===================================================================
RCS file: /home/cvs/OpenXM/src/asir-doc/parts/type.texi,v
retrieving revision 1.2
retrieving revision 1.3
diff -u -p -r1.2 -r1.3
--- OpenXM/src/asir-doc/parts/type.texi	1999/12/10 06:58:49	1.2
+++ OpenXM/src/asir-doc/parts/type.texi	1999/12/21 02:47:32	1.3
@@ -1,68 +1,143 @@
+@comment $OpenXM$
+\BJP
 @node $B7?(B,,, Top
 @chapter $B7?(B
+\E
+\BEG
+@node Data types,,, Top
+@chapter Data types
+\E
 
 @menu
+\BJP
 * Asir $B$G;HMQ2DG=$J7?(B::
 * $B?t$N7?(B::
 * $BITDj85$N7?(B::
+\E
+\BEG
+* Types in Asir::
+* Types of numbers::
+* Types of indeterminates::
+\E
 @end menu
 
+\BJP
 @node Asir $B$G;HMQ2DG=$J7?(B,,, $B7?(B
 @section @b{Asir} $B$G;HMQ2DG=$J7?(B
+\E
+\BEG
+@node Types in Asir,,, Data types
+@section Types in @b{Asir}
+\E
 
 @noindent
+\BJP
 @b{Asir} $B$K$*$$$F$O(B, $B2DFI$J7A<0$GF~NO$5$l$?$5$^$6$^$JBP>]$O(B, $B%Q!<%6$K$h$j(B
 $BCf4V8@8l$KJQ49$5$l(B, $B%$%s%?%W%j%?$K$h$j(B @b{Risa} $B$N7W;;%(%s%8%s$r8F$S=P$7(B
 $B$J$,$iFbIt7A<0$KJQ49$5$l$k(B. $BJQ49$5$l$?BP>]$O(B, $B<!$N$$$:$l$+$N7?$r;}$D(B. 
 $B3FHV9f$O(B, $BAH$_9~$_H!?t(B @code{type()} $B$K$h$jJV$5$l$kCM$KBP1~$7$F$$$k(B. 
 $B3FNc$O(B, @b{Asir} $B$N%W%m%s%W%H$KBP$9$kF~NO$,2DG=$J7A<0$N$$$/$D$+$r(B
 $B<($9(B. 
+\E
+\BEG
+In @b{Asir}, various objects described according to the syntax of
+@b{Asir} are translated to intermediate forms and by @b{Asir}
+interpreter further translated into internal forms with the help of
+basic algebraic engine.  Such an object in an internal form has one of
+the following types listed below.
+In the list, the number coincides with the value returned by the
+built-in function @code{type()}.
+Each example shows possible forms of inputs for @b{Asir}'s prompt.
+\E
 
 @table @code
 @item 0 @b{0}
 
+\BJP
 $B<B:]$K$O(B 0 $B$r<1JL;R$K$b$DBP>]$OB8:_$7$J$$(B. 0 $B$O(B, C $B$K$*$1$k(B 0 $B%]%$%s%?$K(B
 $B$h$jI=8=$5$l$F$$$k(B. $B$7$+$7(B, $BJX59>e(B @b{Asir} $B$N(B @code{type(0)} $B$O(B
 $BCM(B 0 $B$rJV$9(B. 
+\E
+\BEG
+As a matter of fact, no object exists that has 0 as its identification
+number.  The number 0 is implemented as a null (0) pointer of C language.
+For convenience's sake,  a 0 is returned for the input @code{type(0)}.
+\E
 
-@item 1 @b{$B?t(B}
+\JP @item 1 @b{$B?t(B}
+\EG @item 1 @b{number}
 
 @example
 1 2/3  14.5  3+2*@@i 
 @end example
 
-$B?t$O(B, $B$5$i$K$$$/$D$+$N7?$KJ,$1$i$l$k(B. $B$3$l$K$D$$$F$O2<$G=R$Y$k(B. 
+\JP $B?t$O(B, $B$5$i$K$$$/$D$+$N7?$KJ,$1$i$l$k(B. $B$3$l$K$D$$$F$O2<$G=R$Y$k(B. 
+\EG Numbers have sub-types. @xref{Types of numbers}.
 
-@item 2 @b{$BB?9`<0(B} ($B?t$G$J$$(B)
+\JP @item 2 @b{$BB?9`<0(B} ($B?t$G$J$$(B)
+\EG @item 2 @b{polynomial} (but not a number)
 
 @example
 x  afo  (2.3*x+y)^10
 @end example
 
+\BJP
 $BB?9`<0$O(B, $BA4$FE83+$5$l(B, $B$=$N;~E@$K$*$1$kJQ?t=g=x$K=>$C$F(B, $B:F5"E*$K(B
 1 $BJQ?tB?9`<0$H$7$F9_QQ$N=g$K@0M}$5$l$k(B (@xref{$BJ,;6I=8=B?9`<0(B}). 
 $B$3$N;~(B, $B$=$NB?9`<0$K8=$l$k=g=x:GBg$NJQ?t$r(B @b{$B<gJQ?t(B} $B$H8F$V(B. 
+\E
+\BEG
+Every polynomial is maintained internally in its full expanded form,
+represented as a nested univariate polynomial, according to the current
+variable ordering, arranged by the descending order of exponents.
+(@xref{Distributed polynomial}).
+In the representation, the indeterminate (or variable), appearing in
+the polynomial, with maximum ordering is called the @b{main variable}.
+Moreover, we call the coefficient of the maximum degree term of
+the polynomial with respect to the main variable the @b{leading coefficient}.
+\E
 
