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 %% $OpenXM$  %% $OpenXM: OpenXM/doc/OpenXM-specs/cmo-basic1.tex,v 1.15 2016/08/22 05:38:27 takayama Exp $
 //&jp \section{ 数, 多項式 の  CMO 表現 }  //&jp \section{ , 紊綣  CMO 茵 }
 //&eg \section{ CMOexpressions for numbers and polynomials }  //&eg \section{ CMOexpressions for numbers and polynomials }
   \label{sec:basic1}
 /*&C  /*&C
 @../SSkan/plugin/cmotag.h  @../SSkan/plugin/cmotag.h
 \begin{verbatim}  \begin{verbatim}
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 #define     CMO_RING_BY_NAME   26  #define     CMO_RING_BY_NAME   26
 #define     CMO_DISTRIBUTED_POLYNOMIAL 31  #define     CMO_DISTRIBUTED_POLYNOMIAL 31
 #define     CMO_RATIONAL       34  #define     CMO_RATIONAL       34
   #define     CMO_COMPLEX       35
   
   #define     CMO_BIGFLOAT32    52
   
 #define     CMO_INDETERMINATE  60  #define     CMO_INDETERMINATE  60
 #define     CMO_TREE           61  #define     CMO_TREE           61
Line 24 
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 */  */
   
 /*&jp  /*&jp
 以下, グループ CMObject/Basic1, CMObject/Tree  篁ヤ, 違若 CMObject/Basic, CMObject/Tree
 および CMObject/DistributedPolynomial   CMObject/DistributedPolynomial
 に属する CMObject の形式を説明する.  絮 CMObject 綵√茯.
 \noroa{ tagged list を導入すべきか?  cf. SSkan/plugin/cmo.txt }  
   
   \noindent
   {\tt OpenXM/src/ox\_toolkit} {\tt bconv} ＜
   CMO expression binary format 紊с,
   .
 */  */
 /*&eg  /*&eg
 In the sequel, we will explain on the groups  In the sequel, we will explain on the groups
 CMObject/Basic1, CMObject/Tree  CMObject/Basic, CMObject/Tree
 and CMObject/DistributedPolynomial.  and CMObject/DistributedPolynomial.
   
   \noindent
   The program {\tt bconv} at {\tt OpenXM/src/ox\_toolkit}
   translates
   CMO expressions into binary formats.
   It is convinient to understand the binary formats explained in
   this section.
 */  */
   
   /*&C
   \noindent Example:
   \begin{verbatim}
   bash$ ./bconv
   > (CMO_ZZ,123123);
   00 00 00 14 00 00 00 01 00 01 e0 f3
   \end{verbatim}
   */
 /*&jp  /*&jp
   
 \bigbreak  \bigbreak
 \noindent  \noindent
 Group CMObject/Basic1 requires CMObject/Basic0. \\  Group CMObject/Basic requires CMObject/Primitive. \\
 ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/Basic1. \\  ZZ, QQ, Zero, Rational, Indeterminate $\in$ CMObject/Basic. \\
 \begin{eqnarray*}  \begin{eqnarray*}
 \mbox{Zero} &:& ({\tt CMO\_ZERO}) \\  \mbox{Zero} &:& ({\tt CMO\_ZERO}) \\
 & & \mbox{ --- ユニバーサルな ゼロを表す. } \\  & & \mbox{ --- 若泣 若茵. } \\
 \mbox{ZZ}         &:& ({\tt CMO\_ZZ},{\sl int32}\, {\rm f}, {\sl byte}\, \mbox{a[1]}, \ldots  \mbox{ZZ}         &:& ({\tt CMO\_ZZ},{\sl int32}\, {\rm f}, {\sl byte}\, \mbox{a[1]}, \ldots ,
 {\sl byte}\, \mbox{a[m]} ) \\  {\sl byte}\, \mbox{a[$|$f$|$]} ) \\
 &:& \mbox{ --- bignum をあらわす. a[i] についてはあとで説明}\\  &:& \mbox{ --- bignum . a[i] ゃц}\\
 \mbox{QQ}        &:& ({\tt CMO\_QQ}, {\sl ZZ}\, {\rm a}, {\sl ZZ}\, {\rm b}) \\  \mbox{QQ}        &:& ({\tt CMO\_QQ},
 & & \mbox{ --- 有理数 $a/b$ を表す. } \\                        {\sl int32}\, {\rm m}, {\sl byte}\, \mbox{a[1]}, \ldots, {\sl byte}\, \mbox{a[$|$m$|$]},
                         {\sl int32}\, {\rm n}, {\sl byte}\, \mbox{b[1]}, \ldots, {\sl byte}\, \mbox{b[$|$n$|$]})\\
   & & \mbox{ --- $a/b$ 茵. } \\
 \mbox{Rational}        &:& ({\tt CMO\_RATIONAL}, {\sl CMObject}\, {\rm a}, {\sl CMObject}\, {\rm b}) \\  \mbox{Rational}        &:& ({\tt CMO\_RATIONAL}, {\sl CMObject}\, {\rm a}, {\sl CMObject}\, {\rm b}) \\
 & & \mbox{ ---  $a/b$ を表す. } \\  & & \mbox{ ---  $a/b$ 茵. } \\
   \mbox{Bigfloat32}         &:& ({\tt CMO\_BIGFLOAT32},
   {\sl int32}\, {\rm prec}, {\sl int32}\, {\rm sign}, {\sl int32}\, {\rm exp},
   {\sl int32}\, \mbox{a[1]}, \ldots , {\sl int32}\, \mbox{a[k]} ) \\
   &:& \mbox{ --- bigfloat . a[i], k ゃц}\\
   \mbox{Complex}        &:& ({\tt CMO\_COMPLEX}, {\sl CMObject}\, {\rm re}, {\sl CMObject}\, {\rm im}) \\
   & & \mbox{ ---  $a+b\sqrt{-1}$ 茵. } \\
 \mbox{Indeterminate}        &:& ({\tt CMO\_INDETERMINATE}, {\sl Cstring}\, {\rm v}) \\  \mbox{Indeterminate}        &:& ({\tt CMO\_INDETERMINATE}, {\sl Cstring}\, {\rm v}) \\
 & & \mbox{ --- 変数名 $v$ . } \\  & & \mbox{ --- 紊医 $v$ . } \\
 \end{eqnarray*}  \end{eqnarray*}
 */  */
   
