  true
                                                                Type: Boolean
\end{verbatim}
so we can match the signature of integrate:
\begin{verbatim}
  integrate(Fraction Polynomial Integer, Symbol) -> 
       Union(Expression Integer, List Expression Integer)
\end{verbatim}

We can trace this domain and ask to see the output in math form:
\begin{verbatim}
(6) -> )trace IRRF2F )math
 
   Packages traced: 
      IntegrationResultRFToFunction Integer
   Parameterized constructors traced:
      IRRF2F
\end{verbatim}
and now, when we do the integration, we see the output of the trace:
\begin{verbatim}
integrate(1/(a*x+b),x)
1<enter IntegrationResultRFToFunction.integrate,32 : 
          1
 arg1= -------
       a x + b
 arg2= x
 1<enter IntegrationResultRFToFunction.expand,18 : 
        1     a x + b
  arg1= - log(-------)
        a        a
 1>exit  IntegrationResultRFToFunction.expand,18 : 
       a x + b
   log(-------)
          a
  [------------]
         a
1>exit  IntegrationResultRFToFunction.integrate,32 : 
 log(a x + b)
 ------------
       a

        log(a x + b)
   (6)  ------------
              a
                                          Type: Union(Expression Integer,...)
\end{verbatim}

From this we learn that the arguments to integrate are exactly the
arguments we supplied and we know the exact types of the arguments
because they have to match the signature of the function:
\begin{verbatim}
1<enter IntegrationResultRFToFunction.integrate,32 : 
  integrate(, Symbol) -> 
          1
 arg1= -------    <== Fraction Polynomial Integer
       a x + b
 arg2= x          <== Symbol

\end{verbatim}
and returns the result
\begin{verbatim}
1>exit  IntegrationResultRFToFunction.integrate,32 : 
 log(a x + b)
 ------------     <== Union(Expression Integer, List Expression Integer)
       a
\end{verbatim}
\subsection{A simple integral, expansion 2 internalIntegrate}
If we look at the function definition for integrate:
\begin{verbatim}
        integrate(f, x) ==
          l := [mkPrim(real g, x) for g in expand internalIntegrate(f, x)]
          empty? rest l => first l
          l
\end{verbatim}
we can see that there is a call to the function 
\begin{verbatim}
  internalIntegrate(f, x)
\end{verbatim}
and we can compute the types of the arguments since they are exactly
the types of the integrate function itself:
\begin{verbatim}
  internalIntegrate(Fraction Polynomial Integer, Symbol) 
\end{verbatim}
and since the return value will be fed to the expand function we
can look at the signature of expand:
\begin{verbatim}
    expand: IntegrationResult Fraction Polynomial Integer ->
               List Expression Integer
\end{verbatim}
and we can get the full signature for internalIntegrate:
\begin{verbatim}
  internalIntegrate(Fraction Polynomial Integer, Symbol) ->
    IntegrationResult Fraction Polynomial Integer
\end{verbatim}

This comes from the domain
\begin{verbatim}
 RationalFunctionIntegration(F): Exports == Implementation where
   F: Join(IntegralDomain, RetractableTo Integer, CharacteristicZero)
\end{verbatim}
where F is Integer.
\begin{verbatim}
  SE  ==> Symbol
  P   ==> Polynomial F
  Q   ==> Fraction P
  UP  ==> SparseUnivariatePolynomial Q
  QF  ==> Fraction UP
  LGQ ==> List Record(coeff:Q, logand:Q)
  UQ  ==> Union(Record(ratpart:Q, coeff:Q), "failed")
  ULQ ==> Union(Record(mainpart:Q, limitedlogs:LGQ), "failed")

  Exports ==> with
    internalIntegrate: (Q, SE) -> IntegrationResult Q
       ++ internalIntegrate(f, x) returns g such that \spad{dg/dx = f}.
  Implementation ==> add
    import RationalIntegration(Q, UP)
    import IntegrationResultFunctions2(QF, Q)
    import PolynomialCategoryQuotientFunctions(IndexedExponents SE,
                                                       SE, F, P, Q)
    internalIntegrate(f, x) ==
      map(multivariate(#1, x), integrate univariate(f, x))

\end{verbatim}
If we look the signature for internalIntegrate and expand it we see:
\begin{verbatim}
    internalIntegrate: (Q, SE) -> IntegrationResult Q

    internalIntegrate: ( Fraction Polynomial Integer, Symbol) ->
      IntegrationResult Fraction Polynomial Integer

\end{verbatim}
which is exactly what we need. When we look at the function we see:
\begin{verbatim}
    internalIntegrate(f, x) ==
      map(multivariate(#1, x), integrate univariate(f, x))
\end{verbatim}
We can watch the function call by tracing INTRF:
\begin{verbatim}
(7) -> )trace INTRF )math
 
   Packages traced: 
      IntegrationResultRFToFunction Integer, 
            RationalFunctionIntegration Integer
   Parameterized constructors traced:
      IRRF2F, INTRF
\end{verbatim}
and we see:
\begin{verbatim}
(7) -> integrate(1/(a*x+b),x)
1<enter IntegrationResultRFToFunction.integrate,32 : 
          1
 arg1= -------
       a x + b
 arg2= x
 1<enter RationalFunctionIntegration.internalIntegrate,25 : 
           1
  arg1= -------
        a x + b
  arg2= x
 1>exit  RationalFunctionIntegration.internalIntegrate,25 : 
  1     a x + b
  - log(-------)
  a        a
 1<enter IntegrationResultRFToFunction.expand,18 : 
        1     a x + b
  arg1= - log(-------)
        a        a
 1>exit  IntegrationResultRFToFunction.expand,18 : 
       a x + b
   log(-------)
          a
  [------------]
         a
1>exit  IntegrationResultRFToFunction.integrate,32 : 
 log(a x + b)
 ------------
       a

        log(a x + b)
   (7)  ------------
              a
                                          Type: Union(Expression Integer,...)
\end{verbatim}
Now we see that internalIntegrate was called with the arguments
\begin{verbatim}
 1<enter RationalFunctionIntegration.internalIntegrate,25 : 
           1
  arg1= -------      <== Fraction Polynomial Integer
        a x + b
  arg2= x            <== Symbol
\end{verbatim}
and returned the values:
\begin{verbatim}
 1>exit  RationalFunctionIntegration.internalIntegrate,25 : 
  1     a x + b
  - log(-------)     <== IntegrationResult Fraction Polynomial Integer
  a        a
\end{verbatim}
\subsection{A simple integral, expansion 3 univariate}
But the internalIntegrate function does its work by calling yet
other functions, the deepest of which is univariate:
\begin{verbatim}
    internalIntegrate(f, x) ==
      map(multivariate(#1, x), integrate univariate(f, x))
\end{verbatim}
Since univariate uses the arguments to the internalIntegrate function
which has the signature:
\begin{verbatim}
    internalIntegrate: ( Fraction Polynomial Integer, Symbol) ->
\end{verbatim}
we can determine that we need a univariate function with the signature:
\begin{verbatim}
    univariate: ( Fraction Polynomial Integer, Symbol) ->
\end{verbatim}
This function is found in PolynomialCategoryQuotientFunctions, POLYCATQ
which has the form:
\begin{verbatim}
PolynomialCategoryQuotientFunctions(E, V, R, P, F):
 Exports == Implementation where
  E: OrderedAbelianMonoidSup
  V: OrderedSet
  R: Ring
  P: PolynomialCategory(R, E, V)
  F: Field with
    coerce: P -> %
    numer : % -> P
    denom : % -> P

  UP  ==> SparseUnivariatePolynomial F
  RF  ==> Fraction UP

  Exports ==> with
    variables   : F -> List V
      ++ variables(f) returns the list of variables appearing
      ++ in the numerator or the denominator of f.
    mainVariable: F -> Union(V, "failed")
      ++ mainVariable(f) returns the highest variable appearing
      ++ in the numerator or the denominator of f, "failed" if
      ++ f has no variables.
    univariate  : (F, V) -> RF
      ++ univariate(f, v) returns f viewed as a univariate
      ++ rational function in v.
  Implementation ==> add
    P2UP: (P, V) -> UP

    univariate(f, x) == P2UP(numer f, x) / P2UP(denom f, x)

    P2UP(p, x) ==
      map(#1::F,
          univariate(p, x))$SparseUnivariatePolynomialFunctions2(P, F)
\end{verbatim}

So we are calling the function:
\begin{verbatim}
 univariate: ( Fraction Polynomial Integer, Symbol) ->
  Fraction SparseUnivariatePolynomial Field with
   coerce: PolynomialCategory(Ring, OrderedAbelianMonoidSup, OrderedSet) -> %
   numer: % -> PolynomialCategory(Ring, OrderedAbelianMonoidSup, OrderedSet)
   denom: % -> PolynomialCategory(Ring, OrderedAbelianMonoidSup, OrderedSet)
\end{verbatim}
which we can see by tracing that domain:
\begin{verbatim}
(8) -> )trace POLYCATQ )math
 
   Packages traced: 
      IntegrationResultRFToFunction Integer, 
            RationalFunctionIntegration Integer, 
            PolynomialCategoryQuotientFunctions(IndexedExponents 
            Kernel Expression Integer,Kernel Expression Integer,
            Integer,SparseMultivariatePolynomial(Integer,Kernel 
            Expression Integer),Expression Integer), 
            PolynomialCategoryQuotientFunctions(IndexedExponents 
            Symbol,Symbol,Integer,Polynomial Integer,Fraction 
            Polynomial Integer)
   Parameterized constructors traced:
      IRRF2F, INTRF, POLYCATQ
\end{verbatim}
which gives the input:
\begin{verbatim}
  1<enter PolynomialCategoryQuotientFunctions.univariate,16 : 
            1
   arg1= -------  <== Fraction Polynomial Integer
         a x + b
   arg2= x        <== Symbol
\end{verbatim}
and the output
\begin{verbatim}
  1>exit  PolynomialCategoryQuotientFunctions.univariate,16 : 
     1
     -
     a
   -----   <==   Fraction SparseUnivariatePolynomial Field with
       b                   coerce: P -> % 
   ? + -                   numer:  % -> P
       a                   denom:  % -> P
\end{verbatim}

It should be clear that univariate divided the numerator and
denominator by the leading coefficient of the polynomial in
the denominator. It also replaced ``x'' with the variable ``?''.
\subsection{A simple integral, expansion 4 integrate}
When univariate returns, the results are fed to another integrate,
this time from RationalIntegration (INTRAT). This domain looks like:
\begin{verbatim}
RationalIntegration(F, UP): Exports == Implementation where
  F : Join(Field, CharacteristicZero, RetractableTo Integer)
  UP: UnivariatePolynomialCategory F

  RF  ==> Fraction UP
  IR  ==> IntegrationResult RF
  LLG ==> List Record(coeff:RF, logand:RF)
  URF ==> Union(Record(ratpart:RF, coeff:RF), "failed")
  U   ==> Union(Record(mainpart:RF, limitedlogs:LLG), "failed")
  OF  ==> OutputForm

  Exports ==> with
    integrate  : RF -> IR
      ++ integrate(f) returns g such that \spad{g' = f}.

  Implementation ==> add
    import TranscendentalIntegration(F, UP)

    integrate f ==
      rec := monomialIntegrate(f, differentiate)
      integrate(rec.polypart)::RF::IR + rec.ir

\end{verbatim}

This domain was constructed and ``brought into scope'' in
RationalFunctionIntegration(F) with the statement
\begin{verbatim}
 import RationalIntegration(Fraction Polynomial Integer,
         SparseUnivariatePolynomial Fraction Polynomial Integer)
\end{verbatim}
and the function has the signature
\begin{verbatim}
 integrate:
   Fraction SparseUnivariatePolynomial Fraction Polynomial Integer ->
    IntegrationResult Fraction 
                      Fraction Polynomial Integer
\end{verbatim}
\begin{verbatim}
  1<enter RationalIntegration.integrate,32 : 
           1
           -
           a
   arg1= -----  <== Fraction SparseUnivariatePolynomial
             b                 Fraction Polynomial Integer
         ? + -
             a
  1>exit  RationalIntegration.integrate,32 : 
   1         b
   - log(? + -) <== IntegrationResult Fraction SparseUnivariatePolynomial
   a         a                 Fraction Polynomial Integer
\end{verbatim}

\subsection{A simple integral, expansion 5 monomialIntegrate}
The integrate function is defined as:
\begin{verbatim}
 integrate f ==
   print(outputForm("tpdhere INTRAT 1")@OF)$OF
   rec := monomialIntegrate(f, differentiate)
   integrate(rec.polypart)::RF::IR + rec.ir
\end{verbatim}
Notice that while ``f'' is an argument to integrate, the ``differentiate''
function is a free variable. The Axiom compiler will look at all of the
symbols ``in scope'' to find its meaning. This code does an import:
\begin{verbatim}
  import TranscendentalIntegration(Fraction Polynomial Integer,
          SparseUnivariatePolynomial Fraction Polynomial Integer)
\end{verbatim}
which exports monomialIntegrate
\begin{verbatim}
TranscendentalIntegration(F, UP): Exports == Implementation where
  F  : Field
  UP : UnivariatePolynomialCategory F

  RF  ==> Fraction UP
  FF  ==> Record(ratpart:F,  coeff:F)
  UF  ==> Union(FF,  "failed")
  IR  ==> IntegrationResult RF
  REC ==> Record(ir:IR, specpart:RF, polypart:UP)

  Exports ==> with
    monomialIntegrate : (RF, UP -> UP) -> REC
      ++ monomialIntegrate(f, ') returns \spad{[ir, s, p]} such that
      ++ \spad{f = ir' + s + p} and all the squarefree factors of the
      ++ denominator of s are special w.r.t the derivation '.

 Implementation ==> add
  import SubResultantPackage(UP, UP2)
  import MonomialExtensionTools(F, UP)
  import TranscendentalHermiteIntegration(F, UP)
  import CommuteUnivariatePolynomialCategory(F, UP, UP2)

  monomialIntegrate(f, derivation) ==
    zero? f => [0, 0, 0]
    r := HermiteIntegrate(f, derivation)
    zero?(inum := numer(r.logpart)) => 
       [r.answer::IR, r.specpart, r.polypart]
    iden  := denom(r.logpart)
    x := monomial(1, 1)$UP
    resultvec := subresultantVector(UP2UP2 inum -
                     (x::UP2) * UP2UP2 derivation iden, UP2UP2 iden)
    respoly := primitivePart leadingCoefficient resultvec 0
    rec := splitSquarefree(respoly, kappa(#1, derivation))
    logs:List(LOG) := [
        [1, UP2UPR(term.factor),
         UP22UPR swap primitivePart(resultvec(term.exponent),term.factor)]
               for term in factors(rec.special)]
    dlog :=
         ((derivation x) = 1) => r.logpart
         differentiate(mkAnswer(0, logs, empty()),
                       differentiate(#1, derivation))
    (u := retractIfCan(p := r.logpart - dlog)@Union(UP, "failed")) case UP =>
      [mkAnswer(r.answer, logs, empty), r.specpart, r.polypart + u::UP]
    [mkAnswer(r.answer, logs, [[p, dummy]]), r.specpart, r.polypart]

\end{verbatim}
which expands into the type signature:
\begin{verbatim}

  monomialIntegrate:
  (Fraction SparseUnivariatePolynomial Fraction Polynomial Integer,
     SparseUnivariatePolynomial Fraction Polynomial Integer ->
          SparseUnivariatePolynomial Fraction Polynomial Integer) ->
    Record(ir: IntegrationResult Fraction 
                     SparseUnivariatePolynomial Fraction Polynomial Integer,
           specpart: Fraction  
                     SparseUnivariatePolynomial Fraction Polynomial Integer,
           polypart: SparseUnivariatePolynomial Fraction Polynomial Integer)
      ++ monomialIntegrate(f, ') returns \spad{[ir, s, p]} such that
      ++ \spad{f = ir' + s + p} and all the squarefree factors of the
      ++ denominator of s are special w.r.t the derivation '.
\end{verbatim}
we can watch this happen:
\begin{verbatim}
)trace INTTR )math
 
   Function traced: UnivariatePolynomialCategory 
   Packages traced: 
      IntegrationResultRFToFunction Integer, 
            RationalFunctionIntegration Integer, 
            RationalIntegration(Fraction Polynomial Integer,
            SparseUnivariatePolynomial Fraction Polynomial 
            Integer), PolynomialCategoryQuotientFunctions(
            IndexedExponents Kernel Expression Integer,Kernel 
            Expression Integer,Integer,
            SparseMultivariatePolynomial(Integer,Kernel 
            Expression Integer),Expression Integer), 
            PolynomialCategoryQuotientFunctions(IndexedExponents 
            Symbol,Symbol,Integer,Polynomial Integer,Fraction 
            Polynomial Integer), TranscendentalIntegration(
            Fraction Polynomial Integer,
            SparseUnivariatePolynomial Fraction Polynomial 
            Integer)
   Parameterized constructors traced:
      IRRF2F, INTRF, INTRAT, POLYCATQ, INTTR
\end{verbatim}
and we can watch the monomialIntegrate function call
\begin{verbatim}
(34) -> integrate(1/(a*x+b),x)
1<enter IntegrationResultRFToFunction.integrate,32 : 
          1
 arg1= -------
       a x + b
 arg2= x
   "tpdhere IRRF2F 1"
 1<enter RationalFunctionIntegration.internalIntegrate,25 : 
           1
  arg1= -------
        a x + b
  arg2= x
  1<enter PolynomialCategoryQuotientFunctions.univariate,16 : 
            1
   arg1= -------
         a x + b
   arg2= x
  1>exit  PolynomialCategoryQuotientFunctions.univariate,16 : 
     1
     -
     a
   -----
       b
   ? + -
       a
  1<enter RationalIntegration.integrate,32 : 
           1
           -
           a
   arg1= -----  <== Fraction SparseUnivariatePolynomial
             b                 Fraction Polynomial Integer
         ? + -
             a
   1<enter TranscendentalIntegration.monomialIntegrate,81 : 
            1
            -
            a
    arg1= -----  <== Fraction SparseUnivariatePolynomial
              b                 Fraction Polynomial Integer
          ? + -
              a
    arg2= theMap(UPOLYC-;differentiate;2S;37,873)
   1>exit  TranscendentalIntegration.monomialIntegrate,81 : 
         1         b
    [ir= - log(? + -),specpart= 0,polypart= 0]
         a         a
  1>exit  RationalIntegration.integrate,32 : 
   1         b
   - log(? + -)
   a         a
 1>exit  RationalFunctionIntegration.internalIntegrate,25 : 
  1     a x + b
  - log(-------)
  a        a
1>exit  IntegrationResultRFToFunction.integrate,32 : 
 log(a x + b)
 ------------
       a

         log(a x + b)
   (34)  ------------
               a
                                          Type: Union(Expression Integer,...)
(35) -> 
\end{verbatim}
\subsection{A simple integral, expansion 6 HermiteIntegrate}
Since ``f'' is not zero we invoke HermiteIntegrate from the domain
TranscendentalHermiteIntegration which looks like:
\begin{verbatim}
TranscendentalHermiteIntegration(F, UP): Exports == Implementation where
  F  : Field
  UP : UnivariatePolynomialCategory F

  N   ==> NonNegativeInteger
  RF  ==> Fraction UP
  REC ==> Record(answer:RF, lognum:UP, logden:UP)
  HER ==> Record(answer:RF, logpart:RF, specpart:RF, polypart:UP)

  Exports ==> with
    HermiteIntegrate: (RF, UP -> UP) -> HER
         ++ HermiteIntegrate(f, D) returns \spad{[g, h, s, p]}
         ++ such that \spad{f = Dg + h + s + p},
         ++ h has a squarefree denominator normal w.r.t. D,
         ++ and all the squarefree factors of the denominator of s are
         ++ special w.r.t. D. Furthermore, h and s have no polynomial parts.
         ++ D is the derivation to use on \spadtype{UP}.

  Implementation ==> add
    import MonomialExtensionTools(F, UP)

    HermiteIntegrate(f, derivation) ==
      rec := decompose(f, derivation)
      hi  := normalHermiteIntegrate(rec.normal, derivation)
      qr  := divide(hi.lognum, hi.logden)
      [hi.answer, qr.remainder / hi.logden, rec.special, qr.quotient + rec.poly]
\end{verbatim}
The function has the same input signature as monomialIntegrate but a 
different return signature.
\begin{verbatim}
  HermiteIntegrate:
  (Fraction SparseUnivariatePolynomial Fraction Polynomial Integer,
     SparseUnivariatePolynomial Fraction Polynomial Integer ->
          SparseUnivariatePolynomial Fraction Polynomial Integer) ->
   Record(answer:Fraction SparseUnivariatePolynomial 
                                                 Fraction Polynomial Integer,
          logpart:Fraction SparseUnivariatePolynomial 
                                                 Fraction Polynomial Integer,
          specpart:Fraction SparseUnivariatePolynomial 
                                                 Fraction Polynomial Integer, 
          polypart:SparseUnivariatePolynomial Fraction Polynomial Integer)
\end{verbatim}
so we trace this domain
\begin{verbatim}
(37) -> )trace INTHERTR )math
 
   Function traced: UnivariatePolynomialCategory 
   Packages traced: 
      IntegrationResultRFToFunction Integer, 
            RationalFunctionIntegration Integer, 
            RationalIntegration(Fraction Polynomial Integer,
            SparseUnivariatePolynomial Fraction Polynomial 
            Integer), PolynomialCategoryQuotientFunctions(
            IndexedExponents Kernel Expression Integer,Kernel 
            Expression Integer,Integer,
            SparseMultivariatePolynomial(Integer,Kernel 
            Expression Integer),Expression Integer), 
            PolynomialCategoryQuotientFunctions(IndexedExponents 
            Symbol,Symbol,Integer,Polynomial Integer,Fraction 
            Polynomial Integer), TranscendentalIntegration(
            Fraction Polynomial Integer,
            SparseUnivariatePolynomial Fraction Polynomial 
            Integer), TranscendentalHermiteIntegration(Fraction 
            Polynomial Integer,SparseUnivariatePolynomial 
            Fraction Polynomial Integer)
   Parameterized constructors traced:
      IRRF2F, INTRF, INTRAT, POLYCATQ, INTTR, INTHERTR
\end{verbatim}
and now we see
\begin{verbatim}
(38) -> integrate(1/(a*x+b),x)
1<enter IntegrationResultRFToFunction.integrate,32 : 
          1
 arg1= -------
       a x + b
 arg2= x
   "tpdhere IRRF2F 1"
 1<enter RationalFunctionIntegration.internalIntegrate,25 : 
           1
  arg1= -------
        a x + b
  arg2= x
  1<enter RationalIntegration.integrate,32 : 
           1
           -
           a
   arg1= -----
             b
         ? + -
             a
   "tpdhere INTRAT 1"
   1<enter TranscendentalIntegration.monomialIntegrate,81 : 
            1
            -
            a
    arg1= -----
              b
          ? + -
              a
    arg2= theMap(UPOLYC-;differentiate;2S;37,873)
    1<enter TranscendentalHermiteIntegration.HermiteIntegrate,18 : 
             1
             -
             a
     arg1= -----
               b
           ? + -
               a
     arg2= theMap(UPOLYC-;differentiate;2S;37,873)
    1>exit  TranscendentalHermiteIntegration.HermiteIntegrate,18 : 
                           1
                           -
                           a
     [answer= 0,logpart= -----,specpart= 0,polypart= 0]
                             b
                         ? + -
                             a
   1>exit  TranscendentalIntegration.monomialIntegrate,81 : 
         1         b
    [ir= - log(? + -),specpart= 0,polypart= 0]
         a         a
   "tpdhere UPOLYC 1"
  1>exit  RationalIntegration.integrate,32 : 
   1         b
   - log(? + -)
   a         a
 1>exit  RationalFunctionIntegration.internalIntegrate,25 : 
  1     a x + b
  - log(-------)
  a        a
 1<enter IntegrationResultRFToFunction.expand,18 : 
        1     a x + b
  arg1= - log(-------)
        a        a
 1>exit  IntegrationResultRFToFunction.expand,18 : 
       a x + b
   log(-------)
          a
  [------------]
         a
1>exit  IntegrationResultRFToFunction.integrate,32 : 
 log(a x + b)
 ------------
       a

         log(a x + b)
   (38)  ------------
               a
                                          Type: Union(Expression Integer,...)
\end{verbatim}
so HermiteIntegrate did nothing to the input.
Next we call normalHermiteIntegrate which is a local function

\section{Tools}
\subsection{svn}
SVN is a source control system on all platforms. Axiom 'silver' is
maintained in an SVN archive on sourceforge. This can be pulled from:
\begin{verbatim}
svn co https://axiom.svn.sf.net/svnroot/axiom/trunk/axiom axiom
\end{verbatim}
\subsection{git}
Git is a unix-based source code control system. Axiom 'silver' is 
maintained in a git archive. This can be pulled from:
\begin{verbatim}
git-clone ssh://git@axiom-developer.org/home/git/silver
\end{verbatim}
the password for the userid git is linus.