-@item 3 @b{$BM-M}<0(B} ($BB?9`<0$G$J$$(B)
+\JP @item 3 @b{$BM-M}<0(B} ($BB?9`<0$G$J$$(B)
+\EG @item 3 @b{rational expression} (not a polynomial)
 
 @example
 (x+1)/(y^2-y-x)  x/x
 @end example
 
+\BJP
 $BM-M}<0$O(B, $BJ,JlJ,;R$,LsJ,2DG=$G$b(B, $BL@<(E*$K(B @code{red()} $B$,8F$P$l$J$$(B
 $B8B$jLsJ,$O9T$o$l$J$$(B. $B$3$l$O(B, $BB?9`<0$N(B GCD $B1i;;$,6K$a$F=E$$1i;;$G$"$k(B
 $B$?$a$G(B, $BM-M}<0$N1i;;$OCm0U$,I,MW$G$"$k(B. 
+\E
+\BEG
+Note that in @b{Risa/Asir} a rational expression is not simplified
+by reducing the common divisors unless @code{red()} is called
+explicitly, even if it is possible.  This is because the GCD computation
+of polynomials is a considerably heavy operation.  You have to be careful
+enough in operating rational expressions.
+\E
 
-@item 4 @b{$B%j%9%H(B}
+\JP @item 4 @b{$B%j%9%H(B}
+\EG @item 4 @b{list}
 
 @example
 []  [1,2,[3,4],[x,y]]
 @end example
 
+\BJP
 $B%j%9%H$OFI$_=P$7@lMQ$G$"$k(B. @code{[]} $B$O6u%j%9%H$r0UL#$9$k(B. $B%j%9%H$KBP$9$k(B
 $BA`:n$H$7$F$O(B, @code{car()}, @code{cdr()}, @code{cons()} $B$J$I$K$h$kA`:n$NB>$K(B, 
 $BFI$_=P$7@lMQ$NG[Ns$H$_$J$7$F(B, @code{[@var{index}]} $B$rI,MW$J$@$1$D$1$k$3$H$K$h$j(B
 $BMWAG$N<h$j=P$7$r9T$&$3$H$,$G$-$k(B. $BNc$($P(B
+\E
+\BEG
+Lists are all read-only object. A null list is specified by @code{[]}.
+There are operations for lists: @code{car()}, @code{cdr()},
+@code{cons()} etc.  And further more, element referencing by indexing is
+available.  Indexing is done by putting @code{[@var{index}]}'s after a
+program variable as many as are required.
+For example,
+\E
 
 @example
 [0] L = [[1,2,3],[4,[5,6]],7]$
@@ -70,17 +145,28 @@ x  afo  (2.3*x+y)^10
 [5,6]
 @end example
 
+\BJP
 $BCm0U$9$Y$-$3$H$O(B, $B%j%9%H(B, $BG[Ns(B ($B9TNs(B, $B%Y%/%H%k(B) $B6&$K(B, $B%$%s%G%C%/%9$O(B
 0 $B$+$i;O$^$k$3$H$H(B, $B%j%9%H$NMWAG$N<h$j=P$7$r%$%s%G%C%/%9$G9T$&$3$H$O(B, 
 $B7k6I$O@hF,$+$i%]%$%s%?$r$?$I$k$3$H$KAjEv$9$k$?$a(B, $BG[Ns$KBP$9$kA`:n$K(B
 $BHf3S$7$FBg$-$J%j%9%H$G$O;~4V$,$+$+$k>l9g$,$"$k$H$$$&$3$H$G$"$k(B. 
+\E
+\BEG
+Notice that for lists, matrices and vectors, the index begins with
+number 0.  Also notice that referencing list elements is done by
+following pointers from the first element.  Therefore, it sometimes takes
+much more time to perform referencing operations on a large list than
+on a vectors or a matrices with the same size.
+\E
 