   
 /*&eg  /*&eg
   
 \bigbreak  \bigbreak
 \noindent  \noindent
 Group CMObject/Basic1 requires CMObject/Basic0. \\  Group CMObject/Basic requires CMObject/Primitive. \\
 ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/Basic1. \\  ZZ, QQ, Zero, Rational, Indeterminate $\in$ CMObject/Basic. \\
 \begin{eqnarray*}  \begin{eqnarray*}
 \mbox{Zero} &:& ({\tt CMO\_ZERO}) \\  \mbox{Zero} &:& ({\tt CMO\_ZERO}) \\
 & & \mbox{ --- Universal zero } \\  & & \mbox{ --- Universal zero } \\
 \mbox{ZZ}         &:& ({\tt CMO\_ZZ},{\sl int32}\, {\rm f}, {\sl byte}\, \mbox{a[1]}, \ldots  \mbox{ZZ}         &:& ({\tt CMO\_ZZ},{\sl int32}\, {\rm f}, {\sl byte}\, \mbox{a[1]}, \ldots ,
 {\sl byte}\, \mbox{a[m]} ) \\  {\sl byte}\, \mbox{a[$|$m$|$]} ) \\
 &:& \mbox{ --- bignum. The meaning of a[i] will be explained later.}\\  &:& \mbox{ --- bignum. The meaning of a[i] will be explained later.}\\
 \mbox{QQ}        &:& ({\tt CMO\_QQ}, {\sl ZZ}\, {\rm a}, {\sl ZZ}\, {\rm b}) \\  \mbox{QQ}        &:& ({\tt CMO\_QQ},
                         {\sl int32}\, {\rm m}, {\sl byte}\, \mbox{a[1]}, \ldots, {\sl byte}\, \mbox{a[$|$m$|$]},
                         {\sl int32}\, {\rm n}, {\sl byte}\, \mbox{b[1]}, \ldots, {\sl byte}\, \mbox{b[$|$n$|$]})\\
 & & \mbox{ --- Rational number $a/b$. } \\  & & \mbox{ --- Rational number $a/b$. } \\
 \mbox{Rational}        &:& ({\tt CMO\_RATIONAL}, {\sl CMObject}\, {\rm a}, {\sl CMObject}\, {\rm b}) \\  \mbox{Rational}        &:& ({\tt CMO\_RATIONAL}, {\sl CMObject}\, {\rm a}, {\sl CMObject}\, {\rm b}) \\
 & & \mbox{ ---  Rational expression $a/b$. } \\  & & \mbox{ ---  Rational expression $a/b$. } \\
   \mbox{Bigfloat32}         &:& ({\tt CMO\_BIGFLOAT32},
   {\sl int32}\, {\rm prec}, {\sl int32}\, {\rm sign}, {\sl int32}\, {\rm exp},
   {\sl int32}\, \mbox{a[1]}, \ldots , {\sl int32}\, \mbox{a[k]} ) \\
   &:& \mbox{ --- bigfloat. The meaning of a[i], k will be explained later.}\\
   \mbox{Complex}        &:& ({\tt CMO\_COMPLEX}, {\sl CMObject}\, {\rm re}, {\sl CMObject}\, {\rm im}) \\
   & & \mbox{ ---  Complex number $a+b\sqrt{-1}$. } \\
 \mbox{Indeterminate}        &:& ({\tt CMO\_INDETERMINATE}, {\sl Cstring}\, {\rm v}) \\  \mbox{Indeterminate}        &:& ({\tt CMO\_INDETERMINATE}, {\sl Cstring}\, {\rm v}) \\
 & & \mbox{ --- Variable name $v$ . } \\  & & \mbox{ --- Variable name $v$ . } \\
 \end{eqnarray*}  \end{eqnarray*}
Line 79  ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/B
Line 117  ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/B
 /*&C  /*&C
   
 */  */
   /*&C
   
   */
   
 /*&jp  /*&jp
 Indeterminate は変数名をあらわす.  Indeterminate 紊医.
 v はバイト列であればなにを用いてもよいが,  v ゃс違,
 システム毎に変数名として用いられるバイト列は制限がある.  激鴻罸紊医ゃ狗.
 各システム xxx は任意の文字列を各システム固有の変数名へ1対1に変換できるように  激鴻 xxx 篁紙絖激鴻堺紊医1絲1紊с
 実装しないといけない.  絎茖.
 (これを  (
 {\tt Dx} は {\tt \#dx} と変換するなどの  {\tt Dx} {\tt \#dx} 紊
 escape sequence を用いて実現するのは, 無理があるようである.  escape sequence 絎憗, ∞с.
 テーブルを作成する必要があるであろう.)  若篏綽荀с.)
 */  */
 /*&eg  /*&eg
 Indeterminate is a name of a variable.  The name of a variable should be expressed by using Indeterminate.
 v may be any sequence of bytes, but each system has its own  v may be any sequence of bytes, but each system has its own
 restrictions on the names of variables.  restrictions on the names of variables.
 Indeterminates of CMO and internal variable names must be translated  Indeterminates of CMO and internal variable names must be translated
 in one to one correspondence.  in one-to-one correspondence.
 */  */
   
   
 /*&jp  /*&jp
   \subsection{Indeterminate Tree}
   
 \noindent  \noindent
 Group CMObject/Tree requires CMObject/Basic1. \\  Group CMObject/Tree requires CMObject/Basic. \\
 Tree, Lambda $\in$ CMObject/Basic1. \\  Tree, Lambda $\in$ CMObject/Tree. \\
 \begin{eqnarray*}  \begin{eqnarray*}
 \mbox{Tree}        &:& ({\tt CMO\_TREE}, {\sl Cstring}\, {\rm name},  \mbox{Tree}        &:& ({\tt CMO\_TREE}, {\sl Cstring}\, {\rm name},
  {\sl Cstring}\, {\rm cdname}, {\sl List}\, {\rm leaves}) \\   {\sl List}\, {\rm attributes}, {\sl List}\, {\rm leaves}) \\
 & & \mbox{ --- 名前 name の定数または関数. 関数の評価はおこなわない. } \\  & & \mbox{ --- name 絎違障∽. ∽違荅箴＜. } \\
 & & \mbox{ --- cdname は空文字列でなければ name の意味が説明されている }\\  & & \mbox{ --- attributes 腥冴鴻с name 絮с篆. }\\
 & & \mbox{ --- OpenMath CD (content dictionary) の名前. } \\  & & \mbox{ --- 絮с鴻, key ゃ≪с. }\\
 \mbox{Lambda}        &:& ({\tt CMO\_LAMBDA}, {\sl List}\, {\rm args},  \mbox{Lambda}        &:& ({\tt CMO\_LAMBDA}, {\sl List}\, {\rm args},
                           {\sl Tree} {\rm body}) \\                            {\sl Tree} {\rm body}) \\
 & & \mbox{ --- body を args を引数とする関数とする. } \\  & & \mbox{ --- body args 綣違∽違. } \\
 & & \mbox{ --- optional な引数が必要なときは, leaves の後へつづける.} \\  
 \end{eqnarray*}  \end{eqnarray*}
 */  */
 /*&eg  /*&eg
   \subsection{Indeterminate and Tree}
   
 \noindent  \noindent
 Group CMObject/Tree requires CMObject/Basic1. \\  Group CMObject/Tree requires CMObject/Basic. \\
 Tree, Lambda $\in$ CMObject/Basic1. \\  Tree, Lambda $\in$ CMObject/Tree. \\
 \begin{eqnarray*}  \begin{eqnarray*}
 \mbox{Tree}        &:& ({\tt CMO\_TREE}, {\sl Cstring}\, {\rm name},  \mbox{Tree}        &:& ({\tt CMO\_TREE}, {\sl Cstring}\, {\rm name},
  {\sl Cstring}\, {\rm cdname}, {\sl List}\, {\rm leaves}) \\   {\sl List}\, {\rm attributes}, {\sl List}\, {\rm leaves}) \\
 & & \mbox{ --- A function or a constant of name. Functions are not evaluated. } \\  & & \mbox{ --- ``name'' is the name of the node of the tree. } \\
 & & \mbox{ --- cdname may be a null. If it is not null, it is the name of}\\  & & \mbox{ --- Attributes may be a null list. If it is not null, it is a list of}\\
 & & \mbox{ --- the OpenMath CD (content dictionary). } \\  & & \mbox{ --- key and value pairs. } \\
 \mbox{Lambda}        &:& ({\tt CMO\_LAMBDA}, {\sl List}\, {\rm args},  \mbox{Lambda}        &:& ({\tt CMO\_LAMBDA}, {\sl List}\, {\rm args},
                           {\sl Tree} {\rm body}) \\                            {\sl Tree} {\rm body}) \\
 & & \mbox{ --- a function with the arguments body. } \\  & & \mbox{ --- a function with the arguments body. } \\
 & & \mbox{ --- optional arguments come after leaves.} \\  
 \end{eqnarray*}  \end{eqnarray*}
 */  */
   
