===================================================================
RCS file: /home/cvs/OpenXM/src/asir-doc/parts/type.texi,v
retrieving revision 1.3
retrieving revision 1.13
diff -u -p -r1.3 -r1.13
--- OpenXM/src/asir-doc/parts/type.texi	1999/12/21 02:47:32	1.3
+++ OpenXM/src/asir-doc/parts/type.texi	2007/02/15 02:41:38	1.13
@@ -1,4 +1,4 @@
-@comment $OpenXM$
+@comment $OpenXM: OpenXM/src/asir-doc/parts/type.texi,v 1.12 2003/04/20 08:01:27 noro Exp $
 \BJP
 @node $B7?(B,,, Top
 @chapter $B7?(B
@@ -52,7 +52,7 @@ Each example shows possible forms of inputs for @b{Asi
 
 @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
@@ -83,14 +83,14 @@ x  afo  (2.3*x+y)^10
 
 \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}). 
+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}).
+(@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
@@ -207,7 +207,9 @@ on the whole value of that vector.
 [1] for (I=0;I<3;I++)A3[I] = newvect(3);
 [2] for (I=0;I<3;I++)for(J=0;J<3;J++)A3[I][J]=newvect(3);
 [3] A3;
-[ [ [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] ] [ [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] ] [ [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] ] ]
+[ [ [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] ]
+[ [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] ]
+[ [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] ] ]
 [4] A3[0];
 [ [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] ]
 [5] A3[0][0];
@@ -288,10 +290,18 @@ afotake
 newstruct(afo)
 @end example
 
-\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.)
+\BJP
+Asir $B$K$*$1$k9=B$BN$O(B, C $B$K$*$1$k9=B$BN$r4J0W2=$7$?$b$N$G$"$k(B. 
+$B8GDjD9G[Ns$N3F@.J,$rL>A0$G%"%/%;%9$G$-$k%*%V%8%'%/%H$G(B, 
+$B9=B$BNDj5AKh$KL>A0$r$D$1$k(B. 
+\E
 
+\BEG
+The type @b{structure} is a simplified version of that in C language.
+It is defined as a fixed length array and each entry of the array
+is accessed by its name. A name is associated with each structure.
+\E
+
 \JP @item 9 @b{$BJ,;6I=8=B?9`<0(B}
 \EG @item 9 @b{distributed polynomial}
 
@@ -320,13 +330,13 @@ For details @xref{Groebner basis computation}.
 
 \JP @item 11 @b{$B%(%i!<%*%V%8%'%/%H(B}
 \EG @item 11 @b{error object}
-
+@*
 \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.
 
 \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. 
@@ -337,18 +347,28 @@ This is used for basis conversion in finite fields of 
 
 \JP @item 13 @b{MATHCAP $B%*%V%8%'%/%H(B}
 \EG @item 13 @b{MATHCAP object}
-
+@*
 \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}
-
+@*
 \JP quantifier elimination $B$GMQ$$$i$l$k0l3,=R8lO@M}<0(B. 
 \EG This expresses a first order formula used in quantifier elimination.
 
+@item 15 @b{matrix over GF(@var{p})}
+@*
+\JP $B>.I8?tM-8BBN>e$N9TNs(B.
+\EG A matrix over a small finite field.
+
+@item 16 @b{byte array}
+@*
+\JP $BId9f$J$7(B byte $B$NG[Ns(B
+\EG An array of unsigned bytes.
+
 \JP @item -1 @b{VOID $B%*%V%8%'%/%H(B}
 \EG @item -1 @b{VOID object}
-
+@*
 \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<($9(B.
 \BEG
 The object with the object identifier -1 indicates that a return value
@@ -369,7 +389,7 @@ of a function is void.
 @item 0
 \JP @b{$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$k(B. 
@@ -383,7 +403,7 @@ lowest terms.
 @item 1
 \JP @b{$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. $B$?$@$7(B, 
@@ -423,13 +443,13 @@ result shall be computed as a double float number.
 @item 2
 \JP @b{$BBe?tE*?t(B}
 \EG @b{algebraic number}
-
+@*
 \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
@@ -457,7 +477,7 @@ The default precision is about 9 digits, which can be 
 [2] setprec(100);      
 9
 [3] eval(2^(1/2));
-1.41421356237309504880168872420969807856967187537694807317654396116148
+1.41421356237309504880168872420969807856967187537694807317...
 @end example
 
 \BJP
@@ -473,12 +493,12 @@ 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})
+(@xref{eval deval}, @ref{pari}.)
 