\subsection{cvs}
This assumes that you have set up ssh on the Savannah site.
CVS does not use a password. You have to log onto the Savannah
site and set up a public key. This requires you to:
\begin{itemize}
\item set up a local public key:
ssh-keygen -b 1024 -t rsa1
\item open a browser
\item nagivate to the savannah page that has your personal keys
\item open .ssh/identity.pub
\item cut .ssh/identity.pub
\item paste it into your personal key list on savannah
\item go have a beer (the page takes an hour or two to update)
\end{itemize}

Once you have a working key you can do the cvs login.
If it prompts you for a password then the key is not working.
If it prompts you to ``Enter the passphrase for RSA key'' then 
cvs login will work.

I maintain a directory where I work (call this WORK)
\begin{verbatim}
  /home/axiomgnu/new
\end{verbatim}
and a directory for CVS (call this GOLD)
\begin{verbatim}
  /axiom
\end{verbatim}

When I want to export a set of changes I do the following steps:

\noindent
0) MAKE SURE THE ~/.ssh/config FILE IS CORRECT:
\begin{verbatim}
    (you should only need to do this once. 
     you need to change the User= field)

Host *.gnu.org
  Protocol=1
  Compression=yes
  CompressionLevel=3
  User=axiom
  StrictHostKeyChecking=no
  PreferredAuthentications=publickey,password
  NumberOfPasswordPrompts=2
\end{verbatim}

\noindent
1) MAKE SURE THE SHELL VARIABLES ARE OK: 
\begin{verbatim}
    (normally set in .bashrc)

export CVS_RSH=ssh
export CVSROOT=:pserver:axiom@subversions.gnu.org:/cvsroot/axiom
                        ^^^^^
                        change this to your id
\end{verbatim}

\noindent
2) MAKE SURE YOU'RE LOGGED IN:
\begin{verbatim}
 (I keep a session open all the time but it doesn't seem to care
  if you login again. i'm not sure what login does, actually)

cvs login
\end{verbatim}

\noindent
3) GET A FRESH COPY FOR THE FIRST TIME OR AT ANY TIME:
\begin{verbatim}
 (you only need to do this the first time but you can erase
  your whole axiom subtree and refresh it again doing this.
 
  note that i work as root so i can update /.  Most rational
  people are smarter than me and work as a regular user so
  you have to change the instructions for cd. But you knew that)

cd /
cvs co axiom
\end{verbatim}

\noindent
4) MAKE SURE THAT GOLD,	MY LOCAL CVS COPY, IS UP TO DATE:
\begin{verbatim}
 (I maintain an exact copy of the CVS repository and only make
  changes to it when i want to export the changes. that way I
  won't export my working tree by accident. my working tree is
  normally badly broken.

  The update command makes sure that you have all of the changes
  other people might have made and checked in. you have to merge
  your changes so you don't step on other people's work.
  So be sure to run update BEFORE you copy files to GOLD)     

cd /axiom
cvs update
\end{verbatim}

\noindent
5) COPY CHANGED FILES FROM WORK TO THE GOLD TREE:
\begin{verbatim}
 (This is an example for updating the *.daase files.
  You basically are changing your GOLD tree to reflect the
  way you want CVS to look once you check in all of the files.)

cd /home/axiomgnu/new
cp src/share/algebra/*.daase /axiom/src/share/algebra
\end{verbatim}

\noindent
6) IF A FILE IS NEW (e.g. src/interp/foo.lisp.pamphlet) THEN:
\begin{verbatim}
 (If you create a file you need to "put it under CVS control"
  CVS only cares about files you explicitly add or delete.
  If you make a new file and copy it to GOLD you need to do this.

  Don't do the "cvs add" in your WORK directory. The cvs add
  command updates the files in the CVS directory and you won't
  have them in your WORK directory. 

  Notice that you do the "cvs add" in the directory where the
  file was added (hence, the cd commands).

cd /axiom/src/interp
cvs add -m"some pithy comment" foo.lisp.pamphlet
cd /axiom
\end{verbatim}

\noindent
7) IF A FILE IS DELETED (e.g. src/interp/foo.lisp.pamphlet) THEN:
\begin{verbatim}
 (you have to delete the file from the GOLD directory BEFORE you
  do a "cvs remove". The "cvs remove" will update the files in
  the CVS directory

  Notice that you do the "cvs remove" in the directory where the
  file was deleted (hence, the cd commands).

cd /axiom/src/interp
rm foo.lisp.pamphlet
cvs remove foo.lisp.pamphlet
cd /axiom
\end{verbatim}

\noindent
8) IF A DIRECTORY IS NEW (e.g. foodir) THEN:
\begin{verbatim}
 (this will put "foodir" under CVS control. It will also create
  foodir/CVS as a directory with a bunch of control files in the
  foodir/CVS directory. Don't mess with the control files.

 (there are a bunch of special rules about directories.
  empty directories are not downloaded by update.)

 (NOTE: THERE IS NO WAY TO DELETE A DIRECTORY)

cd /axiom/src
mkdir foodir
cvs add -m "pithy comment" foodir
cd /axiom
\end{verbatim}

\noindent
9) EDIT CHANGELOG:
\begin{verbatim}
  changelog is already under CVS control so it will get uploaded
  when you do the checkin.)

cd /axiom
emacs -nw changelog
 (add a date, initials, and pithy comment, save it, and exit)
\end{verbatim}

\noindent
10) CHECK IN THE CHANGES
\begin{verbatim}
 (This will actually change the savannah CVS repository.

  The "cvs ci" command will recurse thru all of the lower
  subdirectories and look for changed files. It will change
  the host versions of those files to agree with your copy.
  If somebody else has changed a file while you were busy
  developing code then the checkin MAY complain (if it can't
  merge the changes)

cd /axiom
cvs ci -m"pithy comment"
\end{verbatim}

Congrats. You've now done your first change to the production
image. Please be very careful as this is a world readable copy.
We don't want to ship nonsense. Test everything. Even trivial
changes before you upload.

\section{Common Lisps}
\subsection{GCL}
Axiom was ported to run under AKCL which was a common lisp 
developed by Bill Schelter. He started with KCL (Kyoto Common
Lisp) and, since he lived and worked in Austin, Texas, named
his version AKCL (Austin-Kyoto Common Lisp). Bill worked under
contract to the Scratchpad group at IBM Research. I was the
primary developer for system internals so Bill and I worked
closely together on a lot of issues. After Axiom was sold to
NAG Bill continued to develop AKCL and it eventually became
GCL (Gnu Common Lisp). 

In order to port Axiom to run on GCL we need to do several things.
First, we need to apply a few patches. These patches enlarge the
default stack size, remove the startup banner, link with Axiom's
socket library, and rename collectfn.

The issue with the stack size is probably bogus. At one point the
system was running out of stack space but the problem was due to 
a recursive expansion of a macro and no amount of stack space
would be sufficient. This patch remains at the moment but should
probably be removed and tested.

The startup banner is an issue because we plan to run under
various frontend programs like Texmacs and the Magnus ZLC.
We need to just output a single prompt.

Axiom has a socket library because at the time it was developed under
AKCL there was no socket code in Lisp. There is still not a standard
common lisp socket library but I believe all common lisps have a way
to manipulate sockets. This code should be rewritten in lisp
and \verb|#+| for each common lisp.

The collectfn file is a major optimization under GCL. When collectfn 
is loaded and the lisp compiler is run then collectfn will output
a .fn file. The second time the compiler is invoked the .fn
file is consulted to determine the actual types of arguments used.
Function calling is highly optimized using this type information
so that fast function calling occurs. Axiom should be built one
time to create the int/*/*.fn files. It should then be rebuilt
using the cached .fn files. I will automate this process into
the Makefiles in the future.

GCL implementation will have a major porting problem to brand new
platforms. The compiler strategy is to output C code, compile it
using GCC, and dynamically link the machine code to the running
image. This requires deep knowledge of the symbol tables used
by the native linker for each system. In general this is a hard
problem that requires a lot of expertise. Bill Schelter and I
spent a lot of time and effort making this work for each port.
The magic knowledge is not written down anywhere and I no longer
remember the details.

\subsection{CCL}
When Axiom was sold to NAG it was ported to CCL (Codemist Common
Lisp) which is not, strictly speaking, a common lisp implementation.
It contains just enough common lisp to support Axiom and, as I'm
a great believer in simple code, it only needed a small subset of
a full common lisp. 

CCL can be considered the best way to get Axiom running on a new
architecture as the porting issues are minimal.

CCL is a byte-interpreter implementation and has both the
positive and negative aspects of that design choice. The positive
aspect is that porting the code to run on new architectures is
very simple. Once the CCL byte-code interpreter is running 
then Axiom is running. The saved-system image is pure byte-codes
and is completely system independent.

The negative aspects are that it is slow and the garbage collector
appears broken. Compiling the Axiom library files on an file-by-file
basis takes about 1 hour on GCL and about 12 hours on CCL. Compiling
all of the Axiom library files in the same image (as opposed to 
starting a new image per file) still takes about 1 hour on GCL.
It never finishes in CCL. Indeed it stops doing useful work after
about the 40th file (out of several hundred).

When Axiom became open source I moved the system back to GCL
because I could not understand how to build a CCL system. I plan
to revisit this in the future and document the process so others
can follow it as well as build Makefiles to automate it.

\subsection{CMU CL}
CMU CL grew out of the Carnegie-Mellon University SPICE project.
That project studied the issues involved in building an optimizing
compiler for common lisp. Axiom, back when it was Scratchpad at IBM,
ran on CMU CL. Indeed, a lot of the lisp-level optimizations are 
due to use of the CMU CL compiler and the disassemble function.

\subsection{Franz Lisp}
Axiom, as Scratchpad, ran on Franz Lisp.

\subsection{Lucid Common Lisp}
Axiom, as Scratchpad, ran on Lucid Common Lisp.

\subsection{Symbolics Common Lisp}
Axiom, as Scratchpad, ran on Symbolics Common Lisp.

\subsection{Golden Common Lisp}
Axiom, as Scratchpad, ran on Golden Common Lisp.
This was a PC version of Common Lisp which appears to have died.

\subsection{VM/LISP 370}
Axiom, as Scratchpad, ran on VM/Lisp 370. This was an IBM version
of lisp and was not a common lisp. The .daase random access
file format is an artifact of running on this lisp.

\subsection{Maclisp}
Axiom, as Scratchpad, ran on Maclisp. This was an early MIT version
of lisp and is not common lisp. Many of the funny function names
that have slightly different semantics than their common lisp
counterparts still exist in the system as macros due to this
lisp. 

\section{Changing GCL versions}

Axiom lives on GNU Common Lisp. Axiom adds C code to the lisp image.
Axiom caches versions to ensure that nothing breaks. Changing GCL
versions has introduced subtle bugs at various times. The steps
necessary to introduce a new version are

\begin{enumerate}
\item Add the latest GCL sources to Axiom
\item Update the patches to the new version
\item create diff -Naur patches to the gcl sources
\item update lsp/Makefile.pamphlet to apply the patch at build
\item add a new chunk to lsp/Makefile.pamphlet to build gcl-2.6.10
\item Change the GCLVERSION to point at the new sources
\item change the Makefile to match Makefile.pamphlet
\end{enumerate}

We assume in the following that \verb|$AXHOME| is the home directory and that
Axiom lives in the silver subdirectory.

In more detail these steps are:

\begin{enumerate}
\item Add the lateset GCL sources to Axiom
\begin{enumerate}
\item Download the latest GCL from gnu.org\\
      For these instructions assume the file is 
\begin{verbatim}
      gcl-Version_2_6_10.tar.gz
\end{verbatim}
\item move the tar file into /tmp\\
      We are going to make changes to the distribution via patches
\item untar the file
\begin{verbatim}
      tar -zxf gcl-Version_2_6_10.tar.gz
\end{verbatim}
\item cd to the untarred directory
\begin{verbatim}
      cd gcl-Version_2_6_10
\end{verbatim}
\item rename the gcl directory\\
      Camm follows a convention that the top level directory in the
      tar file is called gcl. Since we maintain several past versions
      we need to rename this and re-tar it
\begin{verbatim}
      mv gcl gcl-2.6.10
\end{verbatim}
\item rename gcl to use our naming convention
\begin{verbatim}
      tar -zcf gcl-2.6.10.tgz gcl-2.6.10
\end{verbatim}
\item We move the original, renamed, retarred file to the zip directory
\begin{verbatim}
      mv gcl-2.6.10.tgz $AXHOME/silver/zips
\end{verbatim}      
\item We have to make sure to include the new file in the git commit
\begin{verbatim}
     cd $AXHOME/silver
\end{verbatim}      
\item Tell git we care about the file
\begin{verbatim}
     git add zips/gcl-2.6.10.tgz
\end{verbatim}
\end{enumerate}
\item Update the patches to the new version
\begin{enumerate}
\item find the previous patches
\begin{verbatim}
     ls $AXHOME/silver/zips/gcl-2.6.8pre7*patch
\end{verbatim}
\item for each patch do ( Step 3 ; Step 4 )
\end{enumerate}
\item create diff -Naur patches to the gcl sources
\begin{enumerate}
\item assume we are looking at gcl-2.6.8pre7.h.linux.defs.patch\\
     The name tells us what file to patch. From the above we can see
     that when Axiom builds GCL it will
\begin{verbatim}
        cd lsp/gcl-2.6.8pre7
\end{verbatim}
     because GCL is built by the lsp/Makefile.pamphlet. That Makefile
     will do a
\begin{verbatim}
        cd h
        patch <gcl-2.6.8pre7.h.linux.defs.patch
\end{verbatim}
     which will apply the patch .... So we need to make a patch
\item move to the subdirectory containing the file
\begin{verbatim}
      cd /tmp/gcl-Version_2_6_10/gcl-2.6.10/h
\end{verbatim}
\item edit the 'linux.defs' file to create the proper patch
\item save the changed file as linux.defs.tpd
\item in a shell, create a diff -Naur patch by 
\begin{verbatim}
      diff -Naur linux.defs linux.defs.tpd >gcl-2.6.10.h.linux.defs.patch
\end{verbatim}
\item move it to the zips directory
\begin{verbatim}
      mv gcl-2.6.10.h.linux.defs.patch $AXHOME/silver/zips
      cd $AXHOME/silver
      git add zips/gcl-2.6.10.h.linux.defs.patch
\end{verbatim}
\end{enumerate}
\item update lsp/Makefile.pamphlet to apply the patch at build
\begin{enumerate}
\item edit lsp/Makefile.pamphlet
\item search for chunk \verb|gcl-2.6.8pre7.h.linux.defs.patch|
\item copy the chunk and name the new chunk 
\verb|gcl-2.6.10.h.linux.defs.patch|
\end{enumerate}
\item add a new chunk to lsp/Makefile.pamphlet to build gcl-2.6.10
\begin{enumerate}
\item find the subsection ``The GCL-2.6.8pre7 stanza''
\item make a copy named ``The GCL-2.6.10 stanza''
\item add the new patches
\item tell git we care
\begin{verbatim}
      cd $AXHOME/silver
      git add lsp/Makefile.pamphlet
\end{verbatim}
\end{enumerate}
\item Change the GCLVERSION to point at the new sources
\begin{enumerate}
\item emacs \verb|$AXHOME/silver/Makefile.pamphlet|
\item search for \verb|#GCLVERSION|, a Makefile comment line
\item the last line is uncommented. Assume it reads GCLVERSION=gcl-2.6.8pre7\\
      gcl-2.6.8pre7 was is the name of the current version we are
      replacing. We will use this name in the next step
\item put a \# in front of the GCLVERSION variable to comment it out\\
      We maintain the list of old, working patches. We also remember the
      names of the prior GCLVERSIONS in case we have to back up
\item Add a new line reading:
\begin{verbatim}
         GCLVERSION=gcl-2.6.10
\end{verbatim}
      This will cause Axiom to untar this tgz file to get the sources
      and apply the corresponding patches
\end{enumerate}
\item change the Makefile to match Makefile.pamphlet
\begin{itemize}
\item compile the tangle program
\begin{verbatim}
      ( cd books ; gcc -o tangle tangle.c )
\end{verbatim}
\item use books/tangle to extract the new Makefile
\begin{verbatim}
      books/tangle Makefile.pamphlet >Makefile
\end{verbatim}
\end{itemize}
\end{enumerate}

\section{Literate Programming}
The Axiom source code was originally developed at IBM Research.
It was sold to The Numerical Algorithms Group (NAG) and was on
the market as a commercial competitor to Mathematica and Maple.

Axiom was withdrawn from the market in 2000 and released as free
and open source software in 2001. When the Axiom project was
started on savannah, the GNU Free Software Foundation site the
source code had been rewritten into ``pamphlet'' files. The reasons
for this are twofold.

\subsection{Pamphlet files}

When the Axiom code was released it contained few comments. That made
it very difficult to understand what the code actually did. Unlike
commercial software there would be no group of individuals who would
work on the project for its lifetime. Thus there needed to be a way to
capture the expertise and understanding underlying ongoing
development. 

Unlike any other piece of free and open source software Axiom will
still give useful answers 30 years from now. Thus it is important,
and worthwhile, to invest a large amount of effort into documenting
how these answers are arrived at and why the algorithms are written
the way they are.

The pamphlet file format follows Knuth's idea of literate
programming. Knuth made the observation that a program should be a
work designed to be read by humans. Making the program readable by
machine was a secondary consideration. Making documentation primary
and code secondary was a dramatic shift for a programmer.

Knuth created a file format that combined documentation and code.  He
created a tool called ``Web'' which had two basic command, tangle and
weave. The tangle command would be run against a literate document and
extract the source code, the weave command would be run against the
literate document and extract the TeX. 

\subsection{noweb}

Knuth's Web tool was specifically designed to use Pascal code. The
``tangle'' operation would prettyprint the output according to the
style rules of Pascal.

Axiom was written in a variety of languages, such as C and Lisp,
and used tools such as Makefiles which have their own syntax. Thus
Web could not be used directly.

Axiom defines a new latex environment called chunk.
This chunk environment makes the pamphlet file a pure latex file.
This eliminates the need for the weave operation. The tangle
operation only needs to occur while manipulating code, either during
system build or end user interaction. At both of these times the
tangle operation can be built into the system and hidden.

To support extracting chunks from pamphlet files Axiom now has a
new top level command. At the top level one can write:
\begin{verbatim}
  )tangle filename
\end{verbatim}
This will look for ``filename.pamphlet'' and extract the top level
chunk which has the name ``*''.

The latest changeset introduces two related changes, gclweb and
axiom.sty.  Together these changes allow optional syntactic changes to
pamphlets.  These changes will completely eliminate the need to weave
files since now a pamphlet file can be a valid latex file. Tangle is
the only remaining command and it will eventually be an option on
)compile, etc. 

The src/interp/gclweb.lisp file introduces the ability to extract code
from pamphlet files while inside Axiom. The short description is that
gclweb will now automatically distinguish the type of chunk style
(latex or noweb) based on the chunk name. It is a first step to a native
understanding of pamphlet files. Future work involves integrating it
into commands like )compile and adding commands like )tangle. 

Tangle can also be called directly from lisp on a file from within Axiom:
\begin{verbatim}
  )lisp (tangle "filename.pamphlet" "chunkname")
  )lisp (tangle "filename.pamphlet" "chunkname" "filename.spad")
\end{verbatim}

gclweb distinguishes the input syntax by looking at the first character
of the chunkname. If it is a '$<$' then noweb is used, otherwise latex.

The src/doc/axiom.sty.pamphlet introduces the new chunk environment.
This is a completely compatible change and has no impact on existing
pamphlets. The new syntax makes pamphlet files = tex files so there is
no need to use weave. The gclweb change has a compatible tangle
function which can be invoked from inside Axiom.

\begin{verbatim}
  \begin{chunk}{chunkname}
    your code goes here
  \end{chunk}
\end{verbatim}

One feature of the latex chunk style is that latex commands
work within the chunk. To get typeset mathematics use 
\verb|\(| and \verb|\)|

\begin{verbatim}
-- This will typeset in a chunk \( x^2+\epsilon \)
-- And you can format things {\bf bold}
\end{verbatim}

\section{Databases}
\subsection{libcheck}
The databases are built from the .kaf files in the .nrlib directories.
(.kaf files are random access files).

interp.exposed is a file that names all of the CDPs (Category, Domain,
and Packages) and classifies them. Only some CDPs are exposed because
most are used to implement algebra and are not intended to be user level
functions. Exposing all of the functions causes much ambiguity.

There is a function called libcheck (see src/interp/util.lisp.pamphlet)
that will check nrlibs vs interp.exposed. This is only partially 
functional as I see that changes were made to the system which broke
this function.

The libcheck function requires an absolute pathname to the int directory
so call it thus:
\begin{verbatim}
 --> )lisp (libcheck "/axiom/int/algebra")
\end{verbatim}
The main reason this function is broken is that the system now gets
exposure information from src/algebra/exposed.lsp.pamphlet. It appears
that interp.exposed.pamphlet is no longer used (although I made sure
that both files have the same information). I'm going to modify libcheck
to use exposed.lsp in the future and eliminate all references in the
system to interp.exposed.

For the moment, however, the libcheck function is quite useful.
It used to be run during system build because I frequently ran into
database problems and this function would alert me to that fact.
I'll add it back into the Makefile once I elide interp.exposed.

\subsection{asq}
Axiom has several databases which contain information about domains,
categories, and packages. The databases are in a compressed format and
are organized as random-access files using numeric index values so it
is hard to get at the stored information. However, there is a
command-line query function called asq (pronounced ask) that knows the
format of the files and can be used for stand-alone queries. For
instance, if you know the abbreviation for a domain but want to know
what source file generated that domain you can say:
\begin{verbatim}
asq -so FOOBAR
\end{verbatim}
and it will tell you the name of the algebra source file that
defines FOOBAR.

\section{Axiom internal representations}
\begin{verbatim}
PRIMITIVE REPRESENTATIONS OF AXIOM OBJECTS


There are several primitive representations in axiom. These are:

boolean 
 this is represented as a lisp boolean

integer
 this is represented as a lisp integer

small integer
 this is represented as a lisp integer

small float
 this is represented as a lisp float

list
 this is represented as a lisp list

vector
 this is represented as a lisp vector

record
 there are 3 cases:
  records of 1 element are a pair (element . nil)
  records of 2 element are a pair (element1 . element2)
  records of 3 or more are a vectors #<a b c...>

mapping
 mappings are a spadcall objects. they are represented as a pair
  (lispfn . env) 
 where the env is usually a type object. A spadcall rips this 
 pair open and applies the lispfn to its args with env as the 
 last arg.

union
 there are 2 cases
  if the object can be determined by a lisp predicate 
  (eg integer) then the union is just the object (eg 3) 
  itself since we can use lisp to decide which branch of 
  the union the object belongs to. that is, 3 is of the 
  integer branch in union(list,integer)

  if the object cannot be determined then the object is 
  wrapped into a pair where the car of the pair is the 
  union branch name and the cdr of the pair is the object. 
  that is, given union(a:SUP,b:POLY(INT)) x might be (a . x)

  note: if no tags are given in the union the system uses 
  consecutive integers, thus union(SUP,POLY(INT)) will give 
  a pair of (1 . x) or (2 . x) depending on the type of x

other types are built up of compositions of these primitive 
types. a sparse univariate polynomial (SUP) over the integers

  x**2+1

is represented as

Term := Record(k:NonNegativeInteger,c:R)
Rep  := List Term

that is, the representation is a list of terms where each term 
is a record whose first field is a nonnegative integer (the 
exponent) and the second field is a member of the coefficient 
ring. since this is a record of length 2 it is represented as 
a pair. thus, the internal form of this polynomial is:

 ((2 . 1) (0 . 1))

a more complex object (recursively defined) is POLY(INT). given

 x**2+1

as a POLY(INT) we look at its representation and see:

D := SparseUnivariatePolynomial($)
VPoly := Record(v:VarSet,ts:D)
Rep := Union(R,VPoly)

so first we find that we are a member of the second form of the 
union and since this is an untagged union the system uses 2 as 
the tag. thus the first level of internal representation is:

 ( 2 . <a VPoly object> )

next we need to define the VPoly object. VPolys are records of 
length 2 so we know they are represented by a pair. the car of 
the pair is a VarSet. the cdr is a D which is a 
SparseUnivariatePolynomial. Thus we consider this to be a poly 
in x (at the top level) and we get:

 ( 2 . ( x . <an SUP>))

the SUP is over the SparseMultivariatePolynomials (SMP) so the 
representation is recursive. Since an SUP is represented as a 
list of
  
 (non-negative int . coefficient) 

one per term and we have 2 terms we know the next level of 
structure is:

 ( 2 . ( x . (( 2 . <an SMP> ) ( 0 . <an SMP> ))))

the SMP is just the integers so it fits into the first branch 
of the union and each SMP looks like:

 ( uniontag . value )

in this case, being the first branch we get

 ( 2 . ( x . (( 2 . ( 1 . 1 )) ( 0 . ( 1 . 1 )))))

as the internal representation of 

 x**2 + 1

what could be easier?
\end{verbatim}

\section{Spad to internal function calling}

\subsection{getdatabse output}
\begin{verbatim}
   GETDATABASE('Permutation, 'OPERATIONALIST)$Lisp
\end{verbatim}

generates the output
\begin{verbatim}
   (($unique)  
    (~= (((Boolean) $ $) () T ELT))
    (sort (((List $) (List $)) 76 T ELT))
    (sign (((Integer) $) 59 T ELT))
    (sample (($) () T CONST))
    (recip (((Union $ "failed") $) () T ELT))
    (order (((NonNegativeInteger) $) 57 T ELT))
    (orbit (((Set #1) $ #1) 48 T ELT))
    (one? (((Boolean) $) () T ELT))
    (odd? (((Boolean) $) 62 T ELT))
    (numberOfCycles (((NonNegativeInteger) $) 60 T ELT))
    (movedPoints (((Set #1) $) 41 T ELT))
    (min (($ $ $) () (OR (has #1 (Finite)) (has #1 (OrderedSet))) ELT))
    (max (($ $ $) () (OR (has #1 (Finite)) (has #1 (OrderedSet))) ELT))
    (listRepresentation
      (((Record (: preimage (List #1)) (: image (List #1))) $) 35 T ELT))
    (latex (((String) $) () T ELT))
    (inv (($ $) 92 T ELT))
    (hash (((SingleInteger) $) () T ELT))
    (fixedPoints (((Set #1) $) 98 (has #1 (Finite)) ELT))
    (even? (((Boolean) $) 58 T ELT))
    (eval ((#1 $ #1) 46 T ELT))
    (elt ((#1 $ #1) 93 T ELT))
    (degree (((NonNegativeInteger) $) 43 T ELT))
    (cycles (($ (List (List #1))) 84 T ELT))
    (cyclePartition (((Partition) $) 52 T ELT))
    (cycle (($ (List #1)) 21 T ELT))
    (conjugate (($ $ $) () T ELT))
    (commutator (($ $ $) () T ELT))
    (coercePreimagesImages (($ (List (List #1))) 38 T ELT))
    (coerceListOfPairs (($ (List (List #1))) 87 T ELT))
    (coerceImages (($ (List #1)) 95 T ELT))
    (coerce  (((OutputForm) $) 83 T ELT)  (($ (List (List #1))) 65 T ELT)
      (($ (List #1)) 66 T ELT))
    (^  (($ $ (PositiveInteger)) () T ELT)
      (($ $ (NonNegativeInteger)) () T ELT)  (($ $ (Integer)) () T ELT))
    (One (($) 16 T CONST))
    (>= (((Boolean) $ $) () (OR (has #1 (Finite)) (has #1 (OrderedSet))) ELT))
    (> (((Boolean) $ $) () (OR (has #1 (Finite)) (has #1 (OrderedSet))) ELT))
    (= (((Boolean) $ $) 44 T ELT))
    (<= (((Boolean) $ $) () (OR (has #1 (Finite)) (has #1 (OrderedSet))) ELT))
    (< (((Boolean) $ $) 64 T ELT))
    (/ (($ $ $) () T ELT))
    (**  (($ $ (PositiveInteger)) () T ELT)
      (($ $ (NonNegativeInteger)) () T ELT)  (($ $ (Integer)) () T ELT))
    (* (($ $ $) 22 T ELT)))
\end{verbatim}

Sometimes in a getdatabase expression you will see:

\begin{verbatim}
(~= (((Boolean) $ $) () T ELT))
---------------------^^
\end{verbatim}

and in other places there is a number

\begin{verbatim}
      (sign (((Integer) $) 59 T ELT))
---------------------------^^
\end{verbatim}

In general, when a large number appears it is a byte index into
the compress.daase file. 