-@item 5 @b{$B%Y%/%H%k(B}
+\JP @item 5 @b{$B%Y%/%H%k(B}
+\EG @item 5 @b{vector}
 
 @example
 newvect(3)  newvect(2,[a,1])
 @end example
 
+\BJP
 $B%Y%/%H%k$O(B, @code{newvect()} $B$GL@<(E*$K@8@.$9$kI,MW$,$"$k(B. $BA0<T$NNc$G(B
 $B$O(B2 $B@.J,$N(B 0 $B%Y%/%H%k$,@8@.$5$l(B, $B8e<T$G$O(B, $BBh(B 0 $B@.J,$,(B @code{a}, $BBh(B 1 
 $B@.J,$,(B @code{1} $B$N%Y%/%H%k$,@8@.$5$l$k(B. $B=i4|2=$N$?$a$N(B $BBh(B 2 $B0z?t$O(B, $BBh(B
@@ -88,6 +174,32 @@ newvect(3)  newvect(2,[a,1])
 $B$j$J$$J,$O(B 0 $B$,Jd$o$l$k(B. $B@.J,$O(B @code{[@var{index}]} $B$K$h$j<h$j=P$;$k(B. $B<B:](B
 $B$K$O(B, $B3F@.J,$K(B, $B%Y%/%H%k(B, $B9TNs(B, $B%j%9%H$r4^$`G$0U$N7?$NBP>]$rBeF~$G$-$k(B
 $B$N$G(B, $BB?<!85G[Ns$r%Y%/%H%k$GI=8=$9$k$3$H$,$G$-$k(B. 
+\E
+\BEG
+Vector objects are created only by explicit execution of @code{newvect()}
+command.  The first example above creates a null vector object with
+3 elements.  The other example creates a vector object
+with 2 elements which is initialized such that its 0-th element
+is @code{a} and 1st element is @code{1}.
+The second argument for @code{newvect} is used to initialize
+elements of the newly created vector.  A list with size smaller or equal
+to the first argument will be accepted.  Elements of the initializing
+list is used from the left to the right.  If the list is too short to
+specify all the vector elements,
+the unspecified elements are filled with as many 0's as are required.
+Any vector element is designated by indexing, e.g.,
+@code{[@var{index}]}.
+@code{Asir} allows any type, including vector, matrix and list,
+for each respective element of a vector.
+As a matter of course, arrays with arbitrary dimensions can be
+represented by vectors, because each element of a vector can be a vector
+or matrix itself.
+An element designator of a vector can be a left value of assignment
+statement.  This implies that an element designator is treated just like
+a simple program variable.
+Note that an assignment to the element designator of a vector has effect
+on the whole value of that vector.
+\E
 
 @example
 [0] A3 = newvect(3);
@@ -102,17 +214,40 @@ newvect(3)  newvect(2,[a,1])
 [ 0 0 0 ]
 @end example
 
-@item 6 @b{$B9TNs(B}
+\JP @item 6 @b{$B9TNs(B}
+\EG @item 6 @b{matrix}
 
 @example
 newmat(2,2)  newmat(2,3,[[x,y],[z]])
 @end example
 
+\BJP
 $B9TNs$N@8@.$b(B @code{newmat()} $B$K$h$jL@<(E*$K9T$o$l$k(B. $B=i4|2=$b(B, $B0z?t(B
 $B$,%j%9%H$N%j%9%H$H$J$k$3$H$r=|$$$F$O%Y%/%H%k$HF1MM$G(B, $B%j%9%H$N3FMWAG(B
 ($B$3$l$O$^$?%j%9%H$G$"$k(B) $B$O(B, $B3F9T$N=i4|2=$K;H$o$l(B, $BB-$j$J$$ItJ,$K$O(B
 0 $B$,Kd$a$i$l$k(B. $B9TNs$b(B, $B3FMWAG$K$OG$0U$NBP>]$rBeF~$G$-$k(B. $B9TNs$N3F(B
 $B9T$O(B, $B%Y%/%H%k$H$7$F<h$j=P$9$3$H$,$G$-$k(B. 
+\E
+\BEG
+Like vector objects, matrix objects are also created only by explicit
+execution of @code{newmat()} command. Initialization of the matrix
+elements are done in a similar manner with that of the vector elements
+except that the elements are specified by a list of lists.  Each element,
+again a list, is used to initialize each row; if the list is too short
+to specify all the row elements, unspecified elements are filled with
+as many 0's as are required.
+Like vectors, any matrix element is designated by indexing, e.g.,
+@code{[@var{index}][@var{index}]}.
+@code{Asir} also allows any type, including vector, matrix and list,
+for each respective element of a matrix.
+An element designator of a matrix can also be a left value of assignment
+statement.  This implies that an element designator is treated just like
+a simple program variable.
+Note that an assignment to the element designator of a matrix has effect
+on the whole value of that matrix.
+Note also that every row, (not column,) of a matrix can be extracted
+and referred to as a vector.
+\E
 
 @example
 [0] M=newmat(2,3);
@@ -124,80 +259,145 @@ newmat(2,2)  newmat(2,3,[[x,y],[z]])
 5
 @end example
 
-@item 7 @b{$BJ8;zNs(B}
+\JP @item 7 @b{$BJ8;zNs(B}
+\EG @item 7 @b{string}
 
 @example
 ""  "afo"
 @end example
 
+\BJP
 $BJ8;zNs$O(B, $B<g$K%U%!%$%kL>$J$I$KMQ$$$i$l$k(B. $BJ8;zNs$KBP$7$F$O2C;;$N$_$,(B
 $BDj5A$5$l$F$$$F(B, $B7k2L$O(B 2 $B$D$NJ8;zNs$N7k9g$G$"$k(B. 
+\E
+\BEG
+Strings are used mainly for naming files.  It is also used for giving
+comments of the results.  Operator symbol @code{+} denote the
+concatenation  operation of two strings.
+\E
 
 @example
 [0] "afo"+"take";
 afotake
 @end example
 
-@item 8 @b{$B9=B$BN(B}
+\JP @item 8 @b{$B9=B$BN(B}
+\EG @item 8 @b{structure}
 
 @example
 newstruct(afo)
 @end example
 
-$B9=B$BN$K4X$7$F$O(B, $B>O$r2~$a$F2r@b$9$kM=Dj$G$"$k(B. 
+\JP $B9=B$BN$K4X$7$F$O(B, $B>O$r2~$a$F2r@b$9$kM=Dj$G$"$k(B. 
+\EG For type @b{structure}, we shall describe it in a later chapter.
+(Not written yet.)
 