 /*&C  
   
 */  
   
 /*&jp  /*&jp
 数式を処理するシステムでは, Tree 構造が一般にもちいられる.  医激鴻с, Tree 罕筝＜.
 たとえば, $\sin(x+e)$ は,  , $\sin(x+e)$ ,
 {\tt (sin, (plus, x, e))}  {\tt (sin, (plus, x, e))}
 なる Tree であらわすのが一般的である.   Tree с筝с.
 Tree の表現を スタックマシンのレベルでおこなうとすると,  Tree 茵憗 鴻帥激潟с,
 {\tt ox\_BEGIN\_BLOCK}, {\tt ox\_END\_BLOCK} で評価を抑制するのが  {\tt ox\_BEGIN\_BLOCK}, {\tt ox\_END\_BLOCK} ц箴＜吟
 一つの方法である (cf. Postscript の {\tt \{ }, {\tt \} }).  筝ゃ号с (cf. Postscript {\tt \{ }, {\tt \} }).
 たとえば上の方法では  遺号с
 {\tt x, e, plus, sin } を begin block, end block でかこめばよろしい.  {\tt x, e, plus, sin } begin block, end block с違.
 われわれはスタックマシンの実装をなるべく簡単にするという立場をとりたい,  鴻帥激潟絎茖鴻靎≦腴眼,
 また数学オブジェクトを OX スタックマシンと CMObject を混在して表現したく  障医吾с OX 鴻帥激潟 CMObject 羞桁茵憗
 ない.  .
 したがって,  ｃ,
 Tree 構造は Open Math 風の表現をもちいた CMO を導入することにした.  Tree 罕 Open Math 蘂茵憗＜ CMO 絨ャ.
 またこのほうが, われわれの想定するシステム xxx において, Open XM 対応が  障祉, 喝激鴻 xxx , Open XM 絲上
 はるかに容易である.  絎号с.
 なお, Tree は, Open Math では, Symbol, Application のメカニズムに相当する.  , Tree , Open Math с, Symbol, Application ＜冴後.
 */  */
 /*&eg  /*&eg
 In many computer algebra systems, mathematical expressions are usually  In many computer algebra systems, mathematical expressions are usually
Line 164  For example,
Line 202  For example,
 $\sin(x+e)$ is expressed as  $\sin(x+e)$ is expressed as
 {\tt (sin, (plus, x, e))}  {\tt (sin, (plus, x, e))}
 as a tree.  as a tree.
 We can   @@@  Tree may be expressed by putting the expression between
   {\tt SM\_beginBlock} and {\tt SM\_endBlock}, which are
   stack machine commands for delayed evaluation.
   (cf. {\tt \{ }, {\tt \} } in PostScript).
   However it makes the implementation of stack machines complicated.
   It is desirable that CMObject is independent of OX stack machine.
   Therefore we introduce an OpenMath like tree representation for CMO
   Tree object.
   This method allows us to implement tree structure easily
   on individual OpenXM systems.
   Note that CMO Tree corresponds to Symbol and Application in OpenMath.
 */  */
   
   
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 */  */
 /*&jp  /*&jp
 Lambda は関数を定義するための関数である.  Lambda ∽違絎臂∽違с.
 Lisp の Lambda 表現と同じ.  Lisp Lambda 茵憗.
   */
   /*&eg
   Lambda is used to define functions.
   The notion ``lambda'' is borrowed from the language Lisp.
   */
   
 \noindent  \noindent
 例: $sin(x+e)$ の表現.  //&jp 箴: $sin(x+e)$ 茵.
   //&eg Example: the expression of $sin(x+e)$.
 \begin{verbatim}  \begin{verbatim}
 (CMO_TREE, (CMO_STRING, "sin"), (CMO_STRING, "basic"),  (CMO_TREE, (CMO_STRING, "sin"),
       (CMO_LIST,[size=]1,(CMO_LIST,[size=]2,(CMO_STRING, "cdname"),
                                             (CMO_STRING,"basic")))
     (CMO_LIST,[size=]1,      (CMO_LIST,[size=]1,
         (CMO_TREE, (CMO_STRING, "plus"), (CMO_STRING, "basic"),          (CMO_TREE, (CMO_STRING, "plus"), (CMO_STRING, "basic"),
             (CMO_LIST,[size=]2, (CMO_INDETERMINATE,"x"),              (CMO_LIST,[size=]2, (CMO_INDETERMINATE,"x"),
                                 (CMO_TREE,(CMO_STRING, "e"),  自然対数の底  //&jp                  (CMO_TREE,(CMO_STRING, "e"),  九丈違綺
                                           (CMO_STRING, "basic"))  //&eg                  (CMO_TREE,(CMO_STRING, "e"),  the base of natural logarithms
       (CMO_LIST,[size=]1,(CMO_LIST,[size=]2,(CMO_STRING, "cdname"),
                                             (CMO_STRING,"basic")))
         ))          ))
     )      )
 )  )
 \end{verbatim}  \end{verbatim}
   //&jp  Leave , 紊綣篁紙吾с.
   //&eg  Elements of the leave may be any objects including polynomials.
   
 \noindent  \noindent
 Example:  Example:
 \begin{verbatim}  \begin{verbatim}
 sm1> [(plus) (Basic) [(123).. (345)..]] [(class) (tree)] dc ::  sm1> [(plus) [[(cdname) (basic)]] [(123).. (345)..]] [(class) (tree)] dc ::
 Class.tree [    $plus$ , $Basic$ , [    123 , 345 ]  ]  Class.tree [$plus$ , [[$cdname$ , $basic$ ]], [ 123 , 345 ]  ]
 \end{verbatim}  \end{verbatim}
   
   \noindent
   Example:
   \begin{verbatim}
   asir
   [753] taka_cmo100_xml_form(quote(sin(x+1)));
   <cmo_tree>  <cmo_string>"sin"</cmo_string>
    <cmo_list><cmo_int32 for="length">1</cmo_int32>
      <cmo_list><cmo_int32 for="length">2</cmo_int32>
        <cmo_string>"cdname"</cmo_string>
        <cmo_string>"basic"</cmo_string>
      </cmo_list> </cmo_list>
   <cmo_tree>    <cmo_string>"plus"</cmo_string>
     <cmo_list><cmo_int32 for="length">1</cmo_int32>
       <cmo_list><cmo_int32 for="length">2</cmo_int32>
         <cmo_string>"cdname"</cmo_string>
         <cmo_string>"basic"</cmo_string>
       </cmo_list> </cmo_list>
    <cmo_indeterminate> <cmo_string>"x"</cmo_string>  </cmo_indeterminate>
    <cmo_zz>1</cmo_zz>
   </cmo_tree></cmo_tree>
   \end{verbatim}
   
   
 \bigbreak  \bigbreak
 次に, 分散表現多項式に関係するグループを定義しよう.  //&jp 罨＜, ｈ；憜綣≫違若絎臂.
   /*&eg
   Let us define a group for distributed polynomials. In the following,
   DMS stands for Distributed Monomial System.
   */
   