Axiom would not fit on a laptop. We needed smaller databases.
The solution to the problem was to scan the datatases for
common substrings, write the substring to compress.daase,
and replace the substring by the byte offset. 

When reading the database these numbers would be replaced
by the substring from compress.daase using random access
seeks based on the byte offset.

See book volume 5
for an explanation of the database file formats.

HOWEVER, in this case, the number has a different meaning
which I will talk about below.

In summary, this shows what the following incantation means:

\begin{verbatim}
  (sign (((Integer) $) () (has #1 (OrderedIntegralDomain))))
\end{verbatim}

\begin{verbatim}
INTEGER inherits sign from OINTDOM (OrderedIntegralDomain)
OINTDOM inherits sign from ORDRING (OrderedRing)
ORDRING implements sign
   since ORDRING is a category, the actual code lives in
   ORDRING-.nrlib/code.lsp
\end{verbatim}

The code for sign in ORDRING-.nrlib/code.lsp has the signature:
\begin{verbatim}
   (DEFUN |ORDRING-;sign;SI;3| (|x| $) ....)
\end{verbatim}

We can "decode" the meaning of the function name as
\begin{itemize}
\item {\bf ORDRING-} the implementing file
\item {\bf sign} the function name
\item {\bf SI} returns SingleInteger (an old domain name)
\item {\bf 3} the third function in the file
(unique, to distinguish multiple functions with the same name)
\end{itemize}

It takes 2 arguments, 
\begin{itemize}
\item {\bf \verb?|x|?} which should be a SingleInteger
\item {\bf \verb?$?} which is the current domain (ORDRING-)
\end{itemize}

So it looks like I have the following structure
\begin{verbatim}
  (NAME ((TARGETTYPE SOURCETYPE) ?1 CONDITION ?2))
\end{verbatim}

but we are looking up 'sign' in INTEGER so there is
a condition on sign

\begin{verbatim}
   Integer has OrderedRing  ==> true
\end{verbatim}

so that explains the condition field.

Here we show how Axiom finds the function implementation, looks up the
function ``in the domain'', and calls it.

\begin{verbatim}
    (sign (((Integer) $) 59 T ELT))
\end{verbatim}

Now you've asked for 'sign' from domain Permutation

\begin{verbatim}
  (sign (((Integer) $) 59 T ELT))
\end{verbatim}

The implementation for 'sign' is in PERM.nrlib/code.lsp. It reads:

\begin{verbatim}
 (defun |PERM;sign;$I;17| (|p| $)
  (cond
   ((spadcall |p| (qrefelt $ 58)) 1)
   ('t -1)))
\end{verbatim}

which you would read as

\begin{verbatim}
  if (calling function in position 58 of myself) is true
    then return 1
    else return -1
\end{verbatim}

How does Axiom find the function? It is in the infovec which
is the ``information vector'' containing information about the
domain.

First we must make sure that PERM has the necessary domain
information loaded (the 'infovec').

\begin{verbatim}
-> [1,2,3]::PERM(INT)
\end{verbatim}

Now, back to the 'sign' function. You saw this:

\begin{verbatim}
   (sign (((Integer) $) 59 T ELT))  (sample (($) () T CONST))
\end{verbatim}
which is asking you to look up element 59 from the domain (\verb|$|)

Note that \verb|$| {\sl actually} means the infovec. So we are asking:

\begin{verbatim}
   (elt (elt (getf (symbol-plist '|Permutation|) '|infovec|) 0) 59)
\end{verbatim}
which results in:

\begin{verbatim}
   |PERM;sign;$I;17|
\end{verbatim}
so we ``looked up'' the function sign in the domain PERM.

Explaining in more detail, from the inside out by walking the
runtime data structures we see

\begin{verbatim}
   (symbol-plist '|Permutation|)
\end{verbatim}
returns the property list on the symbol {\bf Permutation} which is where
Axiom caches domain information. Almost everything of interest about a
domain resides on the property list, shown here in all its glory.

\begin{verbatim}
(LOADED "/research/test/mnt/ubuntu/algebra/PERM.o" 
 SYSTEM:DEBUG (#:G1567 #:G1568)
 |infovec| (
  #(NIL NIL NIL NIL NIL NIL
    (|local| |#1|)        (QUOTE |Rep|)           (|Boolean|)
    (0 . <)               (|PositiveInteger|)     (6 . |lookup|)
    (|Integer|)           (|List| 6)              (11 . |maxIndex|)
    (16 . |elt|)
    (CONS IDENTITY (FUNCALL (|dispatchFunction|   |PERM;One;$;29|) $))
    (|NonNegativeInteger|) (22 . |last|)          (28 . |first|)
    (34 . |concat|)        |PERM;cycle;L$;26|     |PERM;*;3$;28|
    (40 . =)               (46 . =)               (52 . |elt|)
    (58 . |list|)          (63 . |position|)      (69 . |delete|)
    (|Mapping| 8 13 13)    (|List| 13)            (75 . |sort|)
    (81 . |copy|)          (86 . |member?|)
    (|Record| (|:| |preimage| 13) (|:| |image| 13))
    |PERM;listRepresentation;$R;9|                (92 . |elt|)
    (98 . ~=)              |PERM;coercePreimagesImages;L$;10|
    (|Set| 6)              (104 . |construct|)    |PERM;movedPoints;$S;11|
    (109 . |#|)            |PERM;degree;$Nni;12|  |PERM;=;2$B;13|
    (114 . |brace|)        |PERM;eval;$2S;31|     (119 . |insert!|)
    |PERM;orbit;$SS;14|    (|List| 12)            (|Partition|)
    (125 . |partition|)    |PERM;cyclePartition;$P;15|
    (130 . |convert|)      (135 . |removeDuplicates|)
    (|List| $)             (140 . |lcm|)          |PERM;order;$Nni;16|
    |PERM;even?;$B;18|     |PERM;sign;$I;17|      |PERM;numberOfCycles;$Nni;33|
    (145 . |even?|)        |PERM;odd?;$B;19|      (150 . |maxIndex|)
    |PERM;<;2$B;20|        |PERM;coerce;L$;21|    |PERM;coerce;L$;22|
    (|Record| (|:| |cycl| 30) (|:| |permut| $$))
    (|List| 67)            (155 . |cons|)         (|Mapping| 8 67 67)
    (161 . |sort|)         (|List| $$)            (167 . |nil|)
    (171 . |cons|)         (177 . |reverse|)      |PERM;sort;2L;23|
    (|OutputForm|)         (182 . |coerce|)       (187 . |blankSeparate|)
    (192 . |paren|)        (197 . |outputForm|)   (202 . |hconcat|)
    |PERM;coerce;$Of;24|   |PERM;cycles;L$;25|    (207 . |second|)
    (212 . =)              |PERM;coerceListOfPairs;L$;27|
    (|Vector| 6)           (218 . |construct|)    (223 . |elt|)
    (229 . |new|)          |PERM;inv;2$;30|       |PERM;elt;$2S;32|
    (235 . |coerce|)       (240 . |coerceImages|) (245 . |index|)
    (250 . |complement|)   (255 . |fixedPoints|)  (260 . |conjugate|)
    (265 . +)              (|Union| $ (QUOTE "failed"))
    (|SingleInteger|)      (|String|))
 #(~= 271                      |sort| 277                |sign| 282 
   |sample| 287                |recip| 291               |order| 296
   |orbit| 301                 |one?| 307                |odd?| 312
   |numberOfCycles| 317        |movedPoints| 322         |min| 327
   |max| 333                   |listRepresentation| 339  |latex| 344
   |inv| 349                   |hash| 354                |fixedPoints| 359
   |even?| 364                 |eval| 369                |elt| 375
   |degree| 381                |cycles| 386              |cyclePartition| 391
   |cycle| 396                 |conjugate| 401           |commutator| 407
   |coercePreimagesImages| 413 |coerceListOfPairs| 418   |coerceImages| 423
   |coerce| 428                ^ 443                     |One| 461
    >= 465                     > 471                     = 477
    <= 483                     < 489                     / 495 
    ** 501                     * 519)
 ((|unitsKnown| . 0))
 (#(0 0 0 0 3 0 0 0)
  #(NIL
    |Group&|
    |Monoid&|
    |SemiGroup&|
    |OrderedSet&|
    |SetCategory&|
    |BasicType&|
    NIL)
  #((|PermutationCategory| 6)
    (|Group|)
    (|Monoid|)
    (|SemiGroup|)
    (|OrderedSet|)
    (|SetCategory|)
    (|BasicType|)
    (|CoercibleTo| 77))
 . 
  #( 2  6  8  0  0  9  1  6 10  0 11  1 13 12  0 14  2 13  6  0 12 
    15  2 13  0  0 17 18  2 13  0  0 17 19  2 13  0  0  0 20  2  6
     8  0  0 23  2 13  8  0  0 24  2  7 13  0 12 25  1 13  0  6 26
     2 13 12  6  0 27  2 13  0  0 12 28  2 30  0 29  0 31  1 13  0
     0 32  2 13  8  6  0 33  2 30 13  0 12 36  2  6  8  0  0 37  1
    39  0 13 40  1 39 17  0 42  1 39  0 13 45  2 39  0  6  0 47  1
    50  0 49 51  1 50 49  0 53  1 49  0  0 54  1 12  0 55 56  1 12
     8  0 61  1 30 12  0 63  2 68  0 67  0 69  2 68  0 70  0 71  0
    72  0 73  2 72  0  2  0 74  1 72  0  0 75  1  6 77  0 78  1 77
     0 55 79  1 77  0  0 80  1 77  0 12 81  1 77  0 55 82  1 13  6
     0 85  2 39  8  0  0 86  1 88  0 13 89  2 88  6  0 12 90  2  7
     0 17 13 91  1  6  0 12 94  1  0  0 13 95  1  6  0 10 96  1 39
     0  0 97  1  0 39  0 98  1 50  0  0 99  2 50  0  0  0 100 2  0
     8  0  0  1  1  0 55 55 76  1  0 12  0 59  0  0  0  1  1  0 101
     0  1  1  0 17  0 57  2  0 39  0  6 48  1  0  8  0  1  1  0  8
     0 62  1  0 17  0 60  1  0 39  0 41  2  3  0  0  0  1  2  3  0
     0  0  1  1  0 34  0 35  1  0 103 0  1  1  0  0  0 92  1  0 102
     0  1  1  1 39  0 98  1  0  8  0 58  2  0  6  0  6 46  2  0  6
     0  6 93  1  0 17  0 43  1  0  0 30 84  1  0 50  0 52  1  0  0
    13 21  2  0  0  0  0  1  2  0  0  0  0  1  1  0  0 30 38  1  0
     0 30 87  1  0  0 13 95  1  0  0 13 66  1  0  0 30 65  1  0 77
     0 83  2  0  0  0 12  1  2  0  0  0 17  1  2  0  0  0 10  1  0
     0  0 16  2  3  8  0  0  1  2  3  8  0  0  1  2  0  8  0  0 44
     2  3  8  0  0  1  2  0  8  0  0 64  2  0  0  0  0  1  2  0  0
     0 12  1  2  0  0  0 17  1  2  0  0  0 10  1  2  0  0  0  0 22))
 |lookupComplete|)
   PNAME "Permutation"
   DATABASE
    #S(DATABASE
       ABBREVIATION PERM
       ANCESTORS NIL
       CONSTRUCTOR NIL
       CONSTRUCTORCATEGORY 2444459
       CONSTRUCTORKIND |domain|
       CONSTRUCTORMODEMAP
        (((|Permutation| |#1|)
          (|Join|
           (|PermutationCategory| |#1|)
           (CATEGORY |domain|
            (SIGNATURE |listRepresentation|
              ((|Record| (|:| |preimage| #) (|:| |image| #)) $))
            (SIGNATURE |coercePreimagesImages| ($ (|List| (|List| |#1|))))
            (SIGNATURE |coerce| ($ (|List| (|List| |#1|))))
            (SIGNATURE |coerce| ($ (|List| |#1|)))
            (SIGNATURE |coerceListOfPairs| ($ (|List| (|List| |#1|))))
            (SIGNATURE |degree| ((|NonNegativeInteger|) $))
            (SIGNATURE |movedPoints| ((|Set| |#1|) $))
            (SIGNATURE |cyclePartition| ((|Partition|) $))
            (SIGNATURE |order| ((|NonNegativeInteger|) $))
            (SIGNATURE |numberOfCycles| ((|NonNegativeInteger|) $))
            (SIGNATURE |sign| ((|Integer|) $))
            (SIGNATURE |even?| ((|Boolean|) $))
            (SIGNATURE |odd?| ((|Boolean|) $))
            (SIGNATURE |sort| ((|List| $) (|List| $)))
            (IF (|has| |#1| (|Finite|))
              (SIGNATURE |fixedPoints| ((|Set| |#1|) $)) |noBranch|)
            (IF (|has| |#1| (|IntegerNumberSystem|))
              (SIGNATURE |coerceImages| ($ (|List| |#1|)))
              (IF (|has| |#1| (|Finite|))
                (SIGNATURE |coerceImages| ($ #)) |noBranch|))))
          (|SetCategory|))
         (T |Permutation|))
       COSIG (NIL T) 
       DEFAULTDOMAIN NIL 
       MODEMAPS 2443154 
       NILADIC NIL 
       OBJECT "PERM" 
       OPERATIONALIST 
        ((|$unique|)
         (~= (((|Boolean|) $ $) NIL . #0=(T . #1=(ELT))))
         (|sort| (((|List| $) (|List| $)) 76 . #0#))
         (|sign| (((|Integer|) $) 59 . #0#))
         (|sample| (($) NIL T CONST))
         (|recip| (((|Union| $ "failed") $) NIL . #0#))
         (|order| (((|NonNegativeInteger|) $) 57 . #0#))
         (|orbit| (((|Set| |#1|) $ |#1|) 48 . #0#))
         (|one?| (((|Boolean|) $) NIL . #0#))
         (|odd?| (((|Boolean|) $) 62 . #0#))
         (|numberOfCycles| (((|NonNegativeInteger|) $) 60 . #0#))
         (|movedPoints| (((|Set| |#1|) $) 41 . #0#))
         (|min| (($ $ $) NIL
           (OR (|has| |#1| (|Finite|)) (|has| |#1| (|OrderedSet|))) . #1#))
         (|max| (($ $ $) NIL
           (OR (|has| |#1| (|Finite|)) (|has| |#1| (|OrderedSet|))) . #1#))
         (|listRepresentation|
           (((|Record| (|:| |preimage| (|List| |#1|))
                       (|:| |image| (|List| |#1|))) $) 35 . #0#))
        (|latex| (((|String|) $) NIL . #0#))
        (|inv| (($ $) 92 . #0#))
        (|hash| (((|SingleInteger|) $) NIL . #0#))
        (|fixedPoints| (((|Set| |#1|) $) 98 (|has| |#1| (|Finite|)) . #1#))
        (|even?| (((|Boolean|) $) 58 . #0#))
        (|eval| ((|#1| $ |#1|) 46 . #0#))
        (|elt| ((|#1| $ |#1|) 93 . #0#))
        (|degree| (((|NonNegativeInteger|) $) 43 . #0#))
        (|cycles| (($ (|List| (|List| |#1|))) 84 . #0#))
        (|cyclePartition| (((|Partition|) $) 52 . #0#))
        (|cycle| (($ (|List| |#1|)) 21 . #0#))
        (|conjugate| (($ $ $) NIL . #0#))
        (|commutator| (($ $ $) NIL . #0#))
        (|coercePreimagesImages| (($ (|List| (|List| |#1|))) 38 . #0#))
        (|coerceListOfPairs| (($ (|List| (|List| |#1|))) 87 . #0#))
        (|coerceImages| (($ (|List| |#1|)) 95 . #0#))
        (|coerce| 
          (((|OutputForm|) $) 83 . #0#)
          (($ (|List| (|List| |#1|))) 65 . #0#)
          (($ (|List| |#1|)) 66 . #0#))
        (^ (($ $ (|PositiveInteger|)) NIL . #0#)
           (($ $ (|NonNegativeInteger|)) NIL . #0#)
           (($ $ (|Integer|)) NIL . #0#))
        (|One| (($) 16 T CONST))
        (>= (((|Boolean|) $ $) NIL
           (OR (|has| |#1| (|Finite|)) (|has| |#1| (|OrderedSet|))) . #1#))
        (> (((|Boolean|) $ $) NIL
           (OR (|has| |#1| (|Finite|)) (|has| |#1| (|OrderedSet|))) . #1#))
        (= (((|Boolean|) $ $) 44 . #0#))
        (<= (((|Boolean|) $ $) NIL
           (OR (|has| |#1| (|Finite|)) (|has| |#1| (|OrderedSet|))) . #1#))
        (< (((|Boolean|) $ $) 64 . #0#))
        (/ (($ $ $) NIL . #0#))
        (** (($ $ (|PositiveInteger|)) NIL . #0#)
            (($ $ (|NonNegativeInteger|)) NIL . #0#)
            (($ $ (|Integer|)) NIL . #0#))
        (* (($ $ $) 22 . #0#)))
       DOCUMENTATION 1609893 
       CONSTRUCTORFORM 1609883 
       ATTRIBUTES 1614391 
       PREDICATES 1614406 
       SOURCEFILE "bookvol10.3.pamphlet" 
       PARENTS NIL 
       USERS NIL 
       DEPENDENTS NIL 
       SPARE NIL))
\end{verbatim}

There are many things on the property list which looks like

\begin{verbatim}
   (symbol1 thing1 symbol2 thing2 ... symboln thingn)
\end{verbatim}

In the PERM case we see
  
\begin{verbatim}
  (LOADED "/research/silver/mnt/algebra/PERM.o"
   |infovec| (#<vector> #<vector>...)
   ....)
\end{verbatim}

We can get the \verb?|infovec|? off the property list with the call

  (getf (symbol-plist '|Permutation|) '|infovec|)

is a request to search the property list for the symbol |infovec|
and return the value, which is the domain "information vector".

You can see this vector if you look in PERM.nrlib/code.lsp.
At the bottom of that file you'll see:

\begin{verbatim}
  (SETF (GET (QUOTE |Permutation|) (QUOTE |infovec|) ....)
\end{verbatim}

which uses SETF to put the infovec on the property list of PERM.
This information vector contains information for function lookup.
This vector gets created when we "instantiate" PERM.

The {\bf infovec} is a list with the structure
\begin{verbatim}
  (#<vector 08ea516c> 
   #<vector 08ea5150>
   ((|unitsKnown| . 0))
   (#<vector 08ea50fc>
    #<vector 08ea5134>
    #<vector 08ea5118> . #<vector 08ea50e0>)
  |lookupComplete|)
\end{verbatim}

So, now that we have the infovec, back to the game...

\begin{verbatim}
   (elt (getf (symbol-plist '|Permutation|) '|infovec|) 0)
\end{verbatim}

This gets the 0th element out of the infovec list which is a
vector of the name of every function Permutation implements.
We look up function names in this list, in particular, 59:

\begin{verbatim}
  (elt (elt (getf (symbol-plist '|Permutation|) '|infovec|) 0) 59)
\end{verbatim}
looks into this vector of names at the 59th element which returns

\begin{verbatim}
  |PERM;sign;$I;17|
\end{verbatim}

The SPAD form of this function reads:

\begin{verbatim}
  sign(p) ==
    even? p => 1
    -1
\end{verbatim}

The lisp form (see PERM.nrlib/code.lsp) reads:

\begin{verbatim}
(defun |PERM;sign;$I;17| (|p| $)
 (cond
  ((spadcall |p| (qrefelt $ 58)) 1)
  ('t -1)))
\end{verbatim}

We call the \verb?|PERM;sign;$I;17|? which takes 2 arguments

The first of which is the permutation and the second is the infovec
for the PERM domain.

The \verb|(qrefelt $ 58)| uses the above dance to look up a function
in the infovec at the 58th position... which returns

\begin{verbatim}
  |PERM;even?;$B;18|
\end{verbatim}

The spadcall calls \verb'|PERM;even?;$B;18|' with the value of \verb?|p|?.

If we look in the domain Permutation for the implementation of even?

\begin{verbatim}
  even?(p) == even?(#(p.1) - numberOfCycles p)
\end{verbatim}

which in PERM.nrlib/code.lsp we see

\begin{verbatim}
  (defun |PERM;even?;$b;18|
   (spadcall 
    (- (length (spadcall |p| 1 (qrefelt $ 25)))
       (spadcall |p| (qrefelt $ 60)))
    (qrefelt % 61)))
\end{verbatim}

where
\begin{verbatim}
  (qrefelt $ 25) ==> (52 . |elt|)
  (qrefelt $ 60) ==> |PERM;numberOfCycles;$Nni;33|
  (qrefelt $ 61) ==> (145. |even?|)
\end{verbatim}

So, to summarize, the small magic numbers you see in the results
are indexes into the infovec, which is where Axiom stores things
it needs to look up at runtime, usually function references.

If there is () rather than a number than there is no need to do
a function lookup.

Axiom execution is an alternating series of function lookups in
the infovec followed by a call of that function which results in
a function lookup in the infovec followed by a call of that 
function which results in .....

spadcall is a wrapper macro which takes the arguments and a
function to call. qrefelt does the infovec lookup.

\section{axiom command}
The axiom command will eventually be a shell script. At the moment
it is just a copy of the interpsys image. However the whole Axiom
system consists of several processes and the axiom command starts
these processes. The shell script will transparently replace the
axiom executable image which will be renamed to spadsys.

\section{help command documentation}
Axiom supports a )help command that takes a single argument.
This argument is interpreted as the name of a flat ascii file
which should live in \$AXIOM/doc/src/spadhelp.

\subsection{help documentation for algebra}
The help documentation for algebra files lives within the algebra
pamphlet. The help chunk contains the name of the domain, thus:

\begin{verbatim}
  \begin{chunk}{thisdomain.help}
  ====================================================================
  thisdomain examples
  ====================================================================
  
     (documentation for this domain)
  
    examplefunction foo
     output
                  Type: thetype
  
  See Also:
  o )show thisdomain
  o $AXIOM/bin/src/doc/algebra/thisfile.spad.dvi
  
  \end{chunk}
\end{verbatim}

The documentation starts off with the domain enclosed in two lines
of equal signs. The documentation is free format. Generally the
functions are indented two spaces, the output is indented 3 spaces,
and the Type field has been moved toward the center of the line.