-@item 9 @b{$BJ,;6I=8=B?9`<0(B}
+\JP @item 9 @b{$BJ,;6I=8=B?9`<0(B}
+\EG @item 9 @b{distributed polynomial}
 
 @example
 2*<<0,1,2,3>>-3*<<1,2,3,4>>
 @end example
 
+\BJP
 $B$3$l$O(B, $B$[$H$s$I%0%l%V%J4pDl@lMQ$N7?$G(B, $BDL>o$N7W;;$G$3$N7?$,I,MW$H(B
 $B$J$k$3$H$O$^$:$J$$$,(B, $B%0%l%V%J4pDl7W;;%Q%C%1!<%8<+BN$,%f!<%68@8l(B
 $B$G=q$+$l$F$$$k$?$a(B, $B%f!<%6$,A`:n$G$-$k$h$&FHN)$7$?7?$H$7$F(B  @b{Asir}
 $B$G;HMQ$G$-$k$h$&$K$7$F$"$k(B. $B$3$l$K$D$$$F$O(B @xref{$B%0%l%V%J4pDl$N7W;;(B}.
+\E
+\BEG
+This is the short for `Distributed representation of polynomials.'
+This type is specially devised for computation of Groebner bases.
+Though for ordinary users this type may never be needed, it is provided
+as a distinguished type that user can operate by @code{Asir}.
+This is because the Groebner basis package provided with
+@code{Risa/Asir} is written in the @code{Asir} user language.
+For details @xref{Groebner basis computation}.
+\E
 
-@item 10 @b{$BId9f$J$7%^%7%s(B 32bit $B@0?t(B}
+\JP @item 10 @b{$BId9f$J$7%^%7%s(B 32bit $B@0?t(B}
+\EG @item 10 @b{32bit unsigned integer}
 
-@item 11 @b{$B%(%i!<%*%V%8%'%/%H(B}
+\JP @item 11 @b{$B%(%i!<%*%V%8%'%/%H(B}
+\EG @item 11 @b{error object}
 
-$B0J>eFs$D$O(B, Open XM $B$K$*$$$FMQ$$$i$l$kFC<l%*%V%8%'%/%H$G$"$k(B. 
+\JP $B0J>eFs$D$O(B, Open XM $B$K$*$$$FMQ$$$i$l$kFC<l%*%V%8%'%/%H$G$"$k(B. 
+\EG These are special objects used for OpenXM.
 
-@item 12 @b{GF(2) $B>e$N9TNs(B}
+\JP @item 12 @b{GF(2) $B>e$N9TNs(B}
+\EG @item 12 @b{matrix over GF(2)}
 
+\BJP
 $B8=:_(B, $BI8?t(B 2 $B$NM-8BBN$K$*$1$k4pDlJQ49$N$?$a$N%*%V%8%'%/%H$H$7$FMQ$$$i$l(B
 $B$k(B. 
+\E
+\BEG
+This is used for basis conversion in finite fields of characteristic 2.
+\E
 
-@item 13 @b{MATHCAP $B%*%V%8%'%/%H(B}
+\JP @item 13 @b{MATHCAP $B%*%V%8%'%/%H(B}
+\EG @item 13 @b{MATHCAP object}
 
-Open XM $B$K$*$$$F(B, $B<BAu$5$l$F$$$k5!G=$rAw<u?.$9$k$?$a$N%*%V%8%'%/%H$G$"$k(B. 
+\JP Open XM $B$K$*$$$F(B, $B<BAu$5$l$F$$$k5!G=$rAw<u?.$9$k$?$a$N%*%V%8%'%/%H$G$"$k(B. 
+\EG This object is used to express available funcionalities for Open XM.
 
 @item 14 @b{first order formula}
 
-quantifier elimination $B$GMQ$$$i$l$k0l3,=R8lO@M}<0(B. 
+\JP quantifier elimination $B$GMQ$$$i$l$k0l3,=R8lO@M}<0(B. 
+\EG This expresses a first order formula used in quantifier elimination.
 
-@item -1 @b{VOID $B%*%V%8%'%/%H(B}
+\JP @item -1 @b{VOID $B%*%V%8%'%/%H(B}
+\EG @item -1 @b{VOID object}
 
-$B7?<1JL;R(B -1 $B$r$b$D%*%V%8%'%/%H$O4X?t$NLa$jCM$J$I$,L58z$G$"$k$3$H$r<($9(B.
+\JP $B7?<1JL;R(B -1 $B$r$b$D%*%V%8%'%/%H$O4X?t$NLa$jCM$J$I$,L58z$G$"$k$3$H$r<($9(B.
+\BEG
+The object with the object identifier -1 indicates that a return value
+of a function is void.
+\E
 @end table
 
+\BJP
 @node $B?t$N7?(B,,, $B7?(B
 @section $B?t$N7?(B
+\E
+\BEG
+@node Types of numbers,,, Data types
+@section Types of numbers
+\E
 
 @table @code
 @item 0
-@b{$BM-M}?t(B}
+\JP @b{$BM-M}?t(B}
+\EG @b{rational number}
 
+\BJP
 $BM-M}?t$O(B, $BG$0UB?G\D9@0?t(B (@b{bignum}) $B$K$h$j<B8=$5$l$F$$$k(B. $BM-M}?t$O>o$K(B
 $B4{LsJ,?t$GI=8=$5$l$k(B. 
+\E
+\BEG
+Rational numbers are implemented by arbitrary precision integers
+(@b{bignum}).  A rational number is always expressed by a fraction of
+lowest terms.
+\E
 
 @item 1
-@b{$BG\@:EYIbF0>.?t(B}
+\JP @b{$BG\@:EYIbF0>.?t(B}
+\EG @b{double precision floating point number (double float)}
 