 \medbreak  \medbreak
 \noindent  \noindent
 Group CMObject/DistributedPolynomials requires CMObject/Basic0,  Group CMObject/DistributedPolynomials requires CMObject/Primitive,
 CMObject/Basic1. \\  CMObject/Basic. \\
 Monomial, Monomial32, Coefficient, Dpolynomial, DringDefinition,  Monomial, Monomial32, Coefficient, Dpolynomial, DringDefinition,
 Generic DMS ring, RingByName, DMS of N variables $\in$  Generic DMS ring, RingByName, DMS of N variables $\in$
 CMObject/DistributedPolynomials. \\  CMObject/DistributedPolynomials. \\
   /*&jp
 \begin{eqnarray*}  \begin{eqnarray*}
 \mbox{Monomial} &:& \mbox{Monomial32}\, |\, \mbox{Zero} \\  \mbox{Monomial} &:& \mbox{Monomial32}\, |\, \mbox{Zero} \\
 \mbox{Monomial32}&:& ({\tt CMO\_MONOMIAL32}, {\sl int32}\, n,  \mbox{Monomial32}&:& ({\tt CMO\_MONOMIAL32}, {\sl int32}\, n,
 {\sl int32}\, \mbox{e[1]}, \ldots,  {\sl int32}\, \mbox{e[1]}, \ldots,
 {\sl int32}\, \mbox{e[n]}, \\  {\sl int32}\, \mbox{e[n]}, \\
 & & \ \mbox{Coefficient}) \\  & & \ \mbox{Coefficient}) \\
 & & \mbox{ --- e[i] で, $n$ 変数 monomial  & & \mbox{ --- e[i] , $n$ 紊 monomial
 $x^e = x_1^{e_1} \cdots x_n^{e_n}$ の各指数 $e_i$  $x^e = x_1^{e_1} \cdots x_n^{e_n}$ $e_i$
 をあらわす.} \\  .} \\
 \mbox{Coefficient}&:& \mbox{ZZ} | \mbox{Integer32} \\  \mbox{Coefficient}&:& \mbox{ZZ} | \mbox{Integer32} \\
 \mbox{Dpolynomial}&:& \mbox{Zero} \\  \mbox{Dpolynomial}&:& \mbox{Zero} \\
 & & |\ ({\tt CMO\_DISTRIBUTED\_POLYNOMIAL},{\sl int32} m, \\  & & |\ ({\tt CMO\_DISTRIBUTED\_POLYNOMIAL},{\sl int32}\, m, \\
 & & \ \ \mbox{DringDefinition},  & & \ \ \mbox{DringDefinition},
 [\mbox{Monomial32}|\mbox{Zero}], \\  [\mbox{Monomial32}|\mbox{Zero}], \\
 & &\ \  & &\ \
 \{\mbox{Monomial32}\}) \\  \{\mbox{Monomial32}\}) \\
 & &\mbox{--- m はモノミアルの個数である.}\\  & &\mbox{--- m ≪≪違с.}\\
 \mbox{DringDefinition}  \mbox{DringDefinition}
 &:& \mbox{DMS of N variables} \\  &:& \mbox{DMS of N variables} \\
 & & |\ \mbox{RingByName} \\  & & |\ \mbox{RingByName} \\
 & & |\ \mbox{Generic DMS ring} \\  & & |\ \mbox{Generic DMS ring} \\
 & & \mbox{ --- 分散表現多項式環の定義. } \\  & & \mbox{ --- ｈ；憜綣違絎臂. } \\
 \mbox{Generic DMS ring}  \mbox{Generic DMS ring}
 &:& ({\tt CMO\_DMS\_GENERIC})\  新版はこちら\\  &:& \mbox{({\tt CMO\_DMS\_GENERIC}) --- 亥＜}\\
 \mbox{RingByName}&:& ({\tt CMO\_RING\_BY\_NAME}, {\sl Cstring}\  {\rm s}) \\  \mbox{RingByName}&:& ({\tt CMO\_RING\_BY\_NAME}, {\sl Cstring}\  {\rm s}) \\
 & & \mbox{ --- 名前 s で, 格納された ring 定義.} \\  & & \mbox{ --- s , 主 ring 絎臂.} \\
 \mbox{DMS of N variables}  \mbox{DMS of N variables}
 &:& ({\tt CMO\_DMS\_OF\_N\_VARIABLES}, \\  &:& ({\tt CMO\_DMS\_OF\_N\_VARIABLES}, \\
 & & \ ({\tt CMO\_LIST}, {\sl int32}\, \mbox{m},  & & \ ({\tt CMO\_LIST}, {\sl int32}\, \mbox{m},
Line 241  $x^e = x_1^{e_1} \cdots x_n^{e_n}$ の各指数 $e_i$
Line 327  $x^e = x_1^{e_1} \cdots x_n^{e_n}$ の各指数 $e_i$
 & & \ \ [,{\sl object}\,\mbox{s}, {\sl Cstring}\,\mbox{c},  & & \ \ [,{\sl object}\,\mbox{s}, {\sl Cstring}\,\mbox{c},
           {\sl List}\, \mbox{vlist},            {\sl List}\, \mbox{vlist},
 {\sl List}\, \mbox{wvec}, {\sl List}\, \mbox{outord}]) \\  {\sl List}\, \mbox{wvec}, {\sl List}\, \mbox{outord}]) \\
 & & \mbox{ --- m はあとに続く要素の数} \\  & & \mbox{ --- m 膓頳膣} \\
 & & \mbox{ --- n は変数の数, p は 標数} \\  & & \mbox{ --- n 紊違, p 罔} \\
 & & \mbox{ --- s は ring の名前} \\  & & \mbox{ --- s ring } \\
 & & \mbox{ --- c は係数環, QQ, ZZ の場合は文字列で QQ, ZZ と書く.} \\  & & \mbox{ --- c 篆亥, QQ, ZZ 翫絖 QQ, ZZ 吾.} \\
 & & \mbox{ --- vlist は Indeterminate のリスト(新版). 多項式環の変数リスト} \\  & & \mbox{ --- vlist Indeterminate 鴻(亥). 紊綣違紊違鴻} \\
 & & \mbox{ --- wvec は order をきめる weight vector,} \\  & & \mbox{ --- wvec order weight vector,} \\
 & & \mbox{ --- outord は出力するときの変数順序.} \\  & & \mbox{ --- outord 阪紊育綺.} \\
 \end{eqnarray*}  \end{eqnarray*}
   */
   /*&eg
   \begin{eqnarray*}
   \mbox{Monomial} &:& \mbox{Monomial32}\, |\, \mbox{Zero} \\
   \mbox{Monomial32}&:& ({\tt CMO\_MONOMIAL32}, {\sl int32}\, n,
                         {\sl int32}\, \mbox{e[1]}, \ldots,
                         {\sl int32}\, \mbox{e[n]}, \\
                    & & \ \mbox{Coefficient}) \\
                    & & \mbox{ --- e[i] is the exponent $e_i$ of the monomial
                         $x^e = x_1^{e_1} \cdots x_n^{e_n}$. } \\
   \mbox{Coefficient}&:& \mbox{ZZ} | \mbox{Integer32} \\
   \mbox{Dpolynomial}&:& \mbox{Zero} \\
                    & & |\ ({\tt CMO\_DISTRIBUTED\_POLYNOMIAL},{\sl int32}\, m, \\
                    & & \ \ \mbox{DringDefinition}, [\mbox{Monomial32}|\mbox{Zero}], \\
                    & &\ \
                       \{\mbox{Monomial32}\})  \\
                    & &\mbox{--- m is equal to the number of monomials.}\\
   \mbox{DringDefinition}
                    &:& \mbox{DMS of N variables} \\
                    & & |\ \mbox{RingByName} \\
                    & & |\ \mbox{Generic DMS ring} \\
                    & & \mbox{ --- definition of the ring of distributed polynomials. } \\
   \mbox{Generic DMS ring}
                    &:& ({\tt CMO\_DMS\_GENERIC}) \\
   \mbox{RingByName}&:& ({\tt CMO\_RING\_BY\_NAME}, {\sl Cstring} s) \\
                    & & \mbox{ --- The ring definition referred by the name ``s''.} \\
   \mbox{DMS of N variables}
                    &:& ({\tt CMO\_DMS\_OF\_N\_VARIABLES}, \\
                    & & \ ({\tt CMO\_LIST}, {\sl int32}\, \mbox{m},
                     {\sl Integer32}\,  \mbox{n}, {\sl Integer32}\, \mbox{p} \\
                    & & \ \ [,{\sl Cstring}\,\mbox{s}, {\sl List}\, \mbox{vlist},
                             {\sl List}\, \mbox{wvec}, {\sl List}\, \mbox{outord}]) \\
                    & & \mbox{ --- m is the number of elements.} \\
                    & & \mbox{ --- n is the number of variables, p is the characteristic} \\
                    & & \mbox{ --- s is the name of the ring, vlist is the list of variables.} \\
                    & & \mbox{ --- wvec is the weight vector.} \\
                    & & \mbox{ --- outord is the order of variables to output.} \\
   \end{eqnarray*}
   */
   
 RingByName や DMS of N variables はなくても, DMS を定義できる.  /*&jp
 したがって, これらを実装してないシステムで DMS を扱うものが  RingByName DMS of N variables , DMS 絎臂с.
 あってもかまわない.  ｃ, 絎茖激鴻 DMS 宴
   ｃ障.
   