The ``See Also:'' section lists the domain with the ``show'' command
and the path to the source file in dvi format.

\subsection{Adding help documentation in Makefile}

There is a section in the src/algebra/Makefile.pamphlet that reads:
\begin{verbatim}
SPADHELP=\
 ${HELP}/AssociationList.help  ${HELP}/BalancedBinaryTree.help \
\end{verbatim}
which is essentially a list of all of the algebra help files. Each item
in this list refers to a stanza that looks like:

\begin{verbatim}
${HELP}/AssociationList.help: ${BOOKS}/bookvol10.3.pamphlet
	@echo 7000 create AssociationList.help from \
           ${BOOKS}/bookvol10.3.pamphlet
	@${TANGLE} -R"AssociationList.help" ${BOOKS}/bookvol10.3.pamphlet \
            >${HELP}/AssociationList.help
	@cp ${HELP}/AssociationList.help ${HELP}/ALIST.help
	@${TANGLE} -R"AssociationList.input" ${BOOKS}/bookvol10.3.pamphlet \
            >${INPUT}/AssociationList.input
	@echo "AssociationList (ALIST)" >>${HELPFILE}
\end{verbatim}

Notice that the first line has an connection between the help file and 
the spad file that contains it. 

The second line gives debugging output containing a unique number for
console debugging purposes of failed builds.

The third line extracts the help file. Help files are part of
the algebra books (bookvol10.2, bookvol10.3, and bookvol10.4). The
chunkname is the same as the Category, Domain, or Package.

The fourth line copies the file with the long name of the domain to a file
with the abbreviation of the domain so the user can query the domain
with either form using help.

The fifth line creates a regression test file for the help file. In the
algebra each help file has an associated regression test file to test
all of the function calls shown in the help page. These files are
copied to the intermediate directory for regression testing.

The sixth line adds a line to the HELPFILE (see the variable in
the src/algebra/Makefile). This HELPFILE is concatenated onto the
final help.help file in the MNT/doc/spadhelp directory. Thus, when
a user types )help with no argument they see a list of domains which
contain help information.

\subsection{Using help documentation for regression testing}

The fifth line extracts an input test file for the algebra. In general
each help file is used to create an input test file for regression testing.

There is a Makefile variable called REGRESS in the algebra Makefile:
\begin{verbatim}
REGRESS=\
 AssociationList.regress  BalancedBinaryTree.regress \
\end{verbatim}

This is part of a Makefile that structure within the algebra Makefile.
This Makefile gets extracted by the Makefile in the input subdirectory.
Thus there is a connection between the two Makefiles (algebra and input).
This algebra regression Makefile goes by the chunk name 
{\bf algebra.regress}. It contains a list of regression files and a 
single stanza:
\begin{verbatim}
%.regress: %.input
	@ echo algebra regression testing $*
	@ rm -f $*.output
	@ echo ')read $*.input' | ${TESTSYS} 
	@ echo ')lisp (regress "$*.output")' | ${TESTSYS} \
                | egrep -v '(Timestamp|Version)' | tee $*.regress
\end{verbatim}
The input Makefile extracts {\bf algebra.regress} and then calls 
make to process this file.

This keeps the regression test list in the algebra Makefile.

\subsection{help documentation as algebra test files}


\section{debugsys}
The ``debugsys'' executable is the ``interpsys'' image but it is
built using the interpreted lisp code rather than using compiled
lisp code. This will make it slower but may, in certain cases,
give much better feedback in case of errors. If you find you need
to use debugsys you're really doing deep debugging. It isn't useful
for much else. It can be started by typing:
\begin{verbatim}
export AXIOM=/home/axiomgnu/new/mnt/linux
/home/axiomgnu/new/obj/linux/bin/debugpsys
\end{verbatim}
Notice that this image lives in the ``obj'' subtree. It is not
shipped with the ``final'' system image as only developers could
find it useful.
\subsection{debugging hyperdoc}
Hyperdoc will sometimes exit and also kill the AXIOMsys image with
no error message. One way to get around this is to replace the 
AXIOMsys image with the debugsys image:
\begin{enumerate}
\item mv \$AXIOM/bin/AXIOMsys \$AXIOM/bin/AXIOMsys.backup\\
This keeps the failing axiomsys image around for later restoration.
\item cp obj/sys/bin/debugsys \$AXIOM/bin/AXIOMsys\\
This puts an interpreted version of axiom in place of the compiled form
\item axiom\\
Now we are running a fully interpreted form and the error messages are
much more informative.
\end{enumerate}
\section{Understanding a compiled function}
Suppose we stop a program at a function call to some low level lisp function,
say ONEP. We can do that by entering
\begin{verbatim}
)trace ONEP )break
\end{verbatim}
at the Axiom command prompt. Or at the lisp prompt:
\begin{verbatim}
(trace (ONEP :entry (break)))
\end{verbatim}
Next we execute some function that will 
eventually call ONEP thus:
\begin{verbatim}
p := numeric %pi

Break: onep
Broken at ONEP.  Type :H for Help.
BOOT>>
\end{verbatim}
We have stopped and entered a lisp command prompt. We can enter any
lisp expression here and there are commands that begin with a ``:''
character. ``:b'' requests a backtrace of the call stack, thus:
\begin{verbatim}
BOOT>>:b
Backtrace: funcall > system:top-level > restart > /read > 
      |upLET| > eval > |Pi| > |newGoGet| > |newGoGet| > ONEP
\end{verbatim}
Here we see that the function ONEP was called by the function newGoGet.
Notice that the name is surrounded by vertical bars. Vertical bars are
a common lisp escape sequence used to allow non-standard characters to
occur in symbol names. Common lisp is not case sensitive. Boot code is
case sensitive. Thus symbol names that were written in Boot tend to have
escape sequence characters around the name. 

Now that we see the simple backtrace we can ask for a more complex one.
The command is ``:bt''. It shows more detail about each level of call
on the invocation history stack (ihs) including the function name, its
arguments and the depth of the invocation history stack ([ihs=13]):
\begin{verbatim}
BOOT>>:bt

#0   ONEP {1=nil,} [ihs=13]
#1   newGoGet {g3629=("0" (#<vector 08b34bb4> 45 . |char|)),
       loc1=#<compiled-function |CHAR;cha...} [ihs=12]
#2   newGoGet {g3629=("%pi" (#<vector 08b34bec> 0 . |coerce|)),
       loc1=(#<vector 08b34bec> 0 . |c...} [ihs=11]
#3   Pi {g109299=nil,loc1=nil,loc2=#<hash-table 082992f4>,
       loc3=|Pi|,loc4=15,loc5=#<vecto...} [ihs=10]
#4   EVAL {loc0=nil,loc1=nil,loc2=nil,
       loc3=#<compiled-function |Pi|>} [ihs=9]
#5   upLET {t=(#<vector 08b34d04> #<vector 08b34ce8> 
       (#<vector 08b34ccc> (#<vector 08b34c08...} [ihs=8]
#6   /READ {loc0=#p"/home/axiomgnu/new/src/input/algbrbf.input",
       loc1=nil,loc2=nil,loc3=nil,...} [ihs=7]
#7   RESTART {loc0=((|read| 
      |/home/axiomgnu/new/src/input/algbrbf.input|)),
      loc1=|/home/axiomg...} [ihs=6]
#8   TOP-LEVEL {loc0=nil,loc1=0,loc2=0,loc3=nil,loc4=nil,
      loc5=nil,loc6=nil,loc7=nil,loc8=nil,lo...} [ihs=5]
#9   FUNCALL {loc0=#<compiled-function system:top-level>} [ihs=4]
BOOT>>:bl
>> (LAMBDA-BLOCK ONEP (&REST X) ...)():
X        : (1)
NIL
\end{verbatim}
We can ask to see the local variables that are used at the current 
level of the invocation history stack. The command is ``:bl'' thus:
\begin{verbatim}
BOOT>>:bl
>> (LAMBDA-BLOCK ONEP (&REST X) ...)():
X        : (1)
NIL
\end{verbatim}
We can move up the stack one level at a time looking at the function
that called the current function (the previous function) using ``:p'' thus:
\begin{verbatim}
BOOT>>:p
Broken at |NEWGOGET|.
\end{verbatim}
And again, we can look at the variables that can be accessed locally:
\begin{verbatim}
BOOT>>:bl
>> newGoGet():
Local0(G3629): (0 (#<vector 08b34bb4> 45 . char))
Local(1): #<compiled-function CHAR;char;S$;20>
Local(2): 0
Local(3): #<vector 08b233f0>
Local(4): 1
NIL
\end{verbatim}
Here we see that the function newGoGet is calling 
CHAR;char;S\$;20 which is a mangled form of 
the name of the original spad function. To decode this name 
we can see that the CHAR portion is used to identify the 
domain where the function lives. This domain, CHAR, comes 
from the source file ``string.spad'' which lives in 
``src/algebra/string.spad.pamphlet''. To discover this we use 
the Axiom ``asq'' command with the ``-so'' (sourcefile) option 
at a standard shell prompt (NOT in the lisp prompt) thus:
\begin{verbatim}
asq -so CHAR
string.spad
\end{verbatim}
If we look at the code in the string.spad.pamphlet file we find 
the following code signature:
\begin{verbatim}
 char: String   -> %
  ++ char(s) provides a character from a string s of length one.
\end{verbatim}
and it's implementation code:
\begin{verbatim}
 char(s:String) ==
  (#s) = 1 => s(minIndex s) pretend %
  error "String is not a single character"
\end{verbatim}
The string.spad file can be compiled at the command prompt. 
In particular, we can compile only the CHAR domain out of this 
file thus:
\begin{verbatim}
)co string.spad )con CHAR
\end{verbatim}
This will produce a directory called CHAR.NRLIB containing 3 files:
\begin{verbatim}
ls CHAR.NRLIB
code.lsp index.kaf info
\end{verbatim}
The info file contains information used by the spad compiler. We can
ignore it for now.

The index.kaf file contains information that will go into the various
Axiom database (.daase) files. The kaf file format is a random access
file. The first entry is an integer that will be an index into the
file that can be used in an operating system call to seek. In this
case it will be an index which is the last used byte in the file.
Go to the last expression in the file and we find:
\begin{verbatim}
(
 ("slot1Info" 0 11302)
 ("documentation" 0 9179)
 ("ancestors" 0 9036)
 ("parents" 0 9010)
 ("abbreviation" 0 9005)
 ("predicates" 0 NIL)
 ("attributes" 0 NIL)
 ("signaturesAndLocals" 0 8156)
 ("superDomain" 0 NIL)
 ("operationAlist" 0 7207)
 ("modemaps" 0 6037)
 ("sourceFile" 0 5994)
 ("constructorCategory" 0 5434)
 ("constructorModemap" 0 4840)
 ("constructorKind" 0 4831)
 ("constructorForm" 0 4817)
 ("NILADIC" 0 4768)
 ("compilerInfo" 0 2093)
 ("loadTimeStuff" 0 20))
\end{verbatim}
This is a list of triples. Each triple has two interesting parts, the
name of the data and the seek index of the data in the index.kaf file.
So, for instance, if you want to know what source file contains this
domain you can start at the top of the index.kaf file, move ahead 5994
bytes and you will be at the start of the string:
\begin{verbatim}
"/usr/local/axiom/src/algebra/string.spad"
\end{verbatim}
The information in the index.kaf files are collected into the
special databases (the .daase files). The stand-alone ``asq'' function
can query these databases and answer questions. The kind of questions
you can ask are the names in the list above. 

The third file in the CHAR.NRLIB directory is the code.lsp file. This
is the actual common lisp code that will be executed as a result of
calling the various spad functions. The spad code from the char command
was:
\begin{verbatim}
	char(s:String) ==
	  (#s) = 1 => s(minIndex s) pretend %
	  error "String is not a single character"
\end{verbatim}
which got compiled into the common lisp code:
\begin{verbatim}
(DEFUN |CHAR;char;S$;20| (|s| |$|)
 (COND
  ((EQL (QCSIZE |s|) 1)
    (SPADCALL |s| 
      (SPADCALL |s| (QREFELT |$| 47))
      (QREFELT |$| 48)))
  ((QUOTE T)
    (|error| "String is not a single character")))) 
\end{verbatim}
To understand what is going on here we need to understand the low level
details of Axiom's interface to Common Lisp. The ``Q'' functions are
strongly typed (Quick) versions of standard common lisp functions.
QCSIZE is defined in src/interp/vmlisp.lisp.pamphlet thus:
\begin{verbatim}
(defmacro qcsize (x)
 `(the fixnum (length (the simple-string ,x))))
\end{verbatim}
This macro will compute the length of a string.

QREFELT is defined in the same file as:
\begin{verbatim}
(defmacro qrefelt (vec ind)
 `(svref ,vec ,ind))
\end{verbatim}
This macro will return the element of a vector.

SPADCALL is defined in src/interp/macros.lisp.pamphlet as:
\begin{verbatim}
(defmacro SPADCALL (&rest L)
  (let ((args (butlast l)) (fn (car (last l))) (gi (gensym)))
   `(let ((,gi ,fn)) 
     (the (values t) (funcall (car ,gi) ,@args (cdr ,gi))))
   ))
\end{verbatim}
This macro will call the last value of the argument list as a function
and give it everything but the last argument as arguments to the function.
There are confusing historical reasons for this I won't go into here.

So you can see that these are simply macros that will expand into 
highly optimizable (the optimizations depend on the abilities of the
common lisp compiler) common lisp code. 

The common lisp code computes the length of the string s. If the length
is 1 then we call the minIndex function from string on s. The minIndex
function is found by looking ``in the domain''. The compiler changes the
minIndex function call into a reference into a vector. The 47th element
of the vector contains the function minIndex. 
\begin{verbatim}
      (SPADCALL |s| (QREFELT |$| 47))
\end{verbatim}
This code is equivalent (ignoring the gensyms) to the call
\begin{verbatim}
(minIndex s)
\end{verbatim}
The \$ symbol refers to the domain. At runtime this amounts to a lookup
of the ``infovec''. The compile-time infovec shown here:
\begin{verbatim}
(setf (get
 (QUOTE |Character|)
 (QUOTE |infovec|))
 (LIST
  (QUOTE
   #(NIL
     NIL
     NIL
     NIL 
     NIL
     NIL
     (QUOTE |Rep|)
     (|List| 28)
     (|PrimitiveArray| 28)
     (0 . |construct|)
     (QUOTE |OutChars|)
     (QUOTE |minChar|)
     (|Boolean|)
     |CHAR;=;2$B;1|
     |CHAR;<;2$B;2|
     (|NonNegativeInteger|)
     |CHAR;size;Nni;3|
     (|Integer|)
     |CHAR;char;I$;6|
     (|PositiveInteger|)
     |CHAR;index;Pi$;4|
     |CHAR;ord;$I;7|
     |CHAR;lookup;$Pi;5|
     (5 . |coerce|)
     |CHAR;random;$;8|
     |CHAR;space;$;9|
     |CHAR;quote;$;10|
     |CHAR;escape;$;11|
     (|OutputForm|)
     |CHAR;coerce;$Of;12|
     (|CharacterClass|)
     (10 . |digit|)
     (|Character|)
     (14 . |member?|)
     |CHAR;digit?;$B;13|
     (20 . |hexDigit|)
     |CHAR;hexDigit?;$B;14|
     (24 . |upperCase|)
     |CHAR;upperCase?;$B;15|
     (28 . |lowerCase|)
     |CHAR;lowerCase?;$B;16|
     (32 . |alphabetic|)
     |CHAR;alphabetic?;$B;17|
     (36 . |alphanumeric|)
     |CHAR;alphanumeric?;$B;18|
     (|String|)
     |CHAR;latex;$S;19|
     (40 . |minIndex|)
     (45 . |elt|)
     |CHAR;char;S$;20|
     |CHAR;upperCase;2$;21|
     |CHAR;lowerCase;2$;22|
     (|SingleInteger|)))
   (QUOTE
     #(|~=| 51 |upperCase?| 57 |upperCase| 62 |space| 67 
       |size| 71 |random| 75 |quote| 79 |ord| 83 |min| 88 
       |max| 94 |lowerCase?| 100 |lowerCase| 105 |lookup| 110 
       |latex| 115 |index| 120 |hexDigit?| 125 |hash| 130
       |escape| 135 |digit?| 139 |coerce| 144 |char| 149 
       |alphanumeric?| 159 |alphabetic?| 164 |>=| 169 |>| 175 
       |=| 181 |<=| 187 |<| 193))
   (QUOTE NIL)
   (CONS
    (|makeByteWordVec2| 1 (QUOTE (0 0 0 0 0 0)))
    (CONS
      (QUOTE #(NIL |OrderedSet&| NIL |SetCategory&| 
              |BasicType&| NIL))
      (CONS
        (QUOTE 
         #((|OrderedFinite|)
           (|OrderedSet|)
           (|Finite|)
           (|SetCategory|)
           (|BasicType|)
           (|CoercibleTo| 28)))
        (|makeByteWordVec2| 52 
          (QUOTE 
            (1 8 0 7 9 1 6 0 17 23 0 30 0 31 2 30 12 32 0 33 
             0 30 0 35 0 30 0 37 0 30 0 39 0 30 0 41 0 30 0 
             43 1 45 17 0 47 2 45 32 0 17 48 2 0 12 0 0 1 1 
             0 12 0 38 1 0 0 0 50 0 0 0 25 0 0 15 16 0 0 0 24
             0 0 0 26 1 0 17 0 21 2 0 0 0 0 1 2 0 0 0 0 1 1 0 
             12 0 40 1 0 0 0 51 1 0 19 0 22 1 0 45 0 46 1 0 0 
             19 20 1 0 12 0 36 1 0 52 0 1 0 0 0 27 1 0 12 0 34 
             1 0 28 0 29 1 0 0 45 49 1 0 0 17 18 1 0 12 0 44 1 
             0 12 0 42 2 0 12 0 0 1 2 0 12 0 0 1 2 0 12 0 0 13 
             2 0 12 0 0 1 2 0 12 0 0 14))))))
    (QUOTE |lookupComplete|)))
\end{verbatim}
Which is a 5 element list. This contains all kinds of information used at
runtime by the compiled routines. In particular, functions are looked up
at runtime in the first element of the infovec list. This first element
contains 53 items (in this domain). Item 47 is
\begin{verbatim}
     (40 . |minIndex|)
\end{verbatim}
which is the minIndex function we seek. 

At runtime this infovec lives on the property list of the domain name.
The domain name of CHAR is Character. So we look on the property list
(a lisp a-list) thus:
\begin{verbatim}
BOOT>>(symbol-plist '|Character|)

(SYSTEM:DEBUG (#:G85875)
 |infovec| (#<vector 08b34380>
            #<vector 08b34364>
            NIL
            (#<bit-vector 08b34310>
             #<vector 08b34348>
             #<vector 08b3432c> . #<vector 08b342f4>)
            |lookupComplete|) 
 LOADED "/home/axiomgnu/new/mnt/linux/algebra/CHAR.o" 
 NILADIC T 
 PNAME "Character" 
 DATABASE #S(DATABASE 
             ABBREVIATION CHAR 
             ANCESTORS NIL 
             CONSTRUCTOR NIL 
             CONSTRUCTORCATEGORY 228064 
             CONSTRUCTORKIND |domain| 
             CONSTRUCTORMODEMAP 227069 
             COSIG (NIL) 
             DEFAULTDOMAIN NIL 
             MODEMAPS 227404 
             NILADIC T 
             OBJECT "CHAR" 
             OPERATIONALIST 226402 
             DOCUMENTATION 152634 
             CONSTRUCTORFORM 152626 
             ATTRIBUTES 154726 
             PREDICATES 154731 
             SOURCEFILE "string.spad" 
             PARENTS NIL 
             USERS NIL 
             DEPENDENTS NIL 
             SPARE NIL))
\end{verbatim}
This list is organized contains many runtime lookup items (notice the 
PNAME entry is ``Character'', the LOADED entry says where the file came
from, the DATABASE structure entry has database indicies
(see daase.lisp.pamphlet for the structure definition), etc).

Lets get the property list
\begin{verbatim}
BOOT>>(setq a (symbol-plist '|Character|))

(SYSTEM:DEBUG (#:G85875)
 |infovec| (#<vector 08b34380>
            #<vector 08b34364>
            NIL
            (#<bit-vector 08b34310>
             #<vector 08b34348>
             #<vector 08b3432c> . #<vector 08b342f4>)
            |lookupComplete|)
 LOADED "/home/axiomgnu/new/mnt/linux/algebra/CHAR.o"
 NILADIC T
 PNAME "Character"
 DATABASE #S(DATABASE 
             ABBREVIATION CHAR 
             ANCESTORS NIL 
             CONSTRUCTOR NIL 
             CONSTRUCTORCATEGORY 228064 
             CONSTRUCTORKIND |domain| 
             CONSTRUCTORMODEMAP 227069 
             COSIG (NIL) 
             DEFAULTDOMAIN NIL 
             MODEMAPS 227404 
             NILADIC T 
             OBJECT "CHAR" 
             OPERATIONALIST 226402 
             DOCUMENTATION 152634 
             CONSTRUCTORFORM 152626 
             ATTRIBUTES 154726 
             PREDICATES 154731 
             SOURCEFILE "string.spad" 
             PARENTS NIL 
             USERS NIL 
             DEPENDENTS NIL 
             SPARE NIL))
\end{verbatim}
Next we get the infovec value
\begin{verbatim}
BOOT>>(setq b (fourth a))

(#<vector 08b34380>
 #<vector 08b34364>
 NIL
 (#<bit-vector 08b34310>
  #<vector 08b34348>
  #<vector 08b3432c> . #<vector 08b342f4>)
 |lookupComplete|)
\end{verbatim}
Then we get the function table
\begin{verbatim}
BOOT>>(setq c (car b))

#<vector 08b34380>
\end{verbatim}
In this common lisp (GCL) the array is identified by it's memory address.

Notice that it has the right number of entries:
\begin{verbatim}
BOOT>>(length c)

53
\end{verbatim}
And we  can ask for the 47th entry thus:
\begin{verbatim}
BOOT>>(elt c 47)

(40 . |minIndex|)
\end{verbatim}
Later we end up calling the 48th function (which is elt and
returns the actual character in the string). We ask for it:
\begin{verbatim}
BOOT>>(elt c 48)

(45 . |elt|)
\end{verbatim}
At this point we've reached the metal. Common lisp will evaluate
the macro-expanded functions and execute the proper code. Essentially
the compiler has changed all of our spad code into runtime table lookups.

\section{The axiom.input startup file}
If you add a file in your home directory called ``.axiom.input'' it will
be read and executed when Axiom starts. This is useful for various
reasons including setting various switches. Mine reads:
\begin{verbatim}
)lisp (pprint ``running /root/.axiom.input'')
)set quit unprotected
)set message autoload off
)set message startup off
\end{verbatim}
You can execute any command in .axiom.input. Be aware that this will
ALSO be run while you are doing a ``make'' so be careful what you ask to do.