+\BJP
 $B%^%7%s$NDs6!$9$kG\@:EYIbF0>.?t$G$"$k(B. @b{Asir} $B$N5/F0;~$K$O(B, 
 $BDL>o$N7A<0$GF~NO$5$l$?IbF0>.?t$O$3$N7?$KJQ49$5$l$k(B. $B$?$@$7(B, 
 @code{ctrl()} $B$K$h$j(B @b{bigfloat} $B$,A*Br$5$l$F$$$k>l9g$K$O(B
 @b{bigfloat} $B$KJQ49$5$l$k(B. 
+\E
+\BEG
+The numbers of this type are numbers provided by the computer hardware.
+By default, when @b{Asir} is started, floating point numbers in a
+ordinary form are transformed into numbers of this type.  However,
+they will be transformed into @b{bigfloat} numbers
+when the switch @b{bigfloat} is turned on (enabled) by @code{ctrl()}
+command.
+\E
 
 @example
 [0] 1.2;
@@ -210,23 +410,44 @@ quantifier elimination $B$GMQ$$$i$l$k0l3,=R8lO@M}<0(
 1.20000000000000000513 E-1000
 @end example
 
+\BJP
 $BG\@:EYIbF0>.?t$HM-M}?t$N1i;;$O(B, $BM-M}?t$,IbF0>.?t$KJQ49$5$l$F(B, 
 $BIbF0>.?t$H$7$F1i;;$5$l$k(B. 
+\E
+\BEG
+A rational number shall be converted automatically into a double float
+number before the operation with another double float number and the
+result shall be computed as a double float number.
+\E
 
 @item 2
-@b{$BBe?tE*?t(B}
+\JP @b{$BBe?tE*?t(B}
+\EG @b{algebraic number}
 
-@xref{$BBe?tE*?t$K4X$9$k1i;;(B}.
+\JP @xref{$BBe?tE*?t$K4X$9$k1i;;(B}.
+\EG @xref{Algebraic numbers}.
 
 @item 3
 @b{bigfloat}
 
+\BJP
 @b{bigfloat} $B$O(B, @b{Asir} $B$G$O(B @b{PARI} $B%i%$%V%i%j$K$h$j(B
 $B<B8=$5$l$F$$$k(B. @b{PARI} $B$K$*$$$F$O(B, @b{bigfloat} $B$O(B, $B2>?tIt(B
 $B$N$_G$0UB?G\D9$G(B, $B;X?tIt$O(B 1 $B%o!<%I0JFb$N@0?t$K8B$i$l$F$$$k(B. 
 @code{ctrl()} $B$G(B @b{bigfloat} $B$rA*Br$9$k$3$H$K$h$j(B, $B0J8e$NIbF0>.?t(B
 $B$NF~NO$O(B @b{bigfloat} $B$H$7$F07$o$l$k(B. $B@:EY$O%G%U%)%k%H$G$O(B
 10 $B?J(B 9 $B7eDxEY$G$"$k$,(B, @code{setprec()} $B$K$h$j;XDj2DG=$G$"$k(B. 
+\E
+\BEG
+The @b{bigfloat} numbers of @b{Asir} is realized by @b{PARI} library.
+A @b{bigfloat} number of @b{PARI} has an arbitrary precision mantissa
+part.  However, its exponent part admits only an integer with a single
+word precision.
+Floating point operations will be performed all in @b{bigfloat} after
+activating the @b{bigfloat} switch by @code{ctrl()} command.
+The default precision is about 9 digits, which can be specified by
+@code{setprec()} command.
+\E
 
 @example
 [0] ctrl("bigfloat",1);
@@ -239,95 +460,214 @@ quantifier elimination $B$GMQ$$$i$l$k0l3,=R8lO@M}<0(
 1.41421356237309504880168872420969807856967187537694807317654396116148
 @end example
 
+\BJP
 @code{eval()} $B$O(B, $B0z?t$K4^$^$l$kH!?tCM$r2DG=$J8B$j?tCM2=$9$kH!?t$G$"$k(B. 
 @code{setprec()} $B$G;XDj$5$l$?7e?t$O(B, $B7k2L$N@:EY$rJ]>Z$9$k$b$N$G$O$J$/(B, 
 @b{PARI} $BFbIt$GMQ$$$i$l$kI=8=$N%5%$%:$r<($9$3$H$KCm0U$9$Y$-$G$"$k(B. 
+\E
+\BEG
+Function @code{eval()} evaluates numerically its argument as far as
+possible.
+Notice that the integer given for the argument of @code{setprec()} does
+not guarantee the accuracy of the result,
+but it indicates the representation size of numbers with which internal
+operations of @b{PARI} are performed.
+\E
 (@ref{eval}, @xref{pari})
 
 @item 4
-@b{$BJ#AG?t(B}
+\JP @b{$BJ#AG?t(B}
+\EG @b{complex number}
 
+\BJP
 $BJ#AG?t$O(B, $BM-M}?t(B, $BG\@:EYIbF0>.?t(B, @b{bigfloat} $B$r<BIt(B, $B5uIt$H$7$F(B
 @code{a+b*@@i} (@@i $B$O5u?tC10L(B) $B$H$7$FM?$($i$l$k?t$G$"$k(B. $B<BIt(B, $B5uIt$O(B
 $B$=$l$>$l(B @code{real()}, @code{imag()} $B$G<h$j=P$;$k(B. 
+\E
+\BEG
+A @b{complex} number of @b{Risa/Asir} is a number with the form
+@code{a+b*@@i}, where @@i is the unit of imaginary number, and @code{a}
+and @code{b}
+are either a @b{rational} number, @b{double float} number or
+@b{bigfloat} number, respectively.
+The real part and the imaginary part of a @b{complex} number can be
+taken out by @code{real()} and @code{imag()} respectively.
+\E
 