 @item 4
 \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
@@ -497,7 +517,7 @@ taken out by @code{real()} and @code{imag()} respectiv
 @item 5
 \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
@@ -520,7 +540,7 @@ field operations are executed by using a prime @var{p}
 @item 6
 \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. 
@@ -534,7 +554,7 @@ is an arbitrary prime. An object of this type is obtai
 @item 7
 \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
@@ -544,11 +564,11 @@ g mod f $B$O(B, g, f  $B$r$"$i$o$9(B 2 $B$D$N%S%C
 \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)} 
+Let @var{F} be a finite field of characteristic 2. If [F:GF(2)]
+is equal to @var{n}, then @var{F} is expressed as F=GF(2)[t]/(f(t)),
+where f(t) is an irreducible polynomial over GF(2) 
 of degree @var{n}.
-As an element @var{g} of @var{GF(2)[t]} can be expressed by a bit string,
+As an element @var{g} of 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
@@ -559,21 +579,21 @@ representing @var{g} and @var{f} respectively.
 @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))}.
+@code{@@} represents @var{t mod f} in 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}
-
+@*
 \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}.
@@ -581,7 +601,7 @@ By using @code{@@} one can input an element of @var{F}
 
 @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. 
@@ -595,7 +615,7 @@ hexadecimal (@code{0x} prefix) and binary (@code{0b} p
 @item
 \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. 
@@ -606,22 +626,106 @@ coefficients of a polynomial.
 \E
 