\section{Where are Axiom symbols stored?}

You'd think that your question about where the symbol is interned
would be easy to answer but it is not. The top level loop uses Bill
Burge's dreaded zipper parser. You can see it in action by executing
the following sequence:
\begin{verbatim}

)lisp (setq $DALYMODE t)    
  ; this is a special mode of the top level interpreter. If 
  ; $DALYMODE is true then any top-level form that begins 
  ; with an open-paren is considered a lisp expression.
  ; For almost everything I ever do I end up peeking at the 
  ; lisp so this bit of magic helps.
(trace |intloopProcessString|) 
  ; from int-top.boot.pamphlet
(trace |intloopProcess|)       
  ; the third argument is the "zippered" input
(trace |intloopSpadProcess|)   
  ; now it is all clear, no? sigh.
(trace |phInterpret|)          
  ; from int-top.boot.pamphlet
(trace |intInterpretPform|)   
  ; from intint.lisp.pamphlet
(trace |processInteractive|)   
  ; from i-toplev.boot.pamphlet
(setq |$reportInstantiations| t) 
  ; shows what domains were created
(setq |$monitorNewWorld| t)    
  ; watch the interpreter resolve operations
(trace |processInteractive1|)  
  ; from i-toplev.boot.pamphlet

\end{verbatim}
ah HA! I remember now. There is the notion of a ``frame'' which is
basically a namespace in Axiom or an alist in Common Lisp. It is
possible to maintain different ``frames'' and move among them. There
is the notion of the current frame and it contains all the defined
variables. At any given time the current frame is available as
\$InteractiveFrame. This variable is used in processInteractive1.
If you do:
\begin{verbatim}

a:=7
(pprint |$InteractiveFrame|)

\end{verbatim}
you'll see |a| show up on the alist. When you do the 
\begin{verbatim}

pgr:=MonoidRing(Polynomial PrimeField 5, Permutation Integer)
p:pgr:=1

\end{verbatim}
you'll see |p| show up with 2 other things: (|p| mode value)
where mode is the ``type'' of the variable. The value is the
internal value. In this case MonoidRing has an internal
representation. You can find out what the internal representation
of a MonoidRing is by first asking where the source file is:
\begin{verbatim}

(do this at a shell prompt, not in axiom)
asq -so MonoidRing ==> mring.spad

     -- or -- in Axiom type:

)show MonoidRing

\end{verbatim}
and you'll see a line that reads: 
\begin{verbatim}

Issue )edit (yourpath)/../../src/algebra/mring.spad

\end{verbatim}

If you look in mring.spad.pamphlet you'll see line 91 that reads:
\begin{verbatim}

   Rep := List Term

\end{verbatim}
which says that we will store elements of type MonoidRing as a list
of Term objects. Term is defined in the same file (as a macro, which
is what '$==>$' means in spad files) on line 43:
\begin{verbatim}

   Term ==> Record(coef: R, monom: M)

\end{verbatim}
which means that elements of a MonoidRing are Lists of Records.
The 'R' is defined on line 42 as the first argument to MonoidRing
which in this case is ``Polynomial PrimeField 5''. The ``M'' is also
defined on line 42 as the second argument to MonoidRing and in this
case is ``Permutation Integer''. So the real representation is
\begin{verbatim}

  List Record(coef: Polynomial PrimeField 5, 
              monom: Permutation Integer)

\end{verbatim}
In the \$InteractiveFrame we printed out you can see in the value
field that the value is:
\begin{verbatim}

(|value| 
  (|MonoidRing| (|Polynomial| (|PrimeField| 5))
                (|Permutation| (|Integer|)))
   WRAPPED ((0 . 1) . #<vector 08af33d4>))

\end{verbatim}
which basically means that we know how the MonoidRing was constructed and
what it's current value is. The (0 . 1) likely means that this is the
zeroth (constant) term with a leading coefficient of 1. This is just a
guess as I haven't decoded the representation of either Polynomial PrimeField 
or Permutation Integer. You can do the same deconstruction of these two
domains by setting
\begin{verbatim}

pi:=Permutation Integer
z:pi:=1

pp5:=Polynomial PrimeField 5
w:pp5:=1

and following the same steps as above: 
 (pprint |$InteractiveFrame|)
 )show pi
 (find the source file)
 (find the representation and decode it)

 (pprint |$InteractiveFrame|)
 )show pp5
 (find the source file)
 (find the representation and decode it)

\end{verbatim}
Be sure to set \$DALYMODE to nil if you plan to use Axiom for any
real computation. Otherwise every expression that begins with an
open-paren will go directly to lisp.
\section{Translating individual boot files to common lisp}

If you are making changes to boot code it is sometimes helpful to
check the generated lisp code to ensure it does what you want.
You can convert an individual boot file to common lisp using the
boottran::boottocl function:
\begin{verbatim}
)fin       -- drop into common lisp
(boottran::boottocl "foo.boot") 
\end{verbatim}
when you do this it creates a foo.clisp file in ../../int/interp

Alternatively if you work from the pamphlet file the process is
more painful as you have to do
\begin{verbatim}
)cd (yourpath)/int/interp
)sys tangle ../../src/interp/foo.boot.pamphlet >foo.boot
)fin
(boottran::boottocl "foo.boot") 
(restart)
\end{verbatim}
The )cd step tells axiom to cd to the int/interp subdirectory.
The )sys tangle... extracts the boot file from the pamphlet file
The )fin step drops into common lisp
The (bootran... converts the foo.boot file to foo.clisp
The (restart) re-enters the top level loop

\section{Directories}
For this discussion I assume that you have your system rooted
at /spad and was build to run on linux. These directories
may not yet be in the CVS tree but are documented here so they
make sense when the show up.

\vskip .25in
\noindent
The AXIOM variable

The usual setting of the AXIOM variable is /spad/mnt/linux. The
name is composed of three parts, the rooted path, in this case
/spad, ``mnt'', and the system you are running, in this case
linux. Builds for other systems will have other system names.

\vskip .25in
\noindent
/spad

This is the usual root directory of the Axiom system. The name is
historical, a contraction of Scratchpad. This name can be anything
provided the shell variable AXIOM contains the new prefix. 

\vskip .25in
\noindent
/spad/mnt

This is a directory which contains files which are specific to a
given platform. At a site that contains multiple platforms this
directory will contain a subdirectory for each type of platform
(e.g. linux, rios, ps2, rt, sun, etc).

\vskip .25in
\noindent
/spad/mnt/linux

This directory contains the complete copy of the Axiom system for
the linux system. This is the 'mount point' of the system.
Executable systems (for RedHat) are shipped relative to this
point. In what follows, the ./ refers to /spad/mnt/linux.

\begin{verbatim}
********************************************************
There are several directories explained below. They are:

./bin     -- user executables
./doc     -- system documentation
./algebra -- algebra libraries
./lib     -- system executables
./etc     -- I haven't a clue....
********************************************************
\end{verbatim}

\subsection{The mnt/linux/bin directory}
\vskip .25in
\noindent
./bin

This is a directory of user executable commands, either at the top
level or thru certain Axiom system calls. Support executables live
in ./lib

\vskip .25in
\noindent
./bin/htadd

This adds pages to the Hyperdoc database (ht.db, which lives
in ./doc/hypertex/pages; hypertex, since we have a penchant for these
things, is an historical name for Hyperdoc. The single word 'lawyers'
will probably explain away a lot of name changes.)

\vskip .25in
\noindent
./bin/spadsys

This is the Axiom interpreter. It is one of the functions started when
the user invokes the system using the spadsys command. Normally this 
command is run under the control of sman (./lib/sman) and the console
is under the control of clef (./bin/clef), the wonderous command-line
editor. It is possible to start spadsys standalone but it will not
talk to Hyperdoc or graphics. Users who rlogin or use an ascii-only
terminal (for historical reasons, no doubt) can profit by invoking
spadsys directly rather than using ./bin/axiom

\vskip .25in
\noindent
./bin/axiom

This is a shell script that spins the world. It kicks off a whole tree
of processes necessary to perform the X-related magic we do. It expects
the shell variable AXIOM to be set to the 'mount point' (usually to
/spad/mnt/linux).

\vskip .25in
\noindent
./bin/clef

This is the wonderous command-line editor used by Axiom. It can be used
in a stand-alone fashion if you wish.

\vskip .25in
\noindent
./bin/SPADEDFN

This script is invoked by the spad )fe command. It can be changed to 
invoke your favorite editor. While you may invoke your editor, it may
not run (as in, yes, I can invoke the devil but will he come when I 
call?)

\vskip .25in
\noindent
./bin/viewalone

This is a function to run the graphics in a stand-alone fashion. The
Graphics package (an amazing contribution by several very talented
people, most notably Jim Wen and Jon Steinbach) is a C program that
communicates with Axiom thru sockets. It will, however, perform its
miracles unaided if invoked by the sufficiently chaste...

\vskip .25in
\noindent
./bin/hypertex

This is a function to run Hyperdoc (remember the penchant!) stand-alone.
The Hyperdoc package owes its existence to the efforts of J.M. Wiley and
Scott Morrison. This function works off 'pages' that live in 
hypertex pages directory and are referenced in the ``hyperdoc database''
called ht.db (for historical reasons, but you 
knew that). It is possible for creative plagerists to figure out how
to write their own pages and add them to the database (see htadd above),
thus gaining fame far and wide...

\vskip .25in
\noindent
./bin/sys-init.lsp

This is a file of lisp code that gets loaded
before Axiom starts. Thus, we distribute patches by adding lisp
(load ...) commands to this file. The sufficiently clever should
have a field day with this one. (All others should worship the
sufficiently clever and send them money, eh?)

\vskip .25in
\noindent
./bin/init.lsp

This is a file of lisp code loaded if and only if
you start spadsys in this directory. The user can put a file of this
name in her home directory and it will get loaded at startup with
the probable effect of injecting luser errors into the running system.
sigh.

\subsection{The mnt/linux/doc directory}
\vskip .25in
\noindent
./doc

The doc subdirectory contains system documentation.

\vskip .25in
\noindent
./doc/command.list

This is a file of command completions used by clef when you hit the
tab key. This is a little known feature that will surprise someone
someday (hopefully pleasantly).

\vskip .25in
\noindent
./doc/book

This is an attempt at a book describing Axiom. It represents a 
combination of fantasy, describing what never will be and history
(remember the penchant?) describing what was. Any description matching
what is may be regarded as failure of the imagination and ignored.

\vskip .25in
\noindent
./doc/compguide

This is an attempt to describe a compiler that doesn't exist, never
did exist, and never will exist. It makes for entertaining reading
so we included it.

\vskip .25in
\noindent
./doc/hypertex

This is the fabled Hyperdoc subdirectory where all of the pages and
the database live, along with several other obscure files needed to
make the wizards look good.

\vskip .25in
\noindent
./doc/hypertex/pages

This is where the 'pages' live. Each file ending in .ht contains
several pages related, if only by chance, to the same topic. You
may find it instructive to try to read some of these files. Hyperdoc
was learned by the 'campfire' method (sitting around the fire 
passing along historical facts by word of mouth) and will probably
continue to propagate by the same method. Ye may become th' local
scribe and soothsayer if ye study the writings here below....

\vskip .25in
\noindent
./doc/hypertex/pages/rootpage.ht

This file is the magic 'first page' that gets displayed when
Hyperdoc starts. There is a macro (see ./doc/hypertex/pages/util.ht)
called /localinfo which is intended to allow the luser to add
her own pages without modifying the system copies. How this is
done was lost when the campfire got rained out.

\vskip .25in
\noindent
./doc/hypertex/pages/util.ht

This file contains the macros used to extend the system commands.
The syntax is hard to learn (it was hard to write, it ought to 
be hard to learn, eh?).

\vskip .25in
\noindent
./doc/hypertex/pages/ht.db

This is the Hyperdoc database. It is updated using ./bin/htadd
which must be run whenever a page in this directory gets changed.
The necessary arguments to htadd are obvious to those in the know.

\vskip .25in
\noindent
./doc/hypertex/bitmaps

There are several pretty bitmaps used as cursors, buttons and
general decorations that hide in this directory. 

\vskip .25in
\noindent
./doc/hypertex/ht.files

This is a list of some Hyperdoc files. It seems to have no
purpose in life but it is useful as a koan, as in, What is
the length of half a list?

\vskip .25in
\noindent
./doc/hypertex/ht.db

Another copy of the Hyperdoc database. It isn't clear which
one is the real one so I guess we keep both. Maybe we'll 
figure it out at the friday night campfire provided we don't
get too lit.

\vskip .25in
\noindent
./doc/hypertex/gloss.text

The text used in the glossary. Many magic words lie herein.
Some are spoken only by campfire gurus.

\vskip .25in
\noindent
./doc/library

This is a directory of Hyperdoc pages that can be freely smashed,
trashed and generally played with. It uses the /localinfo connection
to set up a 'library' containing Hyperdoc pages keyed to your 
favorite textbook. It is interesting to set the shell variable\\
HTPATH=/spad/mnt/linux/doc/library:\\
       /spad/mnt/linux/doc/hypertex/pages\\
and then start Hyperdoc. See the file ./doc/library/macros.ht

\vskip .25in
\noindent
./doc/msgs

This directory contains several 'message databases'; the only one
of which we seem to care about being s2-us.msgs but I can't swear
to it.

\vskip .25in
\noindent
./doc/spadhelp

This is a directory containing help information for a copy of the
system that once ran long ago and far away. It is kept for 
historical reasons (programmers NEVER throw anything away).

\vskip .25in
\noindent
./doc/viewports

There are several dozen truly fine pictures in Axiom. We have 
created them and hidden them here. Hyperdoc will insert them
at various places (where the text gets too boring, hopefully)
and you can click on them there. They get snarfed from here.
It is possible to view them with stand-alone graphics but don't
ask me how. I missed that campfire due to poisoned marshmellows.

\vskip .25in
\noindent
./doc/complang

This directory contains fantasy from the past as opposed to facts 
from the future. Ignore it.

\vskip .25in
\noindent
./doc/ug

This directory left intentionally blank :-) (an old IBM joke).

\vskip .25in
\noindent
./doc/tex

These are the files necessary to create the famous goertler document.
If you figure out how to use these please send us the instructions
and we will add a log to the campfire with your name on it (a rare
honor indeed as luser's names rarely reach the inner circle).

\vskip .25in
\noindent
./doc/htex

This directory contains the original tex-like source for the luser's
guide. There are many functions that munch on these between here and
paper but this is approximately where they start. If you do your own
algebra perchance you might document it like this. Figuring out the
syntax will also get your name into the inner circle (probably 
connnected with a smirk :-) )

\vskip .25in
\noindent
./doc/newug

Please don't ask me. I couldn't begin to guess. You wouldn't believe
how many 'new' things there are that really aren't. We have more NEW
things than Madison Avenue has NEW laundry soap.

\vskip .25in
\noindent
./doc/gloss.text

This one is here because it is here. Existentially speaking, of course.

\vskip .25in
\noindent
./doc/submitted

This was what the htex files said before history was rewritten...
(and renamed?)

\subsection{The mnt/linux/algebra directory}
\vskip .25in
\noindent
./algebra

This is where all of the interesting action lives.
Each .NRLIB directory contains 2 files, a code.o and an index.kaf*
file. The code.o contains the executable algebra that gets loaded
into the system. The index.kaf* file contains all kinds of things
like signatures, source paths, properties and dried bat droppings.
The documentation for each of these can be reached by using the
BROWSE feature of Hyperdoc. 

\vskip .25in
\noindent
./algebra/MODEMAP.daase

This is an inverted database that contains information gleaned from
the index.kaf* files. Without this there is no way to figure out
which .NRLIB file to load. This database is opened on startup
and kept open. 

\vskip .25in
\noindent
./algebra/interp.exposed

This is a control file for the interpeter that limits the number
of places to search for function names. 

*********************************************

\subsection{The mnt/linux/etc directory}
\vskip .25in
\noindent
./lib

This directory contains functions that get loaded by the system.
Nothing in here is executable by the user but the system needs
these functions to run.

\vskip .25in
\noindent
./lib/htrefs\\
./lib/htsearch\\
./lib/hthits

These three functions are used to search the Hyperdoc pages.
There is no way in the current system to request a search of
those pages so these files are fascinating examples of history
in the making...

\vskip .25in
\noindent
./lib/hypertex

This is Hyperdoc. What is in a name?

\vskip .25in
\noindent
./lib/sman

This is sman, which comes before all. Methinks the name originated
as a contraction of superman, the name of a stack frame in a system
long ago and far away (VMLisp) chosen because a certain programmer had a
penchant for comic books when he was young.

\vskip .25in
\noindent
./lib/session\\
./lib/spadclient

These two files are processes started by sman for some reason or other.
I can never remember what they do or why. However, the campfire fails
to smoke if they don't work.

\vskip .25in
\noindent
./lib/viewman

This is the controlling function for the graphics. 

\vskip .25in
\noindent
./lib/view2d

This is invoked when a 2 dimensional window is requested.
This is provided mostly for those math majors who never got over
the insights from flatland.

\vskip .25in
\noindent
./lib/view3d

This is invoked when a 3 dimensional window is requested.
Option IBM3634-A is required to convert your 2 dimensional
screen to 3 dimensions for realistic viewing. A mathematically
accurate, if somewhat more achievable, rendering can be had
on a color or monochrome crt without this upgrade.

\vskip .25in
\noindent
./lib/gloss.text\\
./lib/glosskey.text\\
./lib/glossdef.text

These are three files related to the glossary. The first (gloss.text)
is the original glossary text. The second (glosskey.text) is a list
of terms and pointers into glossdef.text. The third (glossdef.text for
those math majors who can't count) is a list of definitions and pointers
back into the second (guess). These files are used by Hyperdoc.

\vskip .25in
\noindent
./lib/browsedb.lisp

This is the original file that creates an in-memory hash table used by
browse. It is used during system build time. We keep it here to ensure
that the bytes on this section of the disk have a well-defined orientation,
allowing us to compute the spin vectors of the individual magnetic domains.
This allows us to give Heisenburg a sense of direction (at least over the
long run).

\vskip .25in
\noindent
./lib/comdb.text\\
./lib/libdb.text

The first file (comdb.text) contains the so-called $++$ (plus
plus) comments from the algebra files. It contains pointers into 
the second file. The second file (libdb.text) contains flags 
(constructor, operation, attribute) and pointers into the first file. 
These files are used by browse in Hyperdoc.

\vskip .25in
\noindent
./lib/loadmprotect\\
./lib/mprotect

This set of two files has been mercifully de-installed from the system.
They will, if used and despite the meaning behind the name, cause 
random system reboots (yeah, HARDWARE reboots. don't ask me how, I'm
just the historian).

\vskip .25in
\noindent
./lib/SPADEDIT\\
./lib/fc\\
./lib/spadbuf\\
./lib/SPADEDFN\\
./lib/obey\\
./lib/ex2ht

I've drawn a blank; intentionally.

\subsection{The mnt/linux/lib directory}
\vskip .25in
\noindent
./etc

This directory intentionally left blank.
We just can't figure out WHY we intended to leave it blank.
Historical reasons, no doubt.

\section{The )set command}
The {\bf )set} command contains many possible options such as:
\begin{verbatim}
                     Current Values of )set Variables                      

Variable  Description                              Current Value
----------------------------------------------------------------
breakmode execute break processing on error          break 
compiler  Library compiler options                   ... 
expose    control interpreter constructor exposure   ... 
functions some interpreter function options          ... 
fortran   view and set options for FORTRAN output    ... 
kernel    library functions built into the kernel for 
          efficiency ... 
hyperdoc  options in using HyperDoc                  ... 
help      view and set some help options             ... 
history   save workspace values in a history file    on 
messages  show messages for various system features  ... 
naglink   options for NAGLink                        ... 
output    view and set some output options           ... 
quit      protected or unprotected quit              unprotected 
streams   set some options for working with streams  ... 
system    set some system development variables      ... 
userlevel operation access level of system user      development 

Variables with current values of ... have further 
sub-options. For example, 
issue )set  system to see what the options are for system .
For more information, issue )help set .
\end{verbatim}
The table that contains these options lives in setvart.boot.pamphlet.
The actual code that implements these options is sprinkled around
but most of the first-level calls resolve to functions in 
setvars.boot.pamphlet. Thus if you plan to add a new output 
style to the system, or figure out where a current style is
broken, these two files are the place to start.

A new )set breakmode command has been implemented to handle the
case that you might want an error message or an error return code
from AXIOMsys. You can set this option with 
\begin{verbatim}
  )set breakmode quit
\end{verbatim}
This will cause AXIOMsys to exit with the return code of 1.
Note that if you invoke the ``axiom'' shell script to start
AXIOMsys you will not see this return code (sman swallows it).

\section{Special Output Formats}
The first level of special output formatting is handled by
functions in setvart.boot.pamphlet. This handles the options
given to the )set command.

\section{Low Level Debugging Techniques}
It should be observed that Axiom is basically Common Lisp and 
some very low level techniques can be used to find where problems
occur in algebra code. This section walks thru a small problem and
illustrates some techniques that can be used to find bugs. The 
point of this exercise is to show a few techniques, not to show a
general method.
\subsection{Finding Anonymous Function Signatures}
This is a technique, adapted from Waldek Hebisch, 
for asking the interpreter to reveal the actual
function that will be called in a given circumstance. Here we have a
function tanint from the domain ElementaryIntegration. 
\begin{verbatim}
     tanint(f, x, k) ==
       eta' := differentiate(eta := first argument k, x)
       r1  := tanintegrate(univariate(f, k), differentiate(#1,
               differentiate(#1, x), monomial(eta', 2) + eta'::UP),
                 rischDEsys(#1, 2 * eta, #2, #3, x, lflimitedint(#1, x, #2),
                    lfextendedint(#1, x, #2)))
       map(multivariate(#1, k), r1.answer) + lfintegrate(r1.a0, x)
\end{verbatim}

We would like to know the type signature of the first argument to the 
inner call to the differentiate function:
\begin{verbatim}
               differentiate(#1, x), monomial(eta', 2) + eta'::UP),
\end{verbatim}

We see that differentiate is called with \verb|#1|, which is Axiom's
notation for an anonymous function. How can we determine the signature?

Axiom has a second notation for anonymous functions using the 
\verb|+->| notation. This notation allows you to explicitly specify
type information. In the above code, we would like to replace the
\verb|#1| variable with the \verb|+->| and explicit type information.

The first step is to look at the output of the Spad compiler. 
The abbreviation for ElementaryIntegration can be found from the 
interpreter by:
\begin{verbatim}
  )show ElementaryIntegration
    Abbreviation for ElementaryIntegration is INTEF 
\end{verbatim}

So the compiler output is in the int/algebra/INTEF.nrlib/code.lsp file.

There we see the definition of the lisp tanint function. Notice that the
\verb|$| is a hidden, internal fourth argument to an Axiom three argument
function. This is the vector of the current domain containing slots where
we can look up information, called the domain vector.

\begin{verbatim}
(DEFUN |INTEF;tanint| (|f| |x| |k| $)
 (PROG (|eta| |eta'| |r1|)
  (RETURN
   (SEQ
    (LETT |eta'|
     (SPADCALL
       (LETT |eta|
        (|SPADfirst|
          (SPADCALL |k| (QREFELT $ 18)))
          |INTEF;tanint|)
       |x|
       (QREFELT $ 19))
     |INTEF;tanint|)
    (LETT |r1|
     (SPADCALL
       (SPADCALL |f| |k| (QREFELT $ 22))
       (CONS (FUNCTION |INTEF;tanint!1|) (VECTOR |eta'| |x| $))
       (CONS (FUNCTION |INTEF;tanint!4|) (VECTOR |x| $ |eta|))
       (QREFELT $ 50))
     |INTEF;tanint|)
    (EXIT
     (SPADCALL
      (SPADCALL 
        (CONS 
          (FUNCTION |INTEF;tanint!5|)
          (VECTOR $ |k|))
        (QCAR |r1|)
        (QREFELT $ 57))
      (SPADCALL (QCDR |r1|) |x| (QREFELT $ 58))
      (QREFELT $ 59))))))) 
\end{verbatim}

The assignment line for \verb|eta'| is:
\begin{verbatim}
       eta' := differentiate(eta := first argument k, x)
\end{verbatim}
which is implemented by the code:
\begin{verbatim}
    (LETT |eta'|
     (SPADCALL
       (LETT |eta|
        (|SPADfirst|
          (SPADCALL |k| (QREFELT $ 18)))
          |INTEF;tanint|)
       |x|
       (QREFELT $ 19))
     |INTEF;tanint|)
\end{verbatim}
from which we see that the inner differentiate is slot 19 in
the domain vector. Every domain has an associated domain vector
which contains references to other functions from other domains,
among other things. The QREFELT function takes the domain vector
\verb|$| and slot number and does a ``quick array reference''. The
return value is a pair, the car of which is a function to call. 
The SPADCALL macro uses the last argument, in this case the result 
of \verb|(QREFELT $ 19)| to find the function to call.

The function from slot 19 can be found with:
\begin{verbatim}
)lisp (setq $dalymode t)
(setf *print-circle* t)
(setf *print-array* nil)
(setf dv (|ElementaryIntegration| (|Integer|) (|Expression| (|Integer|))))
(|replaceGoGetSlot| (cdr (aref dv 19)))
Value = (#<compiled-function |FS-;differentiate;SSS;99|> . #<vector 090cbccc>)
\end{verbatim}

The call of \verb|(setq $dalymode t)| changes the Axiom top level
loop to interpret any input that begins with an open parenthesis to
be interpreted as a lisp s-expression rather than Axiom input. This
saves typing \verb|)lisp| in front of every lisp expression. Be sure
to do a \verb|(setq $dalymode nil)| when you are finished.

The *print-circle* needs to be true because the domain vector contains
circular references to itself and we need to make sure that we check for
this during printing so the print is not infinite.

The *print-array* needs to be nil so that the arrays just print some
identifying information rather than the detailed array contents.

The \verb|(setf dv ...| uses the Lisp internal names for the domains.
In Axiom, the names of types are case-sensitive symbols. These are
represented in lisp surrounded by vertical bars because lisp is not
case sensitive. The dv variable is essentially being set to the Axiom
equivalent of:
\begin{verbatim}
  dv:=ElementaryIntegration(Integer,Expression(Integer))
\end{verbatim}
except we do this in lisp. The end result is that dv will contain the
domain vector for the newly constructed domain. From the lisp code

Consider the call of the form:
\begin{verbatim}
  (SPADCALL A B '(C . D))
\end{verbatim}

The SPADCALL macro takes a set of arguments, the last of which is 
a pair where C is the function to call and D is the domain vector. 
So if we do:
\begin{verbatim}
(macroexpand-1 '(spadcall a b '(c . d)))
Value =
  (LET ((#0=#:G1417 (QUOTE (C . D))))
   (THE (VALUES T) (FUNCALL (CAR #0#) A B (CDR #0#))))
\end{verbatim}
Note that \verb|#0| is a ``pointer'', in this case to the list '(c)
and \verb|#0#| is a use of that pointer. This is done to make sure that
you reference the exact cons cell of the argument.