 @item 5
-@b{$B>.I8?t$NM-8BAGBN$N85(B}
+\JP @b{$B>.I8?t$NM-8BAGBN$N85(B}
+\EG @b{element of a small finite prime field}
 
+\BJP
 $B$3$3$G8@$&>.I8?t$H$O(B, $BI8?t$,(B 2^27 $BL$K~$N$b$N$N$3$H$G$"$k(B. $B$3$N$h$&$JM-8B(B
 $BBN$O(B, $B8=:_$N$H$3$m%0%l%V%J4pDl7W;;$K$*$$$FFbItE*$KMQ$$$i$l(B, $BM-8BBN78?t$N(B
 $BJ,;6I=8=B?9`<0$N78?t$r<h$j=P$9$3$H$GF@$i$l$k(B. $B$=$l<+?H$OB0$9$kM-8BBN$K4X(B
 $B$9$k>pJs$O;}$?$:(B, @code{setmod()} $B$G@_Dj$5$l$F$$$kAG?t(B @var{p} $B$rMQ$$$F(B 
 GF(@var{p}) $B>e$G$N1i;;$,E,MQ$5$l$k(B.
+\E
+\BEG
+Here a small finite fieid means that its characteristic is less than
+2^27. 
+At present small finite fields are used mainly
+for groebner basis computation, and elements in such finite fields
+can be extracted by taking coefficients of distributed polynomials
+whose coefficients are in finite fields. Such an element itself does not
+have any information about the field to which the element belongs, and
+field operations are executed by using a prime @var{p} which is set by
+@code{setmod()}.
+\E
 
 @item 6
-@b{$BBgI8?t$NM-8BAGBN$N85(B}
+\JP @b{$BBgI8?t$NM-8BAGBN$N85(B}
+\EG @b{element of large finite prime field}
 
+\BJP
 $BI8?t$H$7$FG$0U$NAG?t$,$H$l$k(B. 
 $B$3$N7?$N?t$O(B, $B@0?t$KBP$7(B@code{simp_ff} $B$rE,MQ$9$k$3$H$K$h$jF@$i$l$k(B. 
+\E
+\BEG
+This type expresses an element of a finite prime field whose characteristic
+is an arbitrary prime. An object of this type is obtained by applying
+@code{simp_ff} to an integer.
+\E
 
-
 @item 7
-@b{$BI8?t(B 2 $B$NM-8BBN$N85(B}
+\JP @b{$BI8?t(B 2 $B$NM-8BBN$N85(B}
+\EG @b{element of a finite field of characteristic 2}
 
+\BJP
 $BI8?t(B 2 $B$NG$0U$NM-8BBN$N85$rI=8=$9$k(B. $BI8?t(B 2 $B$NM-8BBN(B F $B$O(B, $B3HBg<!?t(B 
 [F:GF(2)] $B$r(B n $B$H$9$l$P(B, GF(2) $B>e4{Ls$J(B n $B<!B?9`<0(B f(t) $B$K$h$j(B
 F=GF(2)[t]/(f(t)) $B$H$"$i$o$5$l$k(B. $B$5$i$K(B, GF(2)[t] $B$N85(B g $B$O(B, f(t) 
-$B$b4^$a$F<+A3$J;EJ}$G%S%C%HNs$H$_$J$5$l$k$?$a(B, $B7A<0>e$O(B, F $B$N85$O(B,
+$B$b4^$a$F<+A3$J;EJ}$G%S%C%HNs$H$_$J$5$l$k$?$a(B, $B7A<0>e$O(B, F $B$N85(B
 g mod f $B$O(B, g, f  $B$r$"$i$o$9(B 2 $B$D$N%S%C%HNs$GI=8=$9$k$3$H$,$G$-$k(B. 
+\E
+\BEG
+This type expresses an element of a finite field of characteristic 2.
+Let @var{F} be a finite field of characteristic 2. If @var{[F:GF(2)]}
+is equal to @var{n}, then @var{F} is expressed as @var{F=GF(2)[t]/(f(t))},
+where @var{f(t)} is an irreducible polynomial over @var{GF(2)} 
+of degree @var{n}.
+As an element @var{g} of @var{GF(2)[t]} can be expressed by a bit string,
+An element @var{g mod f} in @var{F} can be expressed by two bit strings
+representing @var{g} and @var{f} respectively.
+\E
 
-F $B$N85$rF~NO$9$k$$$/$D$+$NJ}K!$,MQ0U$5$l$F$$$k(B. 
+\JP F $B$N85$rF~NO$9$k$$$/$D$+$NJ}K!$,MQ0U$5$l$F$$$k(B. 
+\EG Several methods to input an element of @var{F} are provided.
 