 以下, 以上の CMObject  にたいする,  篁ヤ, 篁ヤ CMObject  ,
 xxx = asir, kan の振舞いを記述する.  xxx = asir, kan 荐菴違.
   */
   /*&eg
   Note that it is possible to define DMS without RingByName and
   DMS of N variables.
   
   In the following we describe how the above CMObjects
   are implemented on Asir and Kan.
   */
   
 \subsection{ Zero}  \subsection{ Zero}
 CMO では ゼロの表現法がなんとうりもあるが,  /*&jp
 どのようなゼロをうけとっても,  CMO с 若茵憝羈.
 システムのゼロに変換できるべきである.  %% 若ｃ,
 (たとえば, asir は 0 はただ一つ.)  %% 激鴻若紊с鴻с.
   */
   /*&eg
   Note that CMO has various representations of zero.
   */
   
   
   //&jp \subsection{ 贋 ZZ }
   //&eg \subsection{ Integer ZZ }
   
 \subsection{ 整数 ZZ }  
   
 \begin{verbatim}  \begin{verbatim}
 #define     CMO_ZZ          20  #define     CMO_ZZ          20
 \end{verbatim}  \end{verbatim}
   
   /*&jp
   膀сOpen xxx 荀靘篁紙紊с贋(bignum)宴ゃ
   茯.  Open XM 荀靘紊膕上墾贋違茵若水 CMO\_ZZ
   GNU MPゃ荐荐, 膃垬腟九上よ；憗
   .  (cf. {\tt kan/sm1} 絽ｃ {\tt
   plugin/cmo-gmp.c}) CMO\_ZZ 罨＜綵√.
   */
   /*&eg
   We describe the bignum (multi-precision integer) representation
   {\tt CMO\_ZZ} in OpenXM.
   The format is similar
   to that in GNU MP. (cf. {\tt plugin/cmo-gmp.c} in the {\tt kan/sm1}
   distribution). CMO\_ZZ is defined as follows.
   */
   
 この節ではOpen xxx 規約における任意の大きさの整数(bignum)の扱いについて  
 説明する.  
 Open XM 規約における多重精度整数を表すデータ型 CMO\_ZZ は GNU MPライブ  
 ラリなどを参考にして設計されていて, 符号付き絶対値表現を用いている.  
 (cf. {\tt kan/sm1} の配布ディレクトリのなかの {\tt plugin/cmo-gmp.c})  
 CMO\_ZZ は次の形式をとる.\\  
 \begin{tabular}{|c|c|c|c|c|}  \begin{tabular}{|c|c|c|c|c|}
 \hline  \hline
 {\tt int32 CMO\_ZZ} & {\tt int32 $f$} & {\tt int32 $b_0$} & $\cdots$ &  {\tt int32 CMO\_ZZ} & {\tt int32 $f$} & {\tt int32 $b_0$} & $\cdots$ &
 {\tt int32 $b_{n}$} \\  {\tt int32 $b_{n}$} \\
 \hline  \hline
 \end{tabular} \\  \end{tabular}
 $f$ は32bit整数である.  
 $b_0, \ldots, b_n$ は unsigned int32 である.  
 $|f|$ は $n+1$ である.  
 この CMO の符号は $f$ の符号で定める.  
 前述したように, 32bit整数の負数は 2 の補数表現で表される.  
   
 Open xxx 規約では上の CMO は以下の整数を意味する.  /*&jp
   $f$ 32bit贋違с.  $b_0, \ldots, b_n$ unsigned int32 с.
   $|f|$ $n+1$ с.   CMO 膃垩 $f$ 膃垩у.  菴違
   , 32bit贋違莢違 2 茖域；憗ц；.
   
   Open xxx 荀靘с筝 CMO 篁ヤ贋違潟. ($R = 2^{32}$)
   */
   /*&eg
   $f$ is a 32bit integer. $b_0, \ldots, b_n$ are unsigned 32bit integers.
   $|f|$ is equal to $n+1$.
   The sign of $f$ represents that of the above integer to be expressed.
   As stated in Section
   \ref{sec:basic0}, a negative 32bit integer is represented by
   two's complement.
   
   In OpenXM the above CMO represents the following integer. ($R = 2^{32}$.)
   */
   
 \[  \[
 \mbox{sgn}(f)\times (b_0 R^{0}+ b_1 R^{1} + \cdots + b_{n-1}R^{n-1} + b_n R^n).  \mbox{sgn}(f)\times (b_0 R^{0}+ b_1 R^{1} + \cdots + b_{n-1}R^{n-1} + b_n R^n).
 \]  \]
 ここで $R = 2^{32}$ である.  
 {\tt int32} を network byte order で表現しているとすると,  /*&jp
 例えば, 整数 $14$ は CMO\_ZZ で表わすと,  \noindent  箴:
   {\tt int32} network byte order ц；
   ,箴, 贋 $14$ CMO\_ZZ ц；,
   */
   /*&eg
   \noindent Example:
   If we express {\tt int32} by the network byte order,
   a CMO\_ZZ $14$ is expressed by
   */
 \[  \[
 \mbox{(CMO\_ZZ, 1, 0, 0, 0, e)},  \mbox{(CMO\_ZZ, 1, 0, 0, 0, e)},
 \]  \]
 と表わす.  //&jp 茵. ゃс
 これはバイト列では  //&eg The corresponding byte sequence is
 \[  \[
 \mbox{\tt 00 00 00 14 00 00 00 01 00 00 00 0e}  \mbox{\tt 00 00 00 14 00 00 00 01 00 00 00 0e}
 \]  \]
 となる.  //&jp .
   
   
 なお ZZ の 0  ( (ZZ) 0 と書く ) は,  //&jp ZZ 0 ( (ZZ) 0 吾 ) , {\tt (CMO\_ZZ, 00,00,00,00)} 茵憗.
 {\tt (CMO\_ZZ, 00,00,00,00)}  //&eg Note that CMO\_ZZ 0 is expressed by {\tt (CMO\_ZZ, 00,00,00,00)}.
 と表現する.  
   
   
 \subsection{ 分散表現多項式 Dpolynomial }  //&jp \subsection{ ｈ；憜綣 Dpolynomial }
   //&eg \subsection{ Distributed polynomial Dpolynomial }
   
 環とそれに属する多項式は次のような考えかたであつかう.  /*&jp
   違絮紊綣罨＜сゃ.
   
 Generic DMS ring に属する元は,  Generic DMS ring 絮,
 変数を $n$ 個持つ 適当な係数集合 $K$ を持つ多項式環 $K[x_1, \ldots, x_n]$  紊違 $n$ 綵篆育 $K$ ゅ綣 $K[x_1, \ldots, x_n]$
 の元である.  с.
 係数集合 $K$ がなにかは, 実際データを読み込み, Coefficient を見た段階で  篆育 $K$ , 絎若帥茯粋昭, Coefficient 荀罧級
 わかる.  .
 この環に属する多項式を CMO 形式でうけとった場合, 各サーバはその  違絮紊綣 CMO 綵√сｃ翫, 泣若
 サーバの対応する Object  に変換しないといけない.  泣若絲上 Object  紊.
 この変換の仕方は, 各サーバ毎にきめる.  紊篁鴻, 泣若罸.
   