 @end itemize
+
+
+@item 8
+\JP @b{$B0L?t(B @var{p^n} $B$NM-8BBN$N85(B}
+\EG @b{element of a finite field of characteristic @var{p^n}}
+
+\BJP
+$B0L?t$,(B @var{p^n} (@var{p} $B$OG$0U$NAG?t(B, @var{n} $B$O@5@0?t(B) $B$O(B, 
+$BI8?t(B @var{p} $B$*$h$S(B GF(@var{p}) $B>e4{Ls$J(B @var{n} $B<!B?9`<0(B m(x)
+$B$r(B @code{setmod_ff} $B$K$h$j;XDj$9$k$3$H$K$h$j@_Dj$9$k(B. 
+$B$3$NBN$N85$O(B m(x) $B$rK!$H$9$k(B GF(@var{p}) $B>e$NB?9`<0$H$7$F(B
+$BI=8=$5$l$k(B. 
+\E
+\BEG
+A finite field of order @var{p^n}, where @var{p} is an arbitrary prime
+and @var{n} is a positive integer, is set by @code{setmod_ff}
+by specifying its characteristic @var{p} and an irreducible polynomial
+of degree @var{n} over GF(@var{p}). An element of this field
+is represented by a polynomial over GF(@var{p}) modulo m(x).
+\E
+
+@item 9
+\JP @b{$B0L?t(B @var{p^n} $B$NM-8BBN$N85(B ($B>.0L?t(B)}
+\EG @b{element of a finite field of characteristic @var{p^n} (small order)}
+
+\BJP
+$B0L?t$,(B @var{p^n} $B$NM-8BBN(B (@var{p^n} $B$,(B @var{2^29} $B0J2<(B, @var{p} $B$,(B @var{2^14} $B0J>e(B
+$B$J$i(B @var{n} $B$O(B 1) $B$O(B, 
+$BI8?t(B @var{p} $B$*$h$S3HBg<!?t(B @var{n} 
+$B$r(B @code{setmod_ff} $B$K$h$j;XDj$9$k$3$H$K$h$j@_Dj$9$k(B. 
+$B$3$NBN$N(B 0 $B$G$J$$85$O(B, @var{p} $B$,(B @var{2^14} $BL$K~$N>l9g(B, 
+GF(@var{p^n}) $B$N>hK!72$N@8@.85$r8GDj$9$k$3$H(B
+$B$K$h$j(B, $B$3$N85$N$Y$-$H$7$FI=$5$l$k(B. $B$3$l$K$h$j(B, $B$3$NBN$N(B 0 $B$G$J$$85(B
+$B$O(B, $B$3$N$Y$-;X?t$H$7$FI=8=$5$l$k(B. @var{p} $B$,(B @var{2^14} $B0J>e(B
+$B$N>l9g$ODL>o$N>jM>$K$h$kI=8=$H$J$k$,(B, $B6&DL$N%W%m%0%i%`$G(B
+$BAPJ}$N>l9g$r07$($k$h$&$K$3$N$h$&$J;EMM$H$J$C$F$$$k(B. 
+
+\E
+\BEG
+A finite field of order @var{p^n}, where @var{p^n} must be less than
+@var{2^29} and @var{n} must be equal to 1 if @var{p} is greater or
+equal to @var{2^14}, 
+is set by @code{setmod_ff}
+by specifying its characteristic @var{p} the extension degree
+@var{n}. If @var{p} is less than @var{2^14}, each non-zero element 
+of this field
+is a power of a fixed element, which is a generator of the multiplicative
+group of the field, and it is represented by its exponent.
+Otherwise, each element is represented by the redue modulo @var{p}.
+This specification is useful for treating both cases in a single 
+program.
+\E
+
+@item 10
+\JP @b{$B0L?t(B @var{p^n} $B$N>.0L?tM-8BBN$NBe?t3HBg$N85(B}
+\EG @b{element of a finite field which is an algebraic extension of a small finite field of characteristic @var{p^n}}
+
+\BJP
+$BA09`$N(B, $B0L?t$,(B @var{p^n} $B$N>.0L?tM-8BBN$N(B @var{m} $B<!3HBg$N85$G$"$k(B.
+$BI8?t(B @var{p} $B$*$h$S3HBg<!?t(B @var{n}, @var{m} 
+$B$r(B @code{setmod_ff} $B$K$h$j;XDj$9$k$3$H$K$h$j@_Dj$9$k(B. $B4pACBN>e$N(B @var{m}
+$B<!4{LsB?9`<0$,<+F0@8@.$5$l(B, $B$=$NBe?t3HBg$N@8@.85$NDj5AB?9`<0$H$7$FMQ$$$i$l$k(B.
+$B@8@.85$O(B @code{@@s} $B$G$"$k(B.
+
+\E
+\BEG
+An extension field @var{K} of the small finite field @var{F} of order @var{p^n}
+is set by @code{setmod_ff}
+by specifying its characteristic @var{p} the extension degree
+@var{n} and @var{m}=[@var{K}:@var{F}]. An irreducible polynomial of degree @var{m}
+over @var{K} is automatically generated and used as the defining polynomial of
+the generator of the extension @var{K/F}. The generator is denoted by @code{@@s}.
+\E
+
+@item 11
+\JP @b{$BJ,;6I=8=$NBe?tE*?t(B}
+\EG @b{algebraic number represented by a distributed polynomial}
+@*
+\JP @xref{$BBe?tE*?t$K4X$9$k1i;;(B}.
+\EG @xref{Algebraic numbers}.
+
+\BJP
+
+\E
+\BEG
+\E
 @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$$$k(B
-modulus $B$G(B, $BB0$9$kBN$,J,$+$j(B, $B$=$NCf$G1i;;$,9T$o$l$k(B. 
+$B>.I8?tM-8BAGBN0J30$NM-8BBN$O(B @code{setmod_ff} $B$G@_Dj$9$k(B. 
+$BM-8BBN$N85$I$&$7$N1i;;$G$O(B,
 $B0lJ}$,M-M}?t$N>l9g$K$O(B, $B$=$NM-M}?t$O<+F0E*$K8=:_@_Dj$5$l$F$$$k(B
 $BM-8BBN$N85$KJQ49$5$l(B, $B1i;;$,9T$o$l$k(B. 
 \E
 \BEG
-The characteristic of a large finite prime field and the defining
-polynomial of a finite field of characteristic 2 are set by @code{setmod_ff}.
+Finite fields other than small finite prime fields are
+set by @code{setmod_ff}.
 Elements of finite fields do not have informations about the modulus.
-Upon an arithmetic operation, the modulus set by @code{setmod_ff} is
-used. If one of the operands is a rational number, it is automatically
+Upon an arithmetic operation, i
+f one of the operands is a rational number, it is automatically
 converted into an element of the finite field currently set and
 the operation is done in the finite field.
 \E
@@ -659,7 +763,7 @@ and further are classified into sub-types of the type 
 @item 0
 \JP @b{$B0lHLITDj85(B}
 \EG @b{ordinary indeterminate}
-
+@*
 \JP $B1Q>.J8;z$G;O$^$kJ8;zNs(B. $BB?9`<0$NJQ?t$H$7$F:G$bIaDL$KMQ$$$i$l$k(B. 
 \BEG
 An object of this sub-type is denoted by a string that start with
@@ -677,7 +781,7 @@ polynomials.
 @item 1
 \JP @b{$BL$Dj78?t(B}
 \EG @b{undetermined coefficient}
-
+@*
 \BJP
 @code{uc()} $B$O(B, @samp{_} $B$G;O$^$kJ8;zNs$rL>A0$H$9$kITDj85$r@8@.$9$k(B. 
 $B$3$l$i$O(B, $B%f!<%6$,F~NO$G$-$J$$$H$$$&$@$1$G(B, $B0lHLITDj85$HJQ$o$i$J$$$,(B, 
@@ -704,7 +808,7 @@ _0
 @item 2
 \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, $BI>2A8e(B
@@ -736,15 +840,15 @@ sin(x)
 @item 3
 \JP @b{$BH!?t;R(B}
 \EG @b{functor}
-
+@*
 \BJP
-$BH!?t8F$S=P$7$O(B, @var{fname(args)} $B$H$$$&7A$G9T$J$o$l$k$,(B, @var{fname} $B$N(B
+$BH!?t8F$S=P$7$O(B, @var{fname}(@var{args}) $B$H$$$&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)}.
+A function call (or a function form) has a form @var{fname}(@var{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