 @itemize @bullet
 @item
 @code{@@}
 
+\BJP
 @code{@@} $B$O$=$N8e$m$K?t;z(B, $BJ8;z$rH<$C$F(B, $B%R%9%H%j$dFC<l$J?t$r$"$i$o$9$,(B, 
 $BC1FH$G8=$l$?>l9g$K$O(B, F=GF(2)[t]/(f(t)) $B$K$*$1$k(B t mod f $B$r$"$i$o$9(B. 
 $B$h$C$F(B, @@ $B$NB?9`<0$H$7$F(B F $B$N85$rF~NO$G$-$k(B. (@@^10+@@+1 $B$J$I(B)
+\E
+\BEG
+@code{@@} represents @var{t mod f} in @var{F=GF(2)[t](f(t))}.
+By using @code{@@} one can input an element of @var{F}. For example
+@code{@@^10+@@+1} represents an element of @var{F}.
+\E
 
 @item
 @code{ptogf2n}
 
-$BG$0UJQ?t$N(B 1 $BJQ?tB?9`<0$r(B, @code{ptogf2n} $B$K$h$jBP1~$9$k(B F $B$N85$KJQ49$9$k(B. 
+\JP $BG$0UJQ?t$N(B 1 $BJQ?tB?9`<0$r(B, @code{ptogf2n} $B$K$h$jBP1~$9$k(B F $B$N85$KJQ49$9$k(B. 
+\BEG
+@code{ptogf2n} converts a univariate polynomial into an element of @var{F}.
+\E
 
 @item
 @code{ntogf2n}
 
+\BJP
 $BG$0U$N<+A3?t$r(B, $B<+A3$J;EJ}$G(B F $B$N85$H$_$J$9(B. $B<+A3?t$H$7$F$O(B, 10 $B?J(B, 
 16 $B?J(B (0x $B$G;O$^$k(B), 2 $B?J(B (0b $B$G;O$^$k(B) $B$GF~NO$,2DG=$G$"$k(B. 
+\E
+\BEG
+As a bit string, a non-negative integer can be regarded as an element
+of @var{F}. Note that one can input a non-negative integer in decimal,
+hexadecimal (@code{0x} prefix) and binary (@code{0b} prefix) formats.
+\E
 
 @item
-@code{$B$=$NB>(B}
+\JP @code{$B$=$NB>(B}
+\EG @code{micellaneous}
 
+\BJP
 $BB?9`<0$N78?t$r4]$4$H(B F $B$N85$KJQ49$9$k$h$&$J>l9g(B, @code{simp_ff}
 $B$K$h$jJQ49$G$-$k(B. 
+\E
+\BEG
+@code{simp_ff} is available if one wants to convert the whole
+coefficients of a polynomial.
+\E
 
 @end itemize
 @end table
 
+\BJP
 $BBgI8?tAGBN$NI8?t(B, $BI8?t(B 2 $B$NM-8BBN$NDj5AB?9`<0$O(B, @code{setmod_ff}
 $B$G@_Dj$9$k(B. 
 $BM-8BBN$N85$I$&$7$N1i;;$G$O(B, @code{setmod_ff} $B$K$h$j@_Dj$5$l$F$$$k(B
 modulus $B$G(B, $BB0$9$kBN$,J,$+$j(B, $B$=$NCf$G1i;;$,9T$o$l$k(B. 
 $B0lJ}$,M-M}?t$N>l9g$K$O(B, $B$=$NM-M}?t$O<+F0E*$K8=:_@_Dj$5$l$F$$$k(B
 $BM-8BBN$N85$KJQ49$5$l(B, $B1i;;$,9T$o$l$k(B. 
+\E
+\BEG
+The characteristic of a large finite prime field and the defining
+polynomial of a finite field of characteristic 2 are set by @code{setmod_ff}.
+Elements of finite fields do not have informations about the modulus.
+Upon an arithmetic operation, the modulus set by @code{setmod_ff} is
+used. If one of the operands is a rational number, it is automatically
+converted into an element of the finite field currently set and
+the operation is done in the finite field.
+\E
 
+\BJP
 @node $BITDj85$N7?(B,,, $B7?(B
 @section $BITDj85$N7?(B
+\E
+\BEG
+@node Types of indeterminates,,, Data types
+@section Types of indeterminates
+\E
 
 @noindent
+\BJP
 $BB?9`<0$NJQ?t$H$J$jF@$kBP>]$r(B@b{$BITDj85(B}$B$H$h$V(B. @b{Asir} $B$G$O(B, 
 $B1Q>.J8;z$G;O$^$j(B, $BG$0U8D$N%"%k%U%!%Y%C%H(B, $B?t;z(B, @samp{_} $B$+$i$J$kJ8;zNs(B
 $B$rITDj85$H$7$F07$&$,(B, $B$=$NB>$K$b%7%9%F%`$K$h$jITDj85$H$7$F07$o$l$k$b$N(B
 $B$,$$$/$D$+$"$k(B. @b{Asir} $B$NFbIt7A<0$H$7$F$O(B, $B$3$l$i$OA4$FB?9`<0$H$7$F$N(B
 $B7?$r;}$D$,(B, $B?t$HF1MM(B, $BITDj85$N7?$K$h$j6hJL$5$l$k(B.
+\E
+\BEG
+An algebraic object is recognized as an indeterminate when it can be
+a (so-called) variable in polynomials.
+An ordinary indeterminate is usually denoted by a string that start with
+a small alphabetical letter followed by an arbitrary number of
+alphabetical letters, digits or @samp{_}.
+In addition to such ordinary indeterminates,
+there are other kinds of indeterminates in a wider sense in @b{Asir}.
+Such indeterminates in the wider sense have type @b{polynomial},
+and further are classified into sub-types of the type @b{indeterminate}.
+\E
 