 Asir の場合は, $K[x_1, \ldots, x_n]$ の元は分散表現多項式に変換される.  Asir 翫, $K[x_1, \ldots, x_n]$ ｈ；憜綣紊.
 \noroa{ でも, order はどうなるの? }  \noroa{ с, order ? }
   
 {\tt kan/sm1} の場合は事情は複雑である.  {\tt kan/sm1} 翫篋茲с.
 {\tt kan/sm1} は, Generic DMS ring にあたる クラスをもたない.  {\tt kan/sm1} , Generic DMS ring 鴻.
 つまり, Default で存在する, $n$ 変数の分散表現多項式環は存在しないわけである.  ゃ障, Default у, $n$ 紊違ｈ；憜綣違絖с.
 したがって, {\tt kan/sm1} では, DMS of N variables が来た場合,  ｃ, {\tt kan/sm1} с, DMS of N variables ャ翫,
 これを CurrentRing の元として読み込む.  CurrentRing の変数の数が $n'$   CurrentRing 茯粋昭.  CurrentRing 紊違違 $n'$
 で, $n' < n$ だと新しい多項式環を生成してデータを読み込む.  , $n' < n$ 違紊綣違若帥茯粋昭.
 Order その他の optional 情報はすべて無視する.  Order 篁 optional 宴鴻∴.
   
 DMS の 2 番目のフィールドで,  DMS 2 ｃ若,
 Ring by Name を用いた場合, 現在の名前空間で変数 yyy に格納された ring object  Ring by Name 翫, 憜腥咲у yyy 主 ring object
 の元として, この多項式を変換しなさいという意味になる.  , 紊綣紊潟.
 {\tt kan/sm1} の場合, 環の定義は ring object として格納されており,  {\tt kan/sm1} 翫, 違絎臂 ring object 主,
 この ring object を 変数 yyy で参照することにより CMO としてうけとった   ring object 紊 yyy ус CMO ｃ
 多項式をこの ring の元として格納できる.  紊綣 ring 主с.
   */
   
   /*&eg
   We treat polynomial rings and their elements as follows.
   
   Generic DMS ring is an $n$-variate polynomial ring $K[x_1, \ldots, x_n]$,
   where $K$ is a coefficient set. $K$ is unknown in advance
   and it is determined when coefficients of an element are received.
   When a server has received an element in Generic DMS ring,
   the server has to translate it into the corresponding local object
   on the server. Each server has its own translation scheme.
   In Asir such an element are translated into a distributed polynomial.
   In {\tt kan/sm1} things are complicated.
   {\tt kan/sm1} does not have any class corresponding to Generic DMS ring.
   {\tt kan/sm1} translates a DMS of N variables into an element of
   the CurrentRing.
   If the CurrentRing is $n'$-variate and $n' < n$, then
   an $n$-variate polynomial ring is newly created.
   
   
   If RingByName ({\tt CMO\_RING\_BY\_NAME}, yyy)
   is specified as the second field of DMS,
   it requests a sever to use a ring object whose name is yyy
   as the destination ring for the translation.
   */
   
 \medbreak \noindent  \medbreak \noindent
 {\bf Example}:  //&jp {\bf Example}: (鴻違茵荐 16 画；荐)
 (すべての数の表記は 16 進表記)  //&eg {\bf Example}: (all numbers are represented in hexadecimal notation)
 {\footnotesize \begin{verbatim}  {\footnotesize \begin{verbatim}
 Z/11Z [6 variables]  Z/11Z [6 variables]
 (kxx/cmotest.sm1) run  (kxx/cmotest.sm1) run
Line 369  ff omc ::
Line 557  ff omc ::
  (CMO_DISTRIBUTED_POLYNOMIAL[1f],[size=]1,(CMO_DMS_GENERIC[18],),   (CMO_DISTRIBUTED_POLYNOMIAL[1f],[size=]1,(CMO_DMS_GENERIC[18],),
   (CMO_MONOMIAL32[13],3*x^2*y),),    (CMO_MONOMIAL32[13],3*x^2*y),),
 \end{verbatim} }  \end{verbatim} }
 length は, monomial の数$+2$ である.  /*&jp
 $ 3 x^2 y$ は 6 変数の多項式環の元としてみなされている.  $ 3 x^2 y$ 6 紊違紊綣違 帥.
 %%Prog: (3x^2 y). cmosave   ===> debug/cmodata1.cmo  */
 %%\\ 反省: 分散多項式の定義で,  /*&eg
 %%{\tt CMO\_LIST} でなく, {\tt CMO\_DMS} がはじめにくるべきだったのでは?  $3 x^2 y$ is regarded as an element of a six-variate polynomial ring.
 %%あたらしい 分散多項式の定義は次のようにすべき:  */
 %% 修正済み. 1999, 9/13  
   
   
   //&jp \subsection{絽域；憜綣絎臂}
   //&eg \subsection{Recursive polynomials}
   
 \subsection{再帰表現多項式の定義}  
   
 \begin{verbatim}  \begin{verbatim}
 #define CMO_RECURSIVE_POLYNOMIAL        27  #define CMO_RECURSIVE_POLYNOMIAL        27
 #define CMO_POLYNOMIAL_IN_ONE_VARIABLE  33  #define CMO_POLYNOMIAL_IN_ONE_VARIABLE  33
 \end{verbatim}  \end{verbatim}
   
 Group CMObject/RecursivePolynomial requires CMObject/Basic0, CMObject/Basic1.\\  Group CMObject/RecursivePolynomial requires CMObject/Primitive, CMObject/Basic.\\
 Polynomial in 1 variable, Coefficient, Name of the main variable,  Polynomial in 1 variable, Coefficient, Name of the main variable,
 Recursive Polynomial, Ring definition for recursive polynomials  Recursive Polynomial, Ring definition for recursive polynomials
 $\in$ CMObject/RecursivePolynomial \\  $\in$ CMObject/RecursivePolynomial \\
   