In Axiom compiler output
\begin{verbatim}
  (SPADCALL eta k (QREFELT $ 19))
\end{verbatim}
approximately translates to
\begin{verbatim}
  (FUNCALL (CAR (QREFELT $ 19)) eta k (CDR (QREFELT $ 19)))
\end{verbatim}
which calls the function from the domain slot 19 on the value
assigned to eta and the variable k and the domain. 
Thus, the full expansion becomes
\begin{verbatim}
  (FUNCALL #<compiled-function |FS-;differentiate;SSS;99|>
    eta k #<vector 090cbccc>)
\end{verbatim}

From this we can see a reference to \verb|FS-;differentiate;SSS;99|
which is the internal name of the differentiate function from the 
\verb|FS-| category.

Note that FunctionSpace is a category. When categories contain
implementation code the compiler generates 2 nrlibs. The Axiom convention for
categorical implementation of code using a trailing ``-'' so the actual
code for \verb|FS-;differentiate;SSS;99| lives in 
int/algebra/FS-.nrlib/code.lsp

We can see that the differentiate function is coming from the category
\begin{verbatim}
)show FS
 FunctionSpace R: OrderedSet  is a category constructor
 Abbreviation for FunctionSpace is FS 

 ....

 differentiate : (%,Symbol) -> % if R has RING
 differentiate : (%,List Symbol) -> % if R has RING
 differentiate : (%,Symbol,NonNegativeInteger) -> % if R has RING
 differentiate : (%,List Symbol,List NonNegativeInteger) -> % if R has RING
\end{verbatim}

From the above signatures we know there is only one differentiate that
is a two argument form so the call 
\begin{verbatim}
               differentiate(#1, x), monomial(eta', 2) + eta'::UP),
\end{verbatim}
must be the first instance.

From the sources (bookvol10.4) we see that the tanint function has 
the signature:
\begin{verbatim}
    tanint      : (F, SE, K) -> IR
\end{verbatim}
and that 
\begin{verbatim}
  SE  ==> Symbol
  F : Join(AlgebraicallyClosedField, TranscendentalFunctionCategory,
           FunctionSpace R)
  K   ==> Kernel F
\end{verbatim}

The differentiate function takes something of type F and a Symbol
and returns something of type F. If we write this as an anonymous
function it becomes:
\begin{verbatim}
   (x2 : F) : F +-> differentiate(x2, x)
\end{verbatim}

Thus, we can rewrite the differentiate call as:
\begin{verbatim}
               differentiate(#1, x), monomial(eta', 2) + eta'::UP),
\end{verbatim}
as
\begin{verbatim}
                   (x2 : F) : F +-> differentiate(x2, x),
                   monomial(eta', 2) + eta'::UP),
\end{verbatim}


Continuing in this way we can fully rewrite the assignments as:
\begin{verbatim}
       r1  := tanintegrate(univariate(f, k),
                (x1 : UP) : UP +-> differentiate(x1,
                   (x2 : F) : F +-> differentiate(x2, x),
                   monomial(eta', 2) + eta'::UP),
                (x6 : Integer, x2 : F, x3 : F) : Union(List F, "failed") +->
                   rischDEsys(x6, 2 * eta, x2, x3, x,
                     (x4 : F, x5 : List F) : U3 +-> lflimitedint(x4, x, x5),
                     (x4 : F, x5 : F) : U2 +-> lfextendedint(x4, x, x5)))
       map((x1 : RF) : F +-> multivariate(x1, k), r1.answer) + _
             lfintegrate(r1.a0, x)
\end{verbatim}

Note that rischDEsys is tricky, because rischDEsys returns only List F,
but tanintegrate expects union. 
\subsection{The example bug}
Axiom can generate TeX output by typing:
\begin{verbatim}
)set output tex on
\end{verbatim}

Here we give an example of TeX output that contains a bug:
\begin{verbatim}
(1) -> )set output tex on
(1) -> radix(10**10,32)
   Loading /axiom/mnt/linux/algebra/RADUTIL.o for package 
      RadixUtilities 
   Loading /axiom/mnt/linux/algebra/RADIX.o 
     for domain RadixExpansion 
   Loading /axiom/mnt/linux/algebra/ANY1.o 
     for package AnyFunctions1 
   Loading /axiom/mnt/linux/algebra/NONE1.o 
     for package NoneFunctions1 
   Loading /axiom/mnt/linux/algebra/ANY.o 
     for domain Any 
   Loading /axiom/mnt/linux/algebra/SEX.o 
     for domain SExpression 

   (1)  9A0NP00
   Loading /axiom/mnt/linux/algebra/TEX.o 
     for domain TexFormat 
   Loading /axiom/mnt/linux/algebra/CCLASS.o 
     for domain CharacterClass 
   Loading /axiom/mnt/linux/algebra/IBITS.o 
     for domain IndexedBits 
   Loading /axiom/mnt/linux/algebra/UNISEG.o 
     for domain UniversalSegment 
$$
9#\A0#\N#\P00 
\leqno(1)
$$
   Loading /axiom/mnt/linux/algebra/VOID.o for domain Void 

                                      Type: RadixExpansion 32
\end{verbatim}
The correct output should be:
\begin{verbatim}
$$
9A0NP00 
\leqno(1)
$$
\end{verbatim}
So we need to figure out where the \#  prefixes are being generated.
In the above code we can see various domains being loaded. These domains
are lisp code. Each domain lives in a subdirectory of its own. For example,
the ANY domain lives in ANY.NRLIB. The ANY.NRLIB directory contains a
common lisp file named code.lsp. The compiled form of this code ANY.o
is loaded whenever the domain Any is referenced. We can look at the lisp 
code:
\begin{verbatim}

(/VERSIONCHECK 2) 

(PUT (QUOTE |ANY;obj;$N;1|)
     (QUOTE |SPADreplace|)
     (QUOTE QCDR)) 

(DEFUN |ANY;obj;$N;1| (|x| $) (QCDR |x|)) 

(PUT (QUOTE |ANY;dom;$Se;2|)
     (QUOTE |SPADreplace|)
     (QUOTE QCAR)) 

(DEFUN |ANY;dom;$Se;2| (|x| $) (QCAR |x|)) 

(PUT (QUOTE |ANY;domainOf;$Of;3|)
     (QUOTE |SPADreplace|)
     (QUOTE QCAR)) 

(DEFUN |ANY;domainOf;$Of;3| (|x| $) (QCAR |x|)) 

(DEFUN |ANY;=;2$B;4| (|x| |y| $)
 (COND
  ((SPADCALL (QCAR |x|) (QCAR |y|) (QREFELT $ 17))
    (EQ (QCDR |x|) (QCDR |y|)))
  ((QUOTE T) (QUOTE NIL)))) 

(DEFUN |ANY;objectOf;$Of;5| (|x| $)
 (|spad2BootCoerce| 
  (QCDR |x|) 
  (QCAR |x|) 
  (SPADCALL 
   (SPADCALL "OutputForm" (QREFELT $ 21))
   (QREFELT $ 23)))) 

(DEFUN |ANY;showTypeInOutput;BS;6| (|b| $)
 (SEQ
  (SETELT $ 10 (SPADCALL |b| (QREFELT $ 9)))
  (EXIT
   (COND 
    (|b| "Type of object will be displayed in 
          output of a member of Any")
    ((QUOTE T) "Type of object will not be displayed in 
                output of a member of Any"))))) 

(DEFUN |ANY;coerce;$Of;7| (|x| $)
 (PROG (|obj1| |p| |dom1| #0=#:G1426 |a| #1=#:G1427)
  (RETURN
   (SEQ 
    (LETT |obj1|
          (SPADCALL |x| (QREFELT $ 24))
          |ANY;coerce;$Of;7|)
    (COND
     ((NULL (SPADCALL (QREFELT $ 10) (QREFELT $ 26)))
       (EXIT |obj1|)))
    (LETT |dom1| 
     (SEQ 
      (LETT |p|
       (|prefix2String| (|devaluate| (QCAR |x|)))
       |ANY;coerce;$Of;7|)
      (EXIT
       (COND
        ((SPADCALL |p| (QREFELT $ 27))
         (SPADCALL |p| (QREFELT $ 23)))
        ((QUOTE T) (SPADCALL |p| (QREFELT $ 29))))))
     |ANY;coerce;$Of;7|)
    (EXIT
     (SPADCALL
      (CONS |obj1|
       (CONS ":"
        (PROGN
         (LETT #0# NIL |ANY;coerce;$Of;7|)
         (SEQ
          (LETT |a| NIL |ANY;coerce;$Of;7|)
          (LETT #1# |dom1| |ANY;coerce;$Of;7|)
          G190
          (COND
           ((OR (ATOM #1#)
                (PROGN
                  (LETT |a| (CAR #1#) |ANY;coerce;$Of;7|)
                  NIL))
             (GO G191)))
          (SEQ 
           (EXIT
            (LETT #0#
             (CONS 
              (SPADCALL |a| (QREFELT $ 30))
               #0#)
             |ANY;coerce;$Of;7|)))
          (LETT #1# (CDR #1#) |ANY;coerce;$Of;7|)
          (GO G190)
          G191
          (EXIT (NREVERSE0 #0#))))))
      (QREFELT $ 31))))))) 

(DEFUN |ANY;any;SeN$;8| (|domain| |object| $)
 (SEQ
  (COND
   ((|isValidType| |domain|) (CONS |domain| |object|))
   ((QUOTE T)
    (SEQ
     (LETT |domain| (|devaluate| |domain|) |ANY;any;SeN$;8|)
     (EXIT
      (COND
       ((|isValidType| |domain|) (CONS |domain| |object|))
       ((QUOTE T)
        (|error| 
"function any must have a domain as first argument"))))))))) 

(DEFUN |Any| NIL
 (PROG NIL 
  (RETURN 
   (PROG (#0=#:G1432) 
    (RETURN 
     (COND 
      ((LETT #0#
            (HGET |$ConstructorCache| (QUOTE |Any|))
            |Any|)
        (|CDRwithIncrement| (CDAR #0#)))
      ((QUOTE T)
        (UNWIND-PROTECT
         (PROG1
          (CDDAR 
           (HPUT |$ConstructorCache| 
                 (QUOTE |Any|) 
                 (LIST (CONS NIL (CONS 1 (|Any;|))))))
          (LETT #0# T |Any|))
         (COND
          ((NOT #0#)
            (HREM |$ConstructorCache| (QUOTE |Any|)))))))))))) 

(DEFUN |Any;| NIL
 (PROG (|dv$| $ |pv$|)
  (RETURN
   (PROGN
    (LETT |dv$| (QUOTE (|Any|)) . #0=(|Any|))
    (LETT $ (make-array 35) . #0#)
    (QSETREFV $ 0 |dv$|)
    (QSETREFV $ 3 (LETT |pv$| (|buildPredVector| 0 0 NIL) . #0#))
    (|haddProp| |$ConstructorCache| (QUOTE |Any|) NIL (CONS 1 $))
    (|stuffDomainSlots| $)
    (QSETREFV $ 6 
      (|Record| (|:| |dm| (|SExpression|)) (|:| |ob| (|None|))))
    (QSETREFV $ 10 (SPADCALL (QUOTE NIL) (QREFELT $ 9)))
    $)))) 

(setf (get (QUOTE |Any|) (QUOTE |infovec|))
 (LIST 
  (QUOTE 
   #(NIL NIL NIL NIL NIL NIL (QUOTE |Rep|)
     (|Boolean|) (|Reference| 7) (0 . |ref|) 
     (QUOTE |printTypeInOutputP|) (|None|) 
     |ANY;obj;$N;1| (|SExpression|) |ANY;dom;$Se;2| 
     (|OutputForm|) |ANY;domainOf;$Of;3| (5 . =) 
     |ANY;=;2$B;4| (|String|) (|Symbol|) (11 . |coerce|) 
     (|List| 20) (16 . |list|) |ANY;objectOf;$Of;5| 
     |ANY;showTypeInOutput;BS;6| (21 . |deref|) 
     (26 . |atom?|) (|List| $) (31 . |list|) 
     (36 . |coerce|) (41 . |hconcat|) |ANY;coerce;$Of;7| 
     |ANY;any;SeN$;8| (|SingleInteger|)))
  (QUOTE #(~= 46 |showTypeInOutput| 52 |objectOf| 57 |obj| 
           62 |latex| 67 |hash| 72 |domainOf| 77 |dom| 82 
           |coerce| 87 |any| 92 = 98))
  (QUOTE NIL)
  (CONS (|makeByteWordVec2| 1 (QUOTE (0 0 0)))
   (CONS (QUOTE #(|SetCategory&| |BasicType&| NIL))
    (CONS 
      (QUOTE #((|SetCategory|) (|BasicType|) (|CoercibleTo| 15)))
      (|makeByteWordVec2| 34 
       (QUOTE (1 8 0 7 9 2 13 7 0 0 17 1 20 0 19 21 1 22 0 20 
               23 1 8 7 0 26 1 13 7 0 27 1 20 28 0 29 1 20 15 
               0 30 1 15 0 28 31 2 0 7 0 0 1 1 0 19 7 25 1 0 
               15 0 24 1 0 11 0 12 1 0 19 0 1 1 0 34 0 1 1 0 
               15 0 16 1 0 13 0 14 1 0 15 0 32 2 0 0 13 11 33 
               2 0 7 0 0 18))))))
  (QUOTE |lookupComplete|))) 

(setf (get (QUOTE |Any|) (QUOTE NILADIC)) T) 
\end{verbatim}
We can ignore this information and focus on the functions that are 
defined in this file. These functions can be traced with the usual
common lisp tracing facility. So lets create a file /tmp/debug.lisp
that contains a trace statement for each DEFUN in ANY.NRLIB/code.lsp.
It looks like:
\begin{verbatim}
(trace |ANY1;retractable?;AB;1|)
(trace |ANY1;coerce;SA;2|)
(trace |ANY1;retractIfCan;AU;3|)
(trace |ANY1;retract;AS;4|)
(trace |AnyFunctions1|)
(trace |AnyFunctions1;|)
\end{verbatim}
We can now restart the axiom system, rerun the failing expression
(this will autoload ANY.o; alternatively we could hand-load the
ANY.NRLIB/code.lsp file), and then load our /tmp/debug.lisp file.
Now all of the functions in the ANY domain are traced and we can
watch the trace occur while the expression is evaluated. In this example
I've created a larger file that traces all of the loaded domains:
\begin{verbatim}
(trace |RADUTIL;radix;FIA;1|)
(trace |RadixUtilities|)
(trace |RadixUtilities;|)

(trace |RADIX;characteristic;Nni;1|)
(trace |RADIX;differentiate;2$;2|)
(trace |RADIX;Zero;$;3|)
(trace |RADIX;One;$;4|)
(trace |RADIX;-;2$;5|)
(trace |RADIX;+;3$;6|)
(trace |RADIX;-;3$;7|)
(trace |RADIX;*;I2$;8|)
(trace |RADIX;*;3$;9|)
(trace |RADIX;/;3$;10|)
(trace |RADIX;/;2I$;11|)
(trace |RADIX;<;2$B;12|)
(trace |RADIX;=;2$B;13|)
(trace |RADIX;numer;$I;14|)
(trace |RADIX;denom;$I;15|)
(trace |RADIX;coerce;$F;16|)
(trace |RADIX;coerce;I$;17|)
(trace |RADIX;coerce;F$;18|)
(trace |RADIX;retractIfCan;$U;19|)
(trace |RADIX;retractIfCan;$U;20|)
(trace |RADIX;ceiling;$I;21|)
(trace |RADIX;floor;$I;22|)
(trace |RADIX;wholePart;$I;23|)
(trace |RADIX;fractionPart;$F;24|)
(trace |RADIX;wholeRagits;$L;25|)
(trace |RADIX;fractRagits;$S;26|)
(trace |RADIX;prefixRagits;$L;27|)
(trace |RADIX;cycleRagits;$L;28|)
(trace |RADIX;wholeRadix;L$;29|)
(trace |RADIX;fractRadix;2L$;30|)
(trace |RADIX;intToExpr|)
(trace |RADIX;exprgroup|)
(trace |RADIX;intgroup|)
(trace |RADIX;overBar|)
(trace |RADIX;coerce;$Of;35|)
(trace |RADIX;checkRagits|)
(trace |RADIX;radixInt|)
(trace |RADIX;radixFrac|)
(trace |RadixExpansion|)
(trace |RadixExpansion;|)

(trace |ANY1;retractable?;AB;1|)
(trace |ANY1;coerce;SA;2|)
(trace |ANY1;retractIfCan;AU;3|)
(trace |ANY1;retract;AS;4|)
(trace |AnyFunctions1|)
(trace |AnyFunctions1;|)

(trace |NONE1;coerce;SN;1|)
(trace |NoneFunctions1|)
(trace |NoneFunctions1;|)

(trace |ANY;obj;$N;1|)
(trace |ANY;dom;$Se;2|)
(trace |ANY;domainOf;$Of;3|)
(trace |ANY;=;2$B;4|)
(trace |ANY;objectOf;$Of;5|)
(trace |ANY;showTypeInOutput;BS;6|)
(trace |ANY;coerce;$Of;7|)
(trace |ANY;any;SeN$;8|)
(trace |Any|)
(trace |Any;|)

(trace |SExpression|)
(trace |SExpression;|)

(trace |TEX;new;$;1|)
(trace |TEX;newWithNum|)
(trace |TEX;coerce;Of$;3|)
(trace |TEX;convert;OfI$;4|)
(trace |TEX;display;$IV;5|)
(trace |TEX;display;$V;6|)
(trace |TEX;prologue;$L;7|)
(trace |TEX;tex;$L;8|)
(trace |TEX;epilogue;$L;9|)
(trace |TEX;setPrologue!;$2L;10|)
(trace |TEX;setTex!;$2L;11|)
(trace |TEX;setEpilogue!;$2L;12|)
(trace |TEX;coerce;$Of;13|)
(trace |TEX;ungroup|)
(trace |TEX;postcondition|)
(trace |TEX;stringify|)
(trace |TEX;lineConcat|)
(trace |TEX;splitLong|)
(trace |TEX;splitLong1|)
(trace |TEX;group|)
(trace |TEX;addBraces|)
(trace |TEX;addBrackets|)
(trace |TEX;parenthesize|)
(trace |TEX;precondition|)
(trace |TEX;formatSpecial|)
(trace |TEX;formatPlex|)
(trace |TEX;formatMatrix|)
(trace |TEX;formatFunction|)
(trace |TEX;formatNullary|)
(trace |TEX;formatUnary|)
(trace |TEX;formatBinary|)
(trace |TEX;formatNary|)
(trace |TEX;formatNaryNoGroup|)
(trace |TEX;formatTex|)
(trace |TexFormat|)
(trace |TexFormat;|)

(trace |CCLASS;digit;$;1|)
(trace |CCLASS;hexDigit;$;2|)
(trace |CCLASS;upperCase;$;3|)
(trace |CCLASS;lowerCase;$;4|)
(trace |CCLASS;alphabetic;$;5|)
(trace |CCLASS;alphanumeric;$;6|)
(trace |CCLASS;=;2$B;7|)
(trace |CCLASS;member?;C$B;8|)
(trace |CCLASS;union;3$;9|)
(trace |CCLASS;intersect;3$;10|)
(trace |CCLASS;difference;3$;11|)
(trace |CCLASS;complement;2$;12|)
(trace |CCLASS;convert;$S;13|)
(trace |CCLASS;convert;$L;14|)
(trace |CCLASS;charClass;S$;15|)
(trace |CCLASS;charClass;L$;16|)
(trace |CCLASS;coerce;$Of;17|)
(trace |CCLASS;#;$Nni;18|)
(trace |CCLASS;empty;$;19|)
(trace |CCLASS;brace;$;20|)
(trace |CCLASS;insert!;C2$;21|)
(trace |CCLASS;remove!;C2$;22|)
(trace |CCLASS;inspect;$C;23|)
(trace |CCLASS;extract!;$C;24|)
(trace |CCLASS;map;M2$;25|)
(trace |CCLASS;map!;M2$;26|)
(trace |CCLASS;parts;$L;27|)
(trace |CharacterClass|)
(trace |CharacterClass;|)

(trace |IBITS;minIndex;$I;1|)
(trace |IBITS;range|)
(trace |IBITS;coerce;$Of;3|)
(trace |IBITS;new;NniB$;4|)
(trace |IBITS;empty;$;5|)
(trace |IBITS;copy;2$;6|)
(trace |IBITS;#;$Nni;7|)
(trace |IBITS;=;2$B;8|)
(trace |IBITS;<;2$B;9|)
(trace |IBITS;and;3$;10|)
(trace |IBITS;or;3$;11|)
(trace |IBITS;xor;3$;12|)
(trace |IBITS;setelt;$I2B;13|)
(trace |IBITS;elt;$IB;14|)
(trace |IBITS;Not;2$;15|)
(trace |IBITS;And;3$;16|)
(trace |IBITS;Or;3$;17|)
(trace |IndexedBits|)
(trace |IndexedBits;|)

(trace |UNISEG;segment;S$;1|)
(trace |UNISEG;segment;2S$;2|)
(trace |UNISEG;BY;$I$;3|)
(trace |UNISEG;lo;$S;4|)
(trace |UNISEG;low;$S;5|)
(trace |UNISEG;hasHi;$B;6|)
(trace |UNISEG;hi;$S;7|)
(trace |UNISEG;high;$S;8|)
(trace |UNISEG;incr;$I;9|)
(trace |UNISEG;SEGMENT;S$;10|)
(trace |UNISEG;SEGMENT;2S$;11|)
(trace |UNISEG;coerce;S$;12|)
(trace |UNISEG;convert;S$;13|)
(trace |UNISEG;=;2$B;14|)
(trace |UNISEG;coerce;$Of;15|)
(trace |UNISEG;expand;$S;16|)
(trace |UNISEG;map;M$S;17|)
(trace |UNISEG;plusInc|)
(trace |UNISEG;expand;LS;19|)
(trace |UNISEG;expand;LS;19!0|)
(trace |UniversalSegment|)
(trace |UniversalSegment;|)

\end{verbatim}
Now we rerun the function and get the trace output
\begin{verbatim}
(2) -> )lisp (load "/axiom/debug.lisp")

Value = T
(2) -> radix(10**10,32)

  1> (|RadixUtilities|)
  <1 (|RadixUtilities| #<vector 08b565cc>)
  1> (|RadixExpansion| 32)
  <1 (|RadixExpansion| #<vector 08b8cc94>)
  1> (|AnyFunctions1| #<vector 08b8cc94>)
  <1 (|AnyFunctions1| #<vector 08b5647c>)
  1> (|RadixExpansion| 32)
  <1 (|RadixExpansion| #<vector 08b8cc94>)
  1> (|RADIX;radixInt| 10000000000 32 #<vector 08b8cc94>)
  <1 (|RADIX;radixInt| (9 10 0 23 25 0 0))
  1> (|RADIX;radixFrac| 0 1 32 #<vector 08b8cc94>)
  <1 (|RADIX;radixFrac| (NIL 0))