 @table @code
 @item 0
-@b{$B0lHLITDj85(B}
+\JP @b{$B0lHLITDj85(B}
+\EG @b{ordinary indeterminate}
 
-$B1Q>.J8;z$G;O$^$kJ8;zNs(B. $BB?9`<0$NJQ?t$H$7$F:G$bIaDL$KMQ$$$i$l$k(B. 
+\JP $B1Q>.J8;z$G;O$^$kJ8;zNs(B. $BB?9`<0$NJQ?t$H$7$F:G$bIaDL$KMQ$$$i$l$k(B. 
+\BEG
+An object of this sub-type is denoted by a string that start with
+a small alphabetical letter followed by an arbitrary number of
+alphabetical letters, digits or @samp{_}.
+This kind of indeterminates are most commonly used for variables of
+polynomials.
+\E
 
 @example
 [0] [vtype(a),vtype(aA_12)];
@@ -335,12 +675,24 @@ modulus $B$G(B, $BB0$9$kBN$,J,$+$j(B, $B$=$NCf$G1
 @end example
 
 @item 1
-@b{$BL$Dj78?t(B}
+\JP @b{$BL$Dj78?t(B}
+\EG @b{undetermined coefficient}
 
+\BJP
 @code{uc()} $B$O(B, @samp{_} $B$G;O$^$kJ8;zNs$rL>A0$H$9$kITDj85$r@8@.$9$k(B. 
 $B$3$l$i$O(B, $B%f!<%6$,F~NO$G$-$J$$$H$$$&$@$1$G(B, $B0lHLITDj85$HJQ$o$i$J$$$,(B, 
 $B%f!<%6$,F~NO$7$?ITDj85$H>WFM$7$J$$$H$$$&@-<A$rMxMQ$7$FL$Dj78?t$N(B
 $B<+F0@8@.$J$I$KMQ$$$k$3$H$,$G$-$k(B. 
+\E
+\BEG
+The function @code{uc()} creates an indeterminate which is denoted by
+a string that begins with @samp{_}.  Such an indeterminate cannot be
+directly input by its name.  Other properties are the same as those of
+@b{ordinary indeterminate}.  Therefore, it has a property that it cannot
+cause collision with the name of ordinary indeterminates input by the
+user.  And this property is conveniently used to create undetermined
+coefficients dynamically by programs.
+\E
 
 @example
 [1] U=uc();                 
@@ -350,13 +702,27 @@ _0
 @end example
 
 @item 2
-@b{$BH!?t7A<0(B}
+\JP @b{$BH!?t7A<0(B}
+\EG @b{function form}
 
+\BJP
 $BAH$_9~$_H!?t(B, $B%f!<%6H!?t$N8F$S=P$7$O(B, $BI>2A$5$l$F2?$i$+$N(B @b{Asir} $B$N(B
 $BFbIt7A<0$KJQ49$5$l$k$,(B, @code{sin(x)}, @code{cos(x+1)} $B$J$I$O(B, $BI>2A8e(B
 $B$b$=$N$^$^$N7A$GB8:_$9$k(B. $B$3$l$OH!?t7A<0$H8F$P$l(B, $B$=$l<+?H$,(B 1 $B$D$N(B
 $BITDj85$H$7$F07$o$l$k(B. $B$^$?$d$dFC<l$JNc$H$7$F(B, $B1_<~N((B @code{@@pi} $B$d(B
 $B<+A3BP?t$NDl(B @code{@@e} $B$bH!?t7A<0$H$7$F07$o$l$k(B. 
+\E
+\BEG
+A function call to a built-in function or to an user defined function
+is usually evaluated by @b{Asir} and retained in a proper internal form.
+Some expressions, however, will remain in the same form after evaluation.
+For example, @code{sin(x)} and @code{cos(x+1)} will remain as if they
+were not evaluated.  These (unevaluated) forms are called
+`function forms' and are treated as if they are indeterminates in a
+wider sense.  Also, special forms such as @code{@@pi} the ratio of
+circumference and diameter, and @code{@@e} Napier's number, will be
+treated as `function forms.'
+\E
 
 @example
 [3] V=sin(x);         
@@ -368,12 +734,22 @@ sin(x)
 @end example
 
 @item 3
-@b{$BH!?t;R(B}
+\JP @b{$BH!?t;R(B}
+\EG @b{functor}
 
+\BJP
 $BH!?t8F$S=P$7$O(B, @var{fname(args)} $B$H$$$&7A$G9T$J$o$l$k$,(B, @var{fname} $B$N(B
 $BItJ,$rH!?t;R$H8F$V(B. $BH!?t;R$K$O(B, $BH!?t$N<oN`$K$h$jAH$_9~$_H!?t;R(B,
 $B%f!<%6Dj5AH!?t;R(B, $B=iEyH!?t;R$J$I$,$"$k$,(B, $BH!?t;R$OC1FH$GITDj85$H$7$F(B
 $B5!G=$9$k(B. 
+\E
+\BEG
+A function call (or a function form) has a form @var{fname(args)}.
+Here, @var{fname} alone is called a @b{functor}.
+There are several kinds of @b{functor}s: built-in functor, user defined
+functor and functor for the elementary functions.  A functor alone is
+treated as an indeterminate in a wider sense.
+\E
 
 @example
 [6] vtype(sin);