   /*&jp
 \begin{eqnarray*}  \begin{eqnarray*}
 \mbox{Polynomial in 1 variable} &:&  \mbox{Polynomial in 1 variable} &:&
 \mbox{({\tt CMO\_POLYNOMIAL\_IN\_ONE\_VARIABLE},\, {\sl int32}\, m, } \\  \mbox{({\tt CMO\_POLYNOMIAL\_IN\_ONE\_VARIABLE},\, {\sl int32}\, m, } \\
 & & \quad \mbox{ Name of the main variable }, \\  & & \quad \mbox{ Name of the main variable }, \\
 & & \quad \mbox{ \{ {\sl int32} e, Coefficient \}} \\  & & \quad \mbox{ \{ {\sl int32} e, Coefficient \}} ) \\
 & & \mbox{ --- m はモノミアルの個数. } \\  & & \mbox{ --- m ≪≪. } \\
 & & \mbox{ --- e, Coefficieint はモノミアルを表現している. } \\  & & \mbox{ --- e, Coefficieint ≪≪茵憗. } \\
 & & \mbox{ --- 順序の高い順にならべる. 普通は巾の高い順.} \\  & & \mbox{ --- 綺蕭鴻. 綏障蕭.} \\
 & & \mbox{ ---  e は 1変数多項式の巾をあらわす. } \\  & & \mbox{ ---  e 1紊医綣綏障. } \\
 \mbox{Coefficient} &:& \mbox{ ZZ} \,|\, \mbox{ QQ } \,|\,  \mbox{Coefficient} &:& \mbox{ ZZ} \,|\, \mbox{ QQ } \,|\,
 \mbox{ integer32  } \,|\,  \mbox{ integer32  } \,|\,
 \mbox{ Polynomial in 1 variable } \\  \mbox{ Polynomial in 1 variable } \\
 & & \quad \,|\, \mbox{Tree} \,|\, \mbox{Zero} \,|\,\mbox{Dpolynomial}\\  & & \quad \,|\, \mbox{Tree} \,|\, \mbox{Zero} \,|\,\mbox{Dpolynomial}\\
 \mbox{Name of the main variable } &:&  \mbox{Name of the main variable } &:&
 \mbox{ {\sl int32} v }   \\  \mbox{ {\sl int32} v }   \\
 & & \mbox{ --- v は 変数番号 (0 からはじまる) を表す. } \\  & & \mbox{ --- v 紊亥 (0 障) 茵. } \\
 \mbox{Recursive Polynomial} &:&  \mbox{Recursive Polynomial} &:&
 \mbox{ ( {\tt CMO\_RECURSIVE\_POLYNOMIAL}, } \\  \mbox{ ( {\tt CMO\_RECURSIVE\_POLYNOMIAL}, } \\
 & & \quad \mbox{ Ring definition for  & & \quad \mbox{ RringDefinition, } \\
 recursive polynomials, } \\  
 & & \quad  & & \quad
 \mbox{ Polynomial in 1 variable}\, | \, \mbox{Coefficient}   \\  \mbox{ Polynomial in 1 variable}\, | \, \mbox{Coefficient} )  \\
 \mbox{Ring definition for recursive polynomials }  \mbox{RringDefinition}
 & : &  \mbox{ {\sl List} v } \\  & : &  \mbox{ {\sl List} v } \\
 & & \quad \mbox{ --- v は, 変数名(indeterminate) のリスト. } \\  & & \quad \mbox{ --- v , 紊医(indeterminate) 障 Tree 鴻. } \\
 & & \quad \mbox{ --- 順序の高い順. } \\  & & \quad \mbox{ --- 綺蕭. } \\
 \end{eqnarray*}  \end{eqnarray*}
   */
   /*&eg
   \begin{eqnarray*}
   \mbox{Polynomial in 1 variable} &:&
   \mbox{({\tt CMO\_POLYNOMIAL\_IN\_ONE\_VARIABLE},\, {\sl int32}\, m, } \\
   & & \quad \mbox{ Name of the main variable }, \\
   & & \quad \mbox{ \{ {\sl int32} e, Coefficient \}} ) \\
   & & \mbox{ --- m is the number of monomials. } \\
   & & \mbox{ --- A pair of e and Coefficient represents a monomial. } \\
   & & \mbox{ --- The pairs of e and Coefficient are sorted in the } \\
   & & \mbox{ \quad decreasing order, usually with respect to e.} \\
   & & \mbox{ ---  e denotes an exponent of a monomial with respect to } \\
   & & \mbox{ \quad the main variable. } \\
   \mbox{Coefficient} &:& \mbox{ ZZ} \,|\, \mbox{ QQ } \,|\,
   \mbox{ integer32  } \,|\,
   \mbox{ Polynomial in 1 variable } \\
   & & \quad \,|\, \mbox{Tree} \,|\, \mbox{Zero} \,|\,\mbox{Dpolynomial}\\
   \mbox{Name of the main variable } &:&
   \mbox{ {\sl int32} v }   \\
   & & \mbox{ --- v denotes a variable number. } \\
   \mbox{Recursive Polynomial} &:&
   \mbox{ ( {\tt CMO\_RECURSIVE\_POLYNOMIAL}, } \\
   & & \quad \mbox{ RringDefinition, } \\
   & & \quad
   \mbox{ Polynomial in 1 variable}\, | \, \mbox{Coefficient} )  \\
   \mbox{RringDefinition}
   & : &  \mbox{ {\sl List} v } \\
   & & \quad \mbox{ --- v is a list of names of indeterminates or trees. } \\
   & & \quad \mbox{ --- It is sorted in the decreasing order. } \\
   \end{eqnarray*}
   */
 \bigbreak  \bigbreak
 \noindent  \noindent
 実例:  Example:
 \begin{verbatim}  \begin{verbatim}
 (CMO_RECURSIEVE_POLYNOMIAL, ("x","y"),  (CMO_RECURSIEVE_POLYNOMIAL, ("x","y"),
 (CMO_POLYNOMIAL_IN_ONE_VARIABLE, 2,      0,  <--- "x"  (CMO_POLYNOMIAL_IN_ONE_VARIABLE, 2,      0,  <--- "x"
Line 432  recursive polynomials, } \\
Line 649  recursive polynomials, } \\
        10, 1,         10, 1,
        5, 31)))         5, 31)))
 \end{verbatim}  \end{verbatim}
 これは,  //&jp ,
 $$   x^3 (1234 y^5 + 17 ) +  x^1 (y^10 + 31 y^5)  $$  //&eg This represents
 をあらわす.  $$   x^3 (1234 y^5 + 17 ) +  x^1 (y^{10} + 31 y^5)  $$
 非可換多項式もこの形式であらわしたいので, 積の順序を上のように  /*&jp
 すること. つまり, 主変数かける係数の順番.  .
   %%紊綣綵√с, 腥綺筝
   %%. ゃ障, 筝糸違篆違.
   */
   /*&eg
   %%We intend to represent non-commutative polynomials with the
   %%same form. In such a case, the order of products are defined
   %%as above, that is a power of the main variable $\times$ a coeffcient.
   
   */
   
 \noindent  \noindent
 \begin{verbatim}  \begin{verbatim}
 sm1  sm1
Line 446  sm1>ff ::
Line 672  sm1>ff ::
 Class.recursivePolynomial h * ((-1)) + (x^2  * (1))  Class.recursivePolynomial h * ((-1)) + (x^2  * (1))
 \end{verbatim}  \end{verbatim}
   
   //&jp \subsection{CPU箴絖 double }
   //&eg \subsection{CPU dependent double}
   
   
 int32 と Integer32 の違い.  
 次にくるデータがかならず int32 とわかっておれば,  
 int32 を用いる.  
 次のデータ型がわからないとき Integer32 を用いる.  
   
   
 \subsection{CPU依存の double }  
   
 \begin{verbatim}  \begin{verbatim}
 #define CMO_64BIT_MACHINE_DOUBLE   40  #define CMO_64BIT_MACHINE_DOUBLE   40
 #define CMO_ARRAY_OF_64BIT_MACHINE_DOUBLE  41  #define CMO_ARRAY_OF_64BIT_MACHINE_DOUBLE  41
Line 464  int32 を用いる.
Line 683  int32 を用いる.
 \end{verbatim}  \end{verbatim}
   
 \noindent  \noindent
 Group CMObject/MachineDouble requires CMObject/Basic0.\\  Group CMObject/MachineDouble requires CMObject/Primitive.\\
 64bit machine double, Array of 64bit machine double  64bit machine double, Array of 64bit machine double
 128bit machine double, Array of 128bit machine double  128bit machine double, Array of 128bit machine double
 $\in$ CMObject/MachineDouble \\  $\in$ CMObject/MachineDouble \\
   