  1> (|RadixExpansion| 32)
  <1 (|RadixExpansion| #<vector 08b8cc94>)
  1> (|RADIX;intgroup| (9 10 0 23 25 0 0) #<vector 08b8cc94>)
    2> (|RADIX;intToExpr| 9 #<vector 08b8cc94>)
    <2 (|RADIX;intToExpr| 9)
    2> (|RADIX;intToExpr| 10 #<vector 08b8cc94>)
    <2 (|RADIX;intToExpr| #\A)
    2> (|RADIX;intToExpr| 0 #<vector 08b8cc94>)
    <2 (|RADIX;intToExpr| 0)
    2> (|RADIX;intToExpr| 23 #<vector 08b8cc94>)
    <2 (|RADIX;intToExpr| #\N)
    2> (|RADIX;intToExpr| 25 #<vector 08b8cc94>)
    <2 (|RADIX;intToExpr| #\P)
    2> (|RADIX;intToExpr| 0 #<vector 08b8cc94>)
    <2 (|RADIX;intToExpr| 0)
    2> (|RADIX;intToExpr| 0 #<vector 08b8cc94>)
    <2 (|RADIX;intToExpr| 0)
  <1 (|RADIX;intgroup| (CONCAT 9 #\A 0 #\N #\P 0 0))
  1> (|RADIX;exprgroup| 
      ((CONCAT 9 #\A 0 #\N #\P 0 0)) #<vector 08b8cc94>)
  <1 (|RADIX;exprgroup| (CONCAT 9 #\A 0 #\N #\P 0 0))
   (2)  9A0NP00
  1> (|TexFormat|)
  <1 (|TexFormat| #<vector 08b24000>)
  1> (|TexFormat|)
  <1 (|TexFormat| #<vector 08b24000>)
  1> (|TEX;newWithNum| 2 #<vector 08b24000>)
  <1 (|TEX;newWithNum| #<vector 08b8c284>)
  1> (|TEX;precondition| 
       (CONCAT 9 #\A 0 #\N #\P 0 0) #<vector 08b24000>)
  <1 (|TEX;precondition| (CONCAT 9 #\A 0 #\N #\P 0 0))
  1> (|TEX;formatTex| 
       (CONCAT 9 #\A 0 #\N #\P 0 0) 0 #<vector 08b24000>)
    2> (|TEX;stringify| CONCAT #<vector 08b24000>)
    <2 (|TEX;stringify| "CONCAT")
    2> (|TEX;formatSpecial| "CONCAT" 
         (9 #\A 0 #\N #\P 0 0) 0 #<vector 08b24000>)
      3> (|TEX;formatNary| "" 
          (9 #\A 0 #\N #\P 0 0) 0 #<vector 08b24000>)
        4> (|TEX;formatNaryNoGroup| "" 
            (9 #\A 0 #\N #\P 0 0) 0 #<vector 08b24000>)
          5> (|TEX;formatTex| 9 0 #<vector 08b24000>)
            6> (|TEX;stringify| 9 #<vector 08b24000>)
            <6 (|TEX;stringify| "9")
          <5 (|TEX;formatTex| "9")
          5> (|TEX;formatTex| #\A 0 #<vector 08b24000>)
            6> (|TEX;stringify| #\A #<vector 08b24000>)
            <6 (|TEX;stringify| "#\\A")
            6> (|IBITS;range| 
                #<bit-vector 0831d930> 35 #<vector 085da658>)
            <6 (|IBITS;range| 35)
          <5 (|TEX;formatTex| "#\\A")
          5> (|TEX;formatTex| 0 0 #<vector 08b24000>)
            6> (|TEX;stringify| 0 #<vector 08b24000>)
            <6 (|TEX;stringify| "0")
          <5 (|TEX;formatTex| "0")
          5> (|TEX;formatTex| #\N 0 #<vector 08b24000>)
            6> (|TEX;stringify| #\N #<vector 08b24000>)
            <6 (|TEX;stringify| "#\\N")
            6> (|IBITS;range| 
                #<bit-vector 0831d930> 35 #<vector 085da658>)
            <6 (|IBITS;range| 35)
          <5 (|TEX;formatTex| "#\\N")
          5> (|TEX;formatTex| #\P 0 #<vector 08b24000>)
            6> (|TEX;stringify| #\P #<vector 08b24000>)
            <6 (|TEX;stringify| "#\\P")
            6> (|IBITS;range| 
                #<bit-vector 0831d930> 35 #<vector 085da658>)
            <6 (|IBITS;range| 35)
          <5 (|TEX;formatTex| "#\\P")
          5> (|TEX;formatTex| 0 0 #<vector 08b24000>)
            6> (|TEX;stringify| 0 #<vector 08b24000>)
            <6 (|TEX;stringify| "0")
          <5 (|TEX;formatTex| "0")
          5> (|TEX;formatTex| 0 0 #<vector 08b24000>)
            6> (|TEX;stringify| 0 #<vector 08b24000>)
            <6 (|TEX;stringify| "0")
          <5 (|TEX;formatTex| "0")
        <4 (|TEX;formatNaryNoGroup| "9#\\A0#\\N#\\P00")
        4> (|TEX;group| "9#\\A0#\\N#\\P00" #<vector 08b24000>)
        <4 (|TEX;group| "{9#\\A0#\\N#\\P00}")
      <3 (|TEX;formatNary| "{9#\\A0#\\N#\\P00}")
    <2 (|TEX;formatSpecial| "{9#\\A0#\\N#\\P00}")
  <1 (|TEX;formatTex| "{9#\\A0#\\N#\\P00}")
  1> (|TEX;postcondition| 
      "{9#\\A0#\\N#\\P00}" #<vector 08b24000>)
    2> (|TEX;ungroup| "{9#\\A0#\\N#\\P00}" #<vector 08b24000>)
    <2 (|TEX;ungroup| "9#\\A0#\\N#\\P00")
  <1 (|TEX;postcondition| "9#\\A0#\\N#\\P00")
$$
  1> (|TEX;splitLong| 
      "9#\\A0#\\N#\\P00" 77 #<vector 08b24000>)
    2> (|TEX;splitLong1| 
        "9#\\A0#\\N#\\P00" 77 #<vector 08b24000>)
      3> (|TEX;lineConcat| 
          "9#\\A0#\\N#\\P00 " NIL #<vector 08b24000>)
      <3 (|TEX;lineConcat| ("9#\\A0#\\N#\\P00 "))
    <2 (|TEX;splitLong1| ("9#\\A0#\\N#\\P00 "))
  <1 (|TEX;splitLong| ("9#\\A0#\\N#\\P00 "))
9#\A0#\N#\P00 
\leqno(2)
$$

                                      Type: RadixExpansion 32
\end{verbatim}
Notice the call that reads:
\begin{verbatim}
    2> (|RADIX;intToExpr| 10 #<vector 08b8cc94>)
    <2 (|RADIX;intToExpr| #\A)
\end{verbatim}
This means that calling |RADIX;intToExpr| with the number 10
and ``the domain vector'' generates the character \#\\A which fails.
If we had the domain vector in a variable we could hand-execute this
algebra function directly and watch it fail. So we go to the file
RADIX.NRLIB/code.lsp which contains the definition of RADIX;intToExpr.
The definition is:
\begin{verbatim}
(DEFUN |RADIX;intToExpr| (|i| $) 
 (COND 
  ((< |i| 10) 
    (SPADCALL |i| (QREFELT $ 66)))
  ((QUOTE T) 
    (SPADCALL 
     (SPADCALL 
       (QREFELT $ 64) 
       (+ (- |i| 10) (SPADCALL (QREFELT $ 64) (QREFELT $ 68)))
       (QREFELT $ 70))
     (QREFELT $ 71))))) 
\end{verbatim}
We can put this definition into our /tmp/debug.lisp file and modify
it to capture the domain vector passed in the \$ variable thus:
\begin{verbatim}
(DEFUN |RADIX;intToExpr| (|i| $) 
 (setq tpd $) 
 (COND 
  ((< |i| 10) 
    (SPADCALL |i| (QREFELT $ 66)))
  ((QUOTE T) 
    (SPADCALL 
     (SPADCALL 
       (QREFELT $ 64) 
       (+ (- |i| 10) (SPADCALL (QREFELT $ 64) (QREFELT $ 68)))
       (QREFELT $ 70))
     (QREFELT $ 71))))) 
\end{verbatim}
Now when this function is executed the tpd variable will contain the
value of \$, the domain vector. So we load debug.lisp again
to redefine RADIX;intToExpr and re-execute the function. The trace
results will be the same but now the global variable tpd will have
the domain vector:
\begin{verbatim}
(4) -> (identity tpd)

Value = #<vector 08b8cc94>
\end{verbatim}
Now we can use common lisp to step the RADIX;intToExpr function:
\begin{verbatim}
(4) -> (step (|RADIX;intToExpr| 10 tpd))

Type ? and a newline for help.
  (|RADIX;intToExpr| 10 ...) ?

Stepper commands:
n (or N or Newline): advances to the next form.
s (or S):	     skips the form.
p (or P):	     pretty-prints the form.
f (or F) FUNCTION:   skips until the FUNCTION is called.
q (or Q):	     quits.
u (or U):	     goes up to the enclosing form.
e (or E) FORM:	     evaluates the FORM and prints the value(s).
r (or R) FORM:	     evaluates the FORM and returns the value(s).
b (or B):	     prints backtrace.
?:		     prints this.

  (|RADIX;intToExpr| 10 ...) 
10 
TPD 
    = #<vector 08b8cc94>
    (SYSTEM::TRACE-CALL (QUOTE #:G1624) ...) 
(QUOTE #:G1624) 
SYSTEM::ARGS 
      = (10 #<vector 08b8cc94>)
      (QUOTE T) 
(QUOTE T) 
(QUOTE (CONS # ...)) 
(QUOTE T) 
(QUOTE (CONS # ...)) 
(LET (#) ...) 
(QUOTE (10 #<vector 08b8cc94>)) 
T 
      = T
  1>  (LET (#) ...) 
(QUOTE (10 #<vector 08b8cc94>)) 
(CONS (QUOTE |RADIX;intToExpr|) ...) 
(QUOTE |RADIX;intToExpr|) 
SYSTEM::ARGLIST 
          = (10 #<vector 08b8cc94>)
        = (|RADIX;intToExpr| 10 ...)
      = (|RADIX;intToExpr| 10 ...)
(|RADIX;intToExpr| 10 ...)
      (SETQ TPD ...) 
$ 
        = #<vector 08b8cc94>
      = #<vector 08b8cc94>
      (COND (# #) ...) 
(< |i| ...) 
|i| 
          = 10
          10 
        = NIL
        (QUOTE T) 
(SPADCALL (SPADCALL # ...) ...) 
(LET (#) ...) 
(QREFELT $ ...) 
(SVREF $ ...) 
$ 
                = #<vector 08b8cc94>
                71 
              = (#<compiled-function |CHAR;coerce;$Of;12|> .
                  #<vector 08b3901c>)
            = (#<compiled-function |CHAR;coerce;$Of;12|> . 
                 #<vector 08b3901c>)
            (THE (VALUES T) ...) 
(FUNCALL (CAR #:G1776) ...) 
(CAR #:G1776) 
#:G1776 
                  = (#<compiled-function |CHAR;coerce;$Of;12|> . 
                      #<vector 08b3901c>)
                = #<compiled-function |CHAR;coerce;$Of;12|>
                (SPADCALL (QREFELT $ ...) ...) 
(LET (#) ...) 
(QREFELT $ ...) 
(SVREF $ ...) 
$ 
                    = #<vector 08b8cc94>
                    70 
                    = (#<compiled-function 
                        |ISTRING;elt;$IC;30|> . 
                        #<vector 08b26850>)
                    = (#<compiled-function 
                        |ISTRING;elt;$IC;30|> . 
                        #<vector 08b26850>)
                    (THE (VALUES T) ...) 
(FUNCALL (CAR #:G1777) ...) 
(CAR #:G1777) 
#:G1777 
                    = (#<compiled-function 
                        |ISTRING;elt;$IC;30|> . 
                        #<vector 08b26850>)
                    = #<compiled-function |ISTRING;elt;$IC;30|>
                    (QREFELT $ ...) 
(SVREF $ ...) 
$ 
                    = #<vector 08b8cc94>
                    64 
                    = "ABCDEFGHIJKLMNOPQRSTUVWXYZ"
                    = "ABCDEFGHIJKLMNOPQRSTUVWXYZ"
                    (+ (- |i| ...) ...) 
(- |i| ...) 
|i| 
                    = 10
                    10 
                    = 0
                    (SPADCALL (QREFELT $ ...) ...) 
(LET (#) ...) 
(QREFELT $ ...) 
(SVREF $ ...) 
$ 
                    = #<vector 08b8cc94>
                    68 
                    = (#<compiled-function 
                        |ISTRING;minIndex;$I;11|> . 
                        #<vector 08b26850>)
                    = (#<compiled-function 
                        |ISTRING;minIndex;$I;11|> . 
                        #<vector 08b26850>)
                    (THE (VALUES T) ...) 
(FUNCALL (CAR #:G1778) ...) 
(CAR #:G1778) 
#:G1778 
                    = (#<compiled-function 
                        |ISTRING;minIndex;$I;11|> . 
                        #<vector 08b26850>)
                    = #<compiled-function 
                        |ISTRING;minIndex;$I;11|>
                    (QREFELT $ ...) 
(SVREF $ ...) 
$ 
                    = #<vector 08b8cc94>
                    64 
                    = "ABCDEFGHIJKLMNOPQRSTUVWXYZ"
                    = "ABCDEFGHIJKLMNOPQRSTUVWXYZ"
                    (CDR #:G1778) 
#:G1778 
                    = (#<compiled-function 
                        |ISTRING;minIndex;$I;11|> . 
                        #<vector 08b26850>)
                    = #<vector 08b26850>
                    = 1
                    = 1
                    = 1
                    = 1
                    = 1
                    (CDR #:G1777) 
#:G1777 
                    = (#<compiled-function 
                        |ISTRING;elt;$IC;30|> . 
                        #<vector 08b26850>)
                    = #<vector 08b26850>
                    = 65
                    = 65
                  = 65
                = 65
                (CDR #:G1776) 
#:G1776 
                  = (#<compiled-function 
                      |CHAR;coerce;$Of;12|> . 
                      #<vector 08b3901c>)
                = #<vector 08b3901c>
              = #\A
            = #\A
          = #\A
        = #\A
      = #\A
  <1  (LET (# #) ...) 
(QUOTE (10 #<vector 08b8cc94>)) 
(QUOTE (#\A)) 
(CONS (QUOTE |RADIX;intToExpr|) ...) 
(QUOTE |RADIX;intToExpr|) 
VALUES 
          = (#\A)
        = (|RADIX;intToExpr| #\A)
      = (|RADIX;intToExpr| #\A)
(|RADIX;intToExpr| #\A)
    = #\A
  = #\A
Value = #\A
(4) -> 
\end{verbatim}
If we examine the source code for this function in int/algebra/radix.spad
we find:
\begin{verbatim}

    ALPHAS : String := "ABCDEFGHIJKLMNOPQRSTUVWXYZ"

    intToExpr(i:I): OUT ==
      -- computes a digit for bases between 11 and 36
      i < 10 => i :: OUT
      elt(ALPHAS,(i-10) + minIndex(ALPHAS)) :: OUT

\end{verbatim}
We do some lookups by hand to find out what functions are being called
from the domain vectors thus:
\begin{verbatim}
(4) -> )lisp (qrefelt tpd 68)

Value = (#<compiled-function 
         |ISTRING;minIndex;$I;11|> . #<vector 08b26850>)

\end{verbatim}




The \#\\A value appears from a call to 
CHAR;coerce;\$Of;12. We can look in CHAR.NRLIB/code.lsp
for this function and continue our descent into the code. The 
function looks like:
\begin{verbatim}
(DEFUN |CHAR;coerce;$Of;12| (|c| $) 
  (ELT (QREFELT $ 10)
       (+ (QREFELT $ 11) (SPADCALL |c| (QREFELT $ 21))))) 
\end{verbatim}
Again we need to get the domain vector, this time from the CHAR domain.
The domain vector has all of the information about a domain including
what functions are referenced and what data values are used. The 
QREFELT is a ``quick elt'' function which resolved to a highly
type optimized function call. The SPADCALL function funcalls the
second argument to SPADCALL with the first argument to SPADCALL
effectively giving: 
\begin{verbatim}
(funcall (qrefelt $ 21) |c|)
\end{verbatim}
So we modify the CHAR;coerce;\$Of;12 function to capture the domain
vector thus:
\begin{verbatim}
(DEFUN |CHAR;coerce;$Of;12| (|c| $)
   (format t "|CHAR;coerce;$Of;12| called")
   (setq tpd1 $)
  (ELT (QREFELT $ 10)
       (+ (QREFELT $ 11) (SPADCALL |c| (QREFELT $ 21))))) 
\end{verbatim}
Again we rerun the failing function and now tpd1 contains the
domain vector for the domain CHAR:

\subsection{Operating system level I/O trace (strace)}
If the bug seems to happen during startup the only method of 
debugging might be to use strace. To do this, replace the 
\$AXIOM/bin/AXIOMsys binary with a shell script. You should:

\begin{enumerate}
\item rename \$AXIOM/bin/AXIOMsys to \$AXIOM/bin/AXIOMsys.bin
\item create the shell script shown below
\item copy the shell script to the file \$AXIOM/bin/AXIOMsys so it will 
execute in place of the normal Axiom image
\item chmod +x \$AXIOM/bin/AXIOMsys to make the script executable
\item start axiom normally
\end{enumerate}

The script reads:
\begin{verbatim}
#!/bin/sh
exec strace -o /tmp/str.$$ /research/test/mnt/ubuntu/bin/AXIOMsys.bin "$@" 
      | tee /tmp/tee.$$
\end{verbatim}

The script will create 2 files in the tmp directory, ``str.NNNNN'' and
``tee.NNNNN'' where NNNNN is the process id assigned to axiom at runtime.

The tee.NNNNN file contains the console output you saw.
The str.NNNNN contains the output of strace which is a list of all of
the system calls and their result.

\section{How to make graphs in algebra books}
dot -Tps <pic >books/ps/domain.ps

where file pic contains something like:

\begin{verbatim}
digraph pic {
 fontsize=10;
 bgcolor="#ECEA81";
 node [shape=box, color=white, style=filled];

"OrderlyDifferentialVariable"
 [color=lightblue,href="bookvol10.3.pdf#nameddest=ODVAR"];

}
\end{verbatim}

In book Volume 10.3 there are .ps files generated for each domain.

These pictures show the category, domain, or package that is the highest
in the algebra tower. That means that the parent domain has to be compiled
before this domain. Since the parent is the highest in the tower then all
of the other domains will already have been compiled.

We derive the dependency information from the algebra Makefile.
In order to insert algebra into the tower we derive its parents
and then create a table of all of the parents. However, the table is
mostly comments except for the highest parent. So if the highest parent
is compiled in layer 12 then the new domain is inserted into layer 13.

The graph information at the end of a domain (e.g. UTSZ) contains only
the uncommented information. It is used to generate the picture. So
for UTSZ we see:

\begin{verbatim}
"UTSZ" [color="#88FF44",href="bookvol10.3.pdf#nameddest=UTSZ"]
"ACFS" [color="#4488FF",href="bookvol10.2.pdf#nameddest=ACFS"]
"UTSZ" -> "ACFS"
\end{verbatim}

This tells us several pieces of information. UTSZ depends on ACFS
as the highest parent. ACFS is in layer 17 so is in layer 18.
ACFS is a category (color 4488FF) and lives in bookvol10.2.
UTSZ is a domain (color 88FF44, note the rotation of hex codes, 
a package is color FF4488) and lives in bookvol10.3.

The necessary stanzas exist after each algebra domain.
Copy the stanza into a digraph block and run the dot function.
So for the UTSZ domain we would create a file (e.g. called 'pic')
that contains a block that reads:
\begin{verbatim}
digraph pic {
 fontsize=10;
 bgcolor="#ECEA81";
 node [shape=box, color=white, style=filled];

"UTSZ" [color="#88FF44",href="bookvol10.3.pdf#nameddest=UTSZ"]
"ACFS" [color="#4488FF",href="bookvol10.2.pdf#nameddest=ACFS"]
"UTSZ" -> "ACFS"

}
\end{verbatim}
and then run the command:
\begin{verbatim}
  dot -Tps <pic >pic.ps
  cp pic.ps ps/v103univariatetaylorseriesczero.ps
\end{verbatim}
The output file ``ps/v103univariatetaylorseriesczero.ps'' is included
after the pagepic of each domain in the book.

\section{Adding or Editing pages in Hyperdoc}
In Axiom it is easy to develop new pages in hyperdoc.
All of the hyperdoc pages live in bookvol7.1.pamphlet.

The structure of each page, say for the Fantastic domain, consists 
of a few lines of latex followed by a literate chunk. All of the
hyperdoc page information goes into the literate chunk and will
be extracted as a hyperdoc page.

\begin{verbatim}
 \section{fantastic.ht}  <-- ordinary latex
 \pagehead{FantasticPage}{fantastic.ht}{Fantastic} <-- special latex
 \pageto{.....           <-- for latex (not hyperdoc) links to other pages
 <<fantastic.ht>>=       <-- this is what htadd looks for

   all the documentation for domain Fantastic

 @                       <-- this is the end of the page for htadd
\end{verbatim}

When you add new pages to bookvol7.1.pamphlet you need to tell the
system about the changes.  The ``htadd'' function will search the
bookvol7.1.pamphlet file for chunks with the name ``*.ht'' and build the
file ht.db.

The ht.db file is used by hyperdoc to find pages. 
Each line in ht.db contains a line that 

\begin{verbatim}
FantasticPage (byteIndex) (lineIndex)
\end{verbatim}

So the two critical files, bookvol7.1.pamphlet and ht.db live in \$AXIOM/doc


There is a very fast cycle for editing.
\begin{verbatim}
cd $AXIOM/doc

while (1) do
   axiom                          <-- at the shell prompt
   (navigate to page in hyperdoc) <-- to check your work
   )lisp (bye)                    <-- at the axiom prompt
   (modify bookvol7.1.pamphlet)   <-- change the page
   rm ht.db                       <-- remove the old database
   htadd bookvol7.1.pamphlet      <-- remake the database
\end{verbatim}
\section{Graphviz file creation}
The graphviz output used on the website is a scaled vector graphics
file (SVG). The dot command to output this file is:
\begin{verbatim}
  dot -Tsvg:cg <pic >pic.svg
\end{verbatim}
The SVG file that gets generated has the following preamble.
\begin{verbatim}
<?xml version="1.0" encoding="UTF-8" standalone="no"?>
<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.0//EN"
 "http://www.w3.org/TR/2001/REC-SVG-20010904/DTD/svg10.dtd" [
 <!ATTLIST svg xmlns:xlink CDATA #FIXED "http://www.w3.org/1999/xlink">
]>
<!-- Generated by dot version 2.8 (Thu Sep 14 20:34:11 UTC 2006)
     For user: (root) root -->
<!-- Title: AxiomSept2008 Pages: 1 -->
<svg width="3960pt" height="2312pt"
 viewBox = "0 0 3960 2312"
 xmlns="http://www.w3.org/2000/svg" 
 xmlns:xlink="http://www.w3.org/1999/xlink">
\end{verbatim}
There are two pieces of information that are important. First,
we need to add the following text by removing the trailing
$>$ character from the svg tag and replacing it with the 
following block. This block exports some javascript functions
that we use to scale the graphics.
\begin{verbatim}
 onload="RunScript(evt)">
<script type="text/ecmascript">
<![CDATA[
var g_element;
var SVGDoc;
var SVGRoot;
function setDimension(w,h) {
  SVGDoc.documentElement.setAttribute("width",w);
  SVGDoc.documentElement.setAttribute("height",h);
}
function setScale(sw,sh) {
  g_element.setAttribute("transform","scale("+sw+" "+sh+")");
}
function RunScript(LoadEvent) {
  top.SVGsetDimension=setDimension;
  top.SVGsetScale=setScale;
  SVGDoc=LoadEvent.target.ownerDocument;
  g_element=SVGDoc.getElementById("graph0");
}
]]>
</script>
\end{verbatim}
A second item of interest is the viewbox line which gives us the
width and height information. We use this information to place
the graph on the web page. A simple example of the web page
looks follows. We need to replace the X and Y sizes with the
sizes from the viewbox above.
\begin{verbatim}
<html>
<head>
<title>Axiom Abbreviated Category and Domain graph</title>
<script type="text/javascript">
var W3CDOM = (document.createElement && document.getElementsByTagName);
window.onload 	= init;
function init(evt) {
	SVGscale(0.5);
}
function SVGscale(scale) {
	window.SVGsetDimension(8162*scale, 3068*scale);
	window.SVGsetScale(scale,scale);	
	var box 	= document.getElementById('svgid');
	box.width  	= 8162*scale;
	box.height 	= 3068*scale;
}
</script>

</head>
<body>
<h1>Axiom Abbreviated Category and Domain graph</h1>	
<div>
	choose here: 
	<a href="#" onclick="SVGscale(0.1);">0.1</a> or
	<a href="#" onclick="SVGscale(0.25);">0.25</a> or
	<a href="#" onclick="SVGscale(0.5);">0.5</a> or 
	<a href="#" onclick="SVGscale(1);">1.0</a> or 
	<a href="#" onclick="SVGscale(1.5);">1.5</a> or ...
</div>
<div>
  <object id='svgid' data="dotabb.svg" type="image/svg+xml" 
   width="8162" height="3068" wmode="transparent" style="overflow:hidden;" />
  </object>
</div>

</body>
</html>
\end{verbatim}

\section{Adding Algebra}
\subsection{Adding algebra to the books}
Assume you have a piece of algebra code that you want to permanently add
to Axiom. This is a fairly complex process since the system automatically
runs regression tests, creates help files, etc. and has other standard
features you need to support. Lets assume your algebra looks like this:
\begin{verbatim}
)abbreviation package INTERGB InterfaceGroebnerPackage
InterfaceGroebnerPackage(K,symb,E,OV,R):Exports == Implementation where
  K    : FIELD
  symb : List Symbol
  E    : OrderedAbelianMonoidSup
  OV   : OrderedSet
  R    : PolynomialCategory(K,E,OV)

  LIST ==> List

  Exports ==>  with
    groebner: LIST R -> LIST R
      
  Implementation ==>  add
     PF ==> PrimeField(q)
     DPF ==> DistributedMultivariatePolynomial(symb,PF)
     D ==> DistributedMultivariatePolynomial(symb,K)
     JCFGBPack ==> GroebnerPackage(PF,E,OV,DPF)
     GBPack ==> GroebnerPackage(K,E,OV,D)  
  
     coerceDtoR: D->R 
     coerceDtoR(pol) == map(#1,pol)$MPolyCatFunctions2(OV,E,E,K,K,D,R)

      groebner(l)==
        ldmp:List D:= [coerceRtoD(pol) for pol in l]
        gg:=groebner(ldmp)$GBPack
        [coerceDtoR(pol) for pol in gg]

\end{verbatim}

There are some things to check and things to change.
\begin{itemize}
\item remove all tabs. Spad is a language that assigns meaning to 
indentation and tabs are not going to survive the build process intact.
\item try to stay within 80 characters. Spad code is printed in the books
so it should try to limit line lengths everywhere.
\item change ``)abbreviation'' to ``)abbrev''. The Makefile will search
for the abbrev line and expect this exact text. 
\begin{verbatim}
)abbrev package INTERGB InterfaceGroebnerPackage
\end{verbatim}
\item make sure there is only a single space between the items in the
abbrev line. The Makefile assumes this.
\item Add the comment header block. The author information is used to
check that all authors are included in the credits. The description
tag is used as output by the ``)describe'' command. For example:
\begin{verbatim}
++ Author: Gaetan Hache
++ Date Created: September 1996
++ Date Last Updated: April, 2010, by Tim Daly
++ Description:
++ Part of the Package for Algebraic Function Fields in one variable PAFF
\end{verbatim}
Other tags can be included but are not used. Do not assume that any
format information will be correctly translated or preserved. Make the
description section be simple text with a single character between the
``++'' and the first word of the text.
\end{itemize}

\begin{enumerate}
\item Choose the right book:
\begin{itemize}
\item Category goes into book bookvol10.2.pamphlet
\item Domain goes into book bookvol10.3.pamphlet
\item Package goes into book bookvol10.4.pamphlet
\item Numerics goes into book bookvol10.5.pamphlet
\end{itemize}
\item Find the right chapter. Chapter ordered by name, not abbreviation
\item Create a new section. The easiest way to do this is to copy another
section and change the names. In any case, your new section will need
\item Create a dividiing line consisting of all \% signs. 
Note that this character is the comment character for TeX.
\item Create the \verb|\section| tag. This tag should contain the exact text of
the )abbrev line from your algebra code. It reads:
\verb|)section{package INTERGB InterfaceGroebnerPackage}|
\item Create a regression test chunk. This chunk derives its name from the
algebra name as in ``InterfaceGroebnerPackage.input''. This chunk will
be automatically extracted and run during regression testing. For the
moment the chunk should be a simple empty input file as in:
\begin{verbatim}
)set break resume
)sys rm -f InterfaceGroebnerPackage.output
)spool InterfaceGroebnerPackage.output
)set message test on
)set message auto off
)clear all

--S 1 of 1
)show InterfaceGroebnerPackage
--E 1

)spool
)lisp (bye)
\end{verbatim}
\item Create a help text chunk. This chunk dervies its name from the
algebra name as in ``InterfaceGroebnerPackage.help''. This chunk will
be automatically extracted and used to build help text during build.
For the moment the chunck should be a simple empty help text file as in:
\begin{verbatim}
====================================================================
 examples InterfaceGroebnerPackage
====================================================================

See Also:
o )show InterfaceGroebnerPackage

\end{verbatim}
\end{enumerate}

At this point we need information from the Axiom interpreter to continue.
Start Axiom and compile your program with ``autoload'' on. For example:
\begin{verbatim}
   axiom -nox
   )set message autoload on
   )co InterfaceGroebnerPackage.spad
\end{verbatim}

You will see output containing lines which detail the category, domains,
and packages needed by your code, for example:
\begin{verbatim}
   Loading /research/test/mnt/ubuntu/algebra/FIELD.o for category Field
      
   Loading /research/test/mnt/ubuntu/algebra/EUCDOM.o for category 
      EuclideanDomain 
   Loading /research/test/mnt/ubuntu/algebra/PID.o for category 
      PrincipalIdealDomain 
   Loading /research/test/mnt/ubuntu/algebra/GCDDOM.o for category 
      GcdDomain 
   Loading /research/test/mnt/ubuntu/algebra/INTDOM.o for category 
      IntegralDomain 
   Loading /research/test/mnt/ubuntu/algebra/COMRING.o for category 
      CommutativeRing 
   Loading /research/test/mnt/ubuntu/algebra/RING.o for category Ring 
   Loading /research/test/mnt/ubuntu/algebra/RNG.o for category Rng 
   Loading /research/test/mnt/ubuntu/algebra/ABELGRP.o for category 
      AbelianGroup 
   Loading /research/test/mnt/ubuntu/algebra/CABMON.o for category 
      CancellationAbelianMonoid 
   Loading /research/test/mnt/ubuntu/algebra/ABELMON.o for category 
      AbelianMonoid 
   Loading /research/test/mnt/ubuntu/algebra/ABELSG.o for category 
      AbelianSemiGroup 
   Loading /research/test/mnt/ubuntu/algebra/SETCAT.o for category 
      SetCategory 
   Loading /research/test/mnt/ubuntu/algebra/BASTYPE.o for category 
      BasicType 
   Loading /research/test/mnt/ubuntu/algebra/KOERCE.o for category 
      CoercibleTo 
   Loading /research/test/mnt/ubuntu/algebra/SGROUP.o for category 
      SemiGroup 
   Loading /research/test/mnt/ubuntu/algebra/MONOID.o for category 
      Monoid 
   Loading /research/test/mnt/ubuntu/algebra/LMODULE.o for category 
      LeftModule 
   Loading /research/test/mnt/ubuntu/algebra/BMODULE.o for category 
      BiModule 
   Loading /research/test/mnt/ubuntu/algebra/RMODULE.o for category 
      RightModule 
   Loading /research/test/mnt/ubuntu/algebra/ALGEBRA.o for category 
      Algebra 
   Loading /research/test/mnt/ubuntu/algebra/MODULE.o for category 
      Module 
   Loading /research/test/mnt/ubuntu/algebra/ENTIRER.o for category 
      EntireRing 
   Loading /research/test/mnt/ubuntu/algebra/UFD.o for category 
      UniqueFactorizationDomain 
   Loading /research/test/mnt/ubuntu/algebra/DIVRING.o for category 
      DivisionRing 
   Loading /research/test/mnt/ubuntu/algebra/OAMONS.o for category 
      OrderedAbelianMonoidSup 
   Loading /research/test/mnt/ubuntu/algebra/OCAMON.o for category 
      OrderedCancellationAbelianMonoid 
   Loading /research/test/mnt/ubuntu/algebra/OAMON.o for category 
      OrderedAbelianMonoid 
   Loading /research/test/mnt/ubuntu/algebra/OASGP.o for category 
      OrderedAbelianSemiGroup 
   Loading /research/test/mnt/ubuntu/algebra/ORDSET.o for category 
      OrderedSet 
   Loading /research/test/mnt/ubuntu/algebra/POLYCAT.o for category 
      PolynomialCategory 
   Loading /research/test/mnt/ubuntu/algebra/PDRING.o for category 
      PartialDifferentialRing 
   Loading /research/test/mnt/ubuntu/algebra/FAMR.o for category 
      FiniteAbelianMonoidRing 
   Loading /research/test/mnt/ubuntu/algebra/AMR.o for category 
      AbelianMonoidRing 
   Loading /research/test/mnt/ubuntu/algebra/CHARZ.o for category 
      CharacteristicZero 
   Loading /research/test/mnt/ubuntu/algebra/CHARNZ.o for category 
      CharacteristicNonZero 
   Loading /research/test/mnt/ubuntu/algebra/FRETRCT.o for category 
      FullyRetractableTo 
   Loading /research/test/mnt/ubuntu/algebra/RETRACT.o for category 
      RetractableTo 
   Loading /research/test/mnt/ubuntu/algebra/EVALAB.o for category 
      Evalable 
   Loading /research/test/mnt/ubuntu/algebra/IEVALAB.o for category 
      InnerEvalable 
   Loading /research/test/mnt/ubuntu/algebra/FLINEXP.o for category 
      FullyLinearlyExplicitRingOver 
   Loading /research/test/mnt/ubuntu/algebra/LINEXP.o for category 
      LinearlyExplicitRingOver 
   Loading /research/test/mnt/ubuntu/algebra/KONVERT.o for category 
      ConvertibleTo 
   Loading /research/test/mnt/ubuntu/algebra/PATMAB.o for category 
      PatternMatchable 
   Loading /research/test/mnt/ubuntu/algebra/PFECAT.o for category 
      PolynomialFactorizationExplicit 
   Loading /research/test/mnt/ubuntu/algebra/FFIELDC.o for category 
      FiniteFieldCategory 
   Loading /research/test/mnt/ubuntu/algebra/FPC.o for category 
      FieldOfPrimeCharacteristic 
   Loading /research/test/mnt/ubuntu/algebra/FINITE.o for category 
      Finite 
   Loading /research/test/mnt/ubuntu/algebra/STEP.o for category 
      StepThrough 
   Loading /research/test/mnt/ubuntu/algebra/DIFRING.o for category 
      DifferentialRing 

   Loading /research/test/mnt/ubuntu/algebra/NNI.o for domain 
      NonNegativeInteger 
   Loading /research/test/mnt/ubuntu/algebra/INT.o for domain Integer 
\end{verbatim}

These are all of the algebra files on which your algebra depends.
Collect all of these names into a list:
\begin{verbatim}
FIELD.o        EUCDOM.o       PID.o          GCDDOM.o       INTDOM.o
COMRING.o      RING.o         RNG.o          ABELGRP.o      CABMON.o
ABELMON.o      ABELSG.o       SETCAT.o       BASTYPE.o      KOERCE.o
SGROUP.o       MONOID.o       LMODULE.o      BMODULE.o      RMODULE.o
ALGEBRA.o      MODULE.o       ENTIRER.o      UFD.o          DIVRING.o
OAMONS.o       OCAMON.o       OAMON.o        OASGP.o        ORDSET.o
POLYCAT.o      PDRING.o       FAMR.o         AMR.o          CHARZ.o
CHARNZ.o       FRETRCT.o      RETRACT.o      EVALAB.o       IEVALAB.o
FLINEXP.o      LINEXP.o       KONVERT.o      PATMAB.o       PFECAT.o
FFIELDC.o      FPC.o          FINITE.o       STEP.o         DIFRING.o
NNI.o          INT.o
\end{verbatim}

The algebra files are arranged into layers. Algebra in a given layer can
only depend on algebra in lower layers. So algebra in layer 4 can only
depend on algebra in layers 0 through 3.

Your algebra depends on all of the above algebra so we first need to
answer the question ``What is the highest layer of algebra my code 
depends upon''.

To answer that question we need to know the layer of each of the above files.
This can be determined by searching the Makefile and finding the layer.
We do this here and annotate the above list, rearranged by layers:

\begin{verbatim}
FIELD.o        EUCDOM.o       PID.o          GCDDOM.o       INTDOM.o
COMRING.o      RING.o         RNG.o          ABELGRP.o      CABMON.o
ABELMON.o      ABELSG.o       SETCAT.o       BASTYPE.o      KOERCE.o
SGROUP.o       MONOID.o       LMODULE.o      BMODULE.o      RMODULE.o
ALGEBRA.o      MODULE.o       ENTIRER.o      UFD.o          DIVRING.o
OAMONS.o       OCAMON.o       OAMON.o        OASGP.o        ORDSET.o
POLYCAT.o      PDRING.o       FAMR.o         AMR.o          CHARZ.o
CHARNZ.o       FRETRCT.o      RETRACT.o      EVALAB.o       IEVALAB.o
FLINEXP.o      LINEXP.o       KONVERT.o      PATMAB.o       PFECAT.o
FFIELDC.o      FPC.o          FINITE.o       STEP.o         DIFRING.o
NNI.o          INT.o
\end{verbatim}

We find that
\begin{verbatim}
layer 0 
EUCDOM.o GCDDOM.o INTDOM.o COMRING.o RING.o RNG.o ABELGRP.o CABMON.o ABELMON.o
ABELSG.o SETCAT.o BASTYPE.o KOERCE.o MONOID.o ENTIRER.o UFD.o DIVRING.o
POLYCAT.o KONVERT.o FFIELDC.o DIFRING.o NNI.o INT.o

layer 1
SGROUP.o LMODULE.o RMODULE.o ORDSET.o RETRACT.o IEVALAB.o FINITE.o STEP.o
PATMAB.o

layer 2
BMODULE.o OASGP.o PDRING.o CHARZ.o CHARNZ.o EVALAB.o LINEXP.o

layer 3
MODULE.o OAMON.o

layer 4
ALGEBRA.o OCAMON.o

layer 5
PID.o OAMONS.o

layer 6
FIELD.o AMR.o FRETRCT.o FLINEXP.o

layer 7
FAMR.o FPC.o

layer 10
PFECAT.o
\end{verbatim}

So we know that our algebra belongs in layer 11 since it depends on 
PFECAT which lives in layer 10.

Now we have all of the information needed to add the algebra to the book.
We create a new section that contains the following parts:
\begin{itemize}
\item the \verb|\section{...}| header
\item the input file
\item the help file
\item the page head markup
\item the pagepic markup
\item the Exports section
\item the theory and discussion section
\item the algebra
\item the dot picture file information
\end{itemize}

We've already explained most of the sections. Now we insert a chunk
for our algebra. The name of this chunk is important because the Makefile
will look for this exact chunk name. In our example case it looks like:
\begin{verbatim}
   \begin{chunk}{package INTERGB InterfaceGroebnerPackage}
\end{verbatim}
which is followed immediately by the algebra starting with the ``abbrev'' 
command.

Make sure that the section names, input file names, help file names,
and chunk names all reflect the new algebra names.

Next we copy the chunkname to the bottom of the file as part of the
algebra chunk.

The next piece to be handled is the ``pagepic'' tag. This inserts a
picture into the book which shows what files our algebra depends upon.
We do this using the graphviz dot program.

We create a file that shows the graph relationship. Since our code
INTERGB depends on PFECAT we show that here. We create a line for our 
code that looks like:
\begin{verbatim}
"INTERGB" [color="#FF4488",href="bookvol10.4.pdf#nameddest=INTERGB"]
\end{verbatim}

This tells us the name of the node, the background color where
\begin{itemize}
\item \#4488FF is a category
\item \#88FF44 is a domain
\item \#FF4488 is a package
\item \#FF8844 is a numeric
\end{itemize}
(note the rotation of bytes) and the html link from the graph to 
the position in the document so the user can click on the picture
and go to the source code. It names the book and the offset.

We have the same information for PFECAT:
\begin{verbatim}
"PFECAT" [color="#4488FF",href="bookvol10.2.pdf#nameddest=PFECAT"]
\end{verbatim}

We combine this information to create a picture file for the graphviz dot
program:
\begin{verbatim}
digraph pic {
 fontsize=10;
 bgcolor="#ECEA81";
 node [shape=box, color=white, style=filled];

"INTERGB" [color="#FF4488",href="bookvol10.4.pdf#nameddest=INTERGB"]
"PFECAT" [color="#4488FF",href="bookvol10.2.pdf#nameddest=PFECAT"]
"INTERGB" -> "PFECAT"

}
\end{verbatim}

Save this file into \verb|$AXIOM/books/pic|. Then run:
\begin{verbatim}
dot -Tps <pic >ps/v104interfacegroebnerpackage.ps
\end{verbatim}

This will generate a postscript file
\verb|$AXIOM/books/ps/v104interfacegroebnerpackage.ps|

Note that the pagepic name should be lower case as this is the name of
a file in the file system. Axiom uses only lower case names as files
to avoid the problem of mixed-case or ambiguous case file systems.

The pagepic file is saved into a subdirectory (ps) under the books
directory. The file contains the name of the book (v104 is volume 10.4),
the name of the algebra in lowercase, and the extension of ps.

Now we modify the pagepic tag in book volume 10.4 to read:

\verb|\pagepic{ps/v104interfacegroebnerpackage.ps}{INTERGB}{1.00}|

This tells us where to find the file, what the abbreviation for the
file will be, and the scale factor to use to display the file (1.00).

We copy the body of the pic file to the book so we can easily
recreate the graphs:
\begin{verbatim}
  \begin{chunk}{INTERGB.dotabb}
  "INTERGB" [color="#FF4488",href="bookvol10.4.pdf#nameddest=INTERGB"]
  "PFECAT" [color="#4488FF",href="bookvol10.2.pdf#nameddest=PFECAT"]
  "INTERGB" -> "PFECAT"
  \end{chunk}
\end{verbatim}

Next we return to the command line and get explicit information about
our new domain:
\begin{verbatim}

(1) -> )show INTERGB
 InterfaceGroebnerPackage(K: Field,
                          symb: List Symbol,
                          E: OrderedAbelianMonoidSup,
                          OV: OrderedSet,
                          R: PolynomialCategory(K,E,OV))  
              is a package constructor
 Abbreviation for InterfaceGroebnerPackage is INTERGB 
 This constructor is exposed in this frame.
 Issue )edit /research/silver/PAFF/PAFF/spad/interGBwoGB.spad to see 
            algebra source code for INTERGB 

------------------------------- Operations --------------------------------
 groebner : List R -> List R          

\end{verbatim}

We need to insert the results of this command into the trivial input file
we have created. The trivial input file will be run during build time and
the results of the command at build time will be compared with the results
we provide. The results we provide are actually comments so they begin with
two dashes, the Axiom comment character. 

The regression programm (``regress'') compares each line generated by the
program with the line stored in the input file. It only compares lines that
start with ``--R''. Each test is surrounded by the comment pair ``--S m of n''
and ``--E m'' which indicates the start and end of test ``m'' respectively.

Since we will eventually compile this domain from the book the final 
output line for the )edit location will be the book. Thus we need to
change the line:
\begin{verbatim}
Issue )edit /PAFF/interGBwoGB.spad to see algebra source code for INTERGB 
\end{verbatim}
to read:
\begin{verbatim}
Issue )edit bookvol10.4.pamphlet to see algebra source code for INTERGB 
\end{verbatim}

So we update our input file section to read:
\begin{verbatim}
)set break resume
)sys rm -f InterfaceGroebnerPackage.output
)spool InterfaceGroebnerPackage.output
)set message test on
)set message auto off
)clear all

--S 1 of 1
)show InterfaceGroebnerPackage
--R InterfaceGroebnerPackage(K: Field,
--R                          symb: List Symbol,
--R                          E: OrderedAbelianMonoidSup,
--R                          OV: OrderedSet,
--R                          R: PolynomialCategory(K,E,OV))  
--R    is a package constructor
--R Abbreviation for InterfaceGroebnerPackage is INTERGB 
--R This constructor is exposed in this frame.
--R Issue )edit bookvol10.4.pamphlet to see algebra source code for INTERGB 
--R
--R------------------------------- Operations --------------------------------
--R groebner : List R -> List R          
--R
--E 1

)spool
)lisp (bye)
\end{verbatim}


There is another section, Exports, which lists all of the unique function
names exported by this algebra. Note that this list is longer than just
the functions locally defined because the algebra inherits functions.
We make a tabular environment with the number of columns specified by
the ``l'' characters (``l'' means left-justify, ``c'' would be center, and
``r''  would be right justify). We include just enough columns to keep from
overflowing the 80 column limit.

For this domain we see generate we only have a single export.
\begin{verbatim}
{\bf Exports:}\\
\cross{INTERGB}{groebner}
\end{verbatim}

The cross function creates a cross-reference entry in the index so you 
can look up the function and find the domain or look up the domain and
find the function.

Now we have finished setting up the algebra.

We have to add the algebra, regression, and help information to the Makefile.

We know from above that the algebra belongs in layer 11 so we find layer 11
and add this abbreviation in alphabetical order. We can use this order since
none of the files depend on each other in this layer. 
The new line looks like:
\begin{verbatim}
  ${OUT}/INMODGCD.o ${OUT}/INNMFACT.o ${OUT}/INPSIGN.o  ${OUT}/INTERGB.o  \
\end{verbatim}
Note that the trailing slash is required by the Makefile in order to continue
the input line.

Take note of the prior algebra abbreviation INPSIGN. We search for that
lower down and find the block reading:
\begin{verbatim}
"INPSIGN" [color="#FF4488",href="bookvol10.4.pdf#nameddest=INPSIGN"]
/*"INPSIGN" -> {"RING"; "RNG"; "ABELGRP"; "CABMON"; "ABELMON"; "ABELSG"}*/
/*"INPSIGN" -> {"SETCAT"; "BASTYPE"; "KOERCE"; "SGROUP"; "MONOID"}*/
/*"INPSIGN" -> {"LMODULE"; "UPOLYC"; "POLYCAT"; "PDRING"; "FAMR"; "AMR"}*/
/*"INPSIGN" -> {"BMODULE"; "RMODULE"; "COMRING"; "ALGEBRA"; "MODULE"}*/
/*"INPSIGN" -> {"CHARZ"; "CHARNZ"; "INTDOM"; "ENTIRER"; "FRETRCT"}*/
/*"INPSIGN" -> {"RETRACT"; "EVALAB"; "IEVALAB"; "FLINEXP"; "LINEXP"}*/
/*"INPSIGN" -> {"ORDSET"; "KONVERT"; "PATMAB"; "GCDDOM"}*/
"INPSIGN" -> "PFECAT"
/*"INPSIGN" -> {"UFD"; "ELTAB"; "DIFRING"; "DIFEXT"; "STEP"; "EUCDOM"}*/
/*"INPSIGN" -> {"PID"; "FIELD"; "DIVRING"; "INT"; "INS-"}*/
\end{verbatim}

This information is used to create the total graph of the algebra.
Most of this information is kept for future graph use but commented out.
We only uncomment lines that are direct dependence relationships. As you
can see from the above INPSIGN depends only on PFECAT.

First we create the signature line which is the same one used above for
the pagepic graph.

We also need to create a similar block for our code. We use the list generated
above, replacing the ``.o'' with semicolons.

The total result is:
\begin{verbatim}
"INTERGB" [color="#FF4488",href="bookvol10.4.pdf#nameddest=INTERGB"]
/*"INTERGB" -> {
/*"INTERGB" -> {"FIELD"; "EUCDOM"; "PID"; "GCDDOM"; "INTDOM"; "COMRING"}*/
/*"INTERGB" -> {"RING"; "RNG"; "ABELGRP"; "CABMON"; "ABELMON"; "ABELSG"}*/
/*"INTERGB" -> {"SETCAT"; "BASTYPE"; "KOERCE"; "SGROUP"; "MONOID"}*/
/*"INTERGB" -> {"LMODULE"; "BMODULE"; "RMODULE"; "ALGEBRA"; "MODULE"}*/
/*"INTERGB" -> {"ENTIRER"; "UFD"; "DIVRING"; "OAMONS"; "OCAMON"; "OAMON"}*/
/*"INTERGB" -> {"OASGP"; "ORDSET"; "POLYCAT"; "PDRING"; "FAMR"; "AMR"}*/
/*"INTERGB" -> {"CHARZ"; "CHARNZ"; "FRETRCT"; "RETRACT"; "EVALAB"}*/
/*"INTERGB" -> {"IEVALAB"; "FLINEXP"; "LINEXP"; "KONVERT"; "PATMAB"}*/
"INTERGB" -> "PFECAT"
/*"INTERGB" -> {"FFIELDC"; "FPC"; "FINITE"; "STEP"; "DIFRING"; "NNI"; "INT"}*/
\end{verbatim}

Help files are automatically extracted from the books using the
lisp function ``makeHelpFiles'' which lives in books/tangle.lisp.
This will find all of the .help chunks in books of interest
and write each chunk to the target directory in its own filename.
So if a chunk name is somedomain.help we create the help file
somedomain.help containing the chunk value.


Now we create the regression hook. Look for the list REGRESS and add the 
line containing the algebra thus:
\begin{verbatim}
 InterfaceGroebnerPackage.regress \
\end{verbatim}

This will cause regression to be run on the InterfaceGroebnerPackage.input
file.

We need to modify book volume 5 (the interpreter) to add this algebra
to the list of exposed algebra. To do this, add the line:
\begin{verbatim}
   (|InterfaceGroebnerPackage| . INTERGB)
\end{verbatim}
to the variable \verb|$globalExposureGroupAlist| under the ``basic''
sublist. 

Now the algebra should build and have the new algebra available.
There are ways this can fail which we will cover in more detail.

Once the code compiles cleanly there are a few ways to check that it works.
First, check that the file int/input/InterfaceGroebnerPackage.regress shows
no failures. Second, run the command
\begin{verbatim}
  )show InterfaceGroebnerPackage
\end{verbatim}
to see if the domain exists. Run the command
\begin{verbatim}
  )help InterfaceGroebnerPackage
\end{verbatim}
to see that the help file exists. Run the command
\begin{verbatim}
  )describe InterfaceGroebnerPackage
\end{verbatim}
to see that the description paragraph exists.

\subsection{Creating a stand-alone pamphlet file}
Suppose we want to add a new algebra file from a new pamphlet.
For example, we want to add BLAS1.spad in books/newbook.pamphlet.
The explanation for the steps follow. The steps are:
\begin{enumerate}
\item write the algebra code
\item create the pamphlet file called newbook.pamphlet
\item create a new section in newbook.pamphlet
\item create the chunk name stanza in newbook.pamphlet
\item insert BLAS1.spad into the new stanza
\item create the dotabb stanza in newbook.pamphlet
\item update src/algebra/Makefile.pamphlet to insert BLAS1.spad 
\item update src/algebra/Makefile.pamphlet to insert BLAS1 into graph
\item update the dotabb stanza in newbook.pamphlet
\item update src/Makefile.pamphlet to copy the newbook file to src/algebra
\item update \$globalExposureGroupAlist in bookvol5 to add BLAS1 to basic
\end{enumerate}

\section{Makefile}
This book is actually a literate program\cite{Knut92} and can contain 
executable source code. In particular, the Makefile for this book
is part of the source of the book and is included below. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cleardoublepage
\phantomsection
\addcontentsline{toc}{chapter}{Bibliography}
\bibliographystyle{axiom}
\bibliography{axiom}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cleardoublepage
\phantomsection
\addcontentsline{toc}{chapter}{Index}
\printindex
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}