   /*&jp
 \begin{eqnarray*}  \begin{eqnarray*}
 \mbox{64bit machine double} &:&  \mbox{64bit machine double} &:&
 \mbox{({\tt CMO\_64BIT\_MACHINE\_DOUBLE}, } \\  \mbox{({\tt CMO\_64BIT\_MACHINE\_DOUBLE}, } \\
 & & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte}} s8)\\  & & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte} s8})\\
 & & \mbox{ --- s1, $\ldots$, s8 は {\tt double} (64bit). } \\  & & \mbox{ --- s1, $\ldots$, s8 {\tt double} (64bit). } \\
 & & \mbox{ --- この表現はCPU依存である.}\\  & & \mbox{ --- 茵憗CPU箴絖с.}\\
 &&  \mbox{\quad\quad mathcap に CPU 情報を付加しておく.} \\  &&  \mbox{\quad\quad byte order negotiation .} \\
 \mbox{Array of 64bit machine double} &:&  \mbox{Array of 64bit machine double} &:&
 \mbox{({\tt CMO\_ARRAY\_OF\_64BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\  \mbox{({\tt CMO\_ARRAY\_OF\_64BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\
 & & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte}}\, s8[1], \ldots , {\sl byte}\, s8[m])\\  & & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte}}\, s8[1], \ldots , {\sl byte}\, s8[m])\\
 & & \mbox{ --- s*[1], $\ldots$ s*[m] は m 個の double (64bit) である. } \\  & & \mbox{ --- s*[1], $\ldots$ s*[m] m double (64bit) с. } \\
 & & \mbox{ --- この表現はCPU依存である.}\\  & & \mbox{ --- 茵憗CPU箴絖с.}\\
 & & \mbox{ \quad\quad mathcap に CPU 情報を付加しておく.} \\  & & \mbox{ \quad\quad byte order negotiation .} \\
 \mbox{128bit machine double} &:&  \mbox{128bit machine double} &:&
 \mbox{({\tt CMO\_128BIT\_MACHINE\_DOUBLE}, } \\  \mbox{({\tt CMO\_128BIT\_MACHINE\_DOUBLE}, } \\
 & & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte}} s16)\\  & & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte} s16})\\
 & & \mbox{ --- s1, $\ldots$, s16 は {\tt long double} (128bit). } \\  & & \mbox{ --- s1, $\ldots$, s16 {\tt long double} (128bit). } \\
 & & \mbox{ --- この表現はCPU依存である.}\\  & & \mbox{ --- 茵憗CPU箴絖с.}\\
 &&  \mbox{\quad\quad mathcap に CPU 情報を付加しておく.} \\  &&  \mbox{\quad\quad byte order negotiation .} \\
 \mbox{Array of 128bit machine double} &:&  \mbox{Array of 128bit machine double} &:&
 \mbox{({\tt CMO\_ARRAY\_OF\_128BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\  \mbox{({\tt CMO\_ARRAY\_OF\_128BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\
 & & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte}} s16[1], \ldots , {\sl byte} s16[m])\\  & & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte} s16[1], \ldots , {\sl byte} s16[m]})\\
 & & \mbox{ --- s*[1], $\ldots$ s*[m] は m 個の long double (128bit) である. } \\  & & \mbox{ --- s*[1], $\ldots$ s*[m] m long double (128bit) с. } \\
 & & \mbox{ --- この表現はCPU依存である.}\\  & & \mbox{ --- 茵憗CPU箴絖с.}\\
 & & \mbox{ \quad\quad mathcap に CPU 情報を付加しておく.}  & & \mbox{ \quad\quad byte order negotiation .}
 \end{eqnarray*}  \end{eqnarray*}
   */
   /*&eg
   \begin{eqnarray*}
   \mbox{64bit machine double} &:&
   \mbox{({\tt CMO\_64BIT\_MACHINE\_DOUBLE}, } \\
   & & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte} s8})\\
   & & \mbox{ --- s1, $\ldots$, s8 {\tt double} (64bit). } \\
   & & \mbox{ --- Encoding depends on CPU.}\\
   &&  \mbox{\quad\quad Need the byte order negotiation.} \\
   \mbox{Array of 64bit machine double} &:&
   \mbox{({\tt CMO\_ARRAY\_OF\_64BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\
   & & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte}}\, s8[1], \ldots , {\sl byte}\, s8[m])\\
   & & \mbox{ --- s*[1], $\ldots$ s*[m] are 64bit double's. } \\
   & & \mbox{ --- Encoding depends on CPU.}\\
   & & \mbox{\quad\quad Need the byte order negotiation.} \\
   \mbox{128bit machine double} &:&
   \mbox{({\tt CMO\_128BIT\_MACHINE\_DOUBLE}, } \\
   & & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte} s16})\\
   & & \mbox{ --- s1, $\ldots$, s16 {\tt long double} (128bit). } \\
   & & \mbox{ --- Encoding depends on CPU.}\\
   & & \mbox{\quad\quad Need the byte order negotiation.} \\
   \mbox{Array of 128bit machine double} &:&
   \mbox{({\tt CMO\_ARRAY\_OF\_128BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\
   & & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte} s16[1], \ldots , {\sl byte} s16[m]})\\
   & & \mbox{ --- s*[1], $\ldots$ s*[m] are 128bit long double's. } \\
   & & \mbox{ --- Encoding depends on CPU.}\\
   & & \mbox{\quad\quad Need the byte order negotiation.} \\
   \end{eqnarray*}
   */
   
 \bigbreak  \bigbreak
 次に IEEE 準拠の float および Big float を定義しよう.  
 \begin{verbatim}  \begin{verbatim}
 #define CMO_BIGFLOAT   50  
 #define CMO_IEEE_DOUBLE_FLOAT 51  #define CMO_IEEE_DOUBLE_FLOAT 51
 \end{verbatim}  \end{verbatim}
   
 IEEE 準拠の float については,  /*&jp
 IEEE 754  double precision floating-point format  IEEE 羣 float ゃ, IEEE 754 double precision floating-point
 (64 bit) の定義を見よ.  format (64 bit) 絎臂荀.
   
 \noindent  256.100006 Intel Pentium double64 с茵憗
 Group CMObject/Bigfloat requires CMObject/Basic0, CMObject/Basic1.\\  {\tt cd 0c 80 43 } \\
 Bigfloat  256.100006 PowerPC (Mac) double64 с茵憗
 $\in$ CMObject/Bigfloat \\  {\tt 43 80 0c cd }.
   箴с帥 byte 綺с.
   潟吾潟鴻帥若 byte order negotiation byte 綺絎.
   
 \begin{eqnarray*}  
 \mbox{Bigfloat} &:&  
 \mbox{({\tt CMO\_BIGFLOAT}, } \\  
 & & \quad \mbox{ {\sl ZZ} a , {\sl ZZ} e})\\  
 & & \mbox{ --- $a \times 2^e$ をあらわす. } \\  
 \end{eqnarray*}  
   
   */
   /*&eg
   See IEEE 754 double precision floating-point (64 bit) for the details of
   float compliant to the IEEE standard.
   
   The internal expression of 256.100006 in the Intel Pentium  is
   {\tt cd 0c 80 43 } \\
   The internal expression of 256.100006 in the PowerPC (Mac) is
   {\tt 43 80 0c cd }.
   As you have seen in this example,
   the orders of the bytes are opposite each other.
   The byte order is specified by the byte order negotiation protocol
   when the engine starts.
   
 */  */
   
   \subsection{Bigfloat32}
   /*&jp
   int32 堺 bigfloat 茵憝号ゃ菴違鴻.
   綵√ mpfr 32bit CPU т戎茵憗演с.
   */
   /*&eg
   This subsection describes our format for bigfloat in terms of the int32.
   This format is identical to the internal format of mpfr on 32 bit CPU's.
   */
   Ref: {\tt OpenXM/src/mpfr/bfsize/bfsize.c}
   
   \begin{verbatim}
   #define     CMO_BIGFLOAT32          52
   \end{verbatim}
   
   
   /*&jp
   Bigfloat32 罨＜綵√ int32 с.
   */
   /*&eg
   The bigfloat32 is an array of int32 numbers of the following format.
   */
   
   \begin{center}
   {\sl int32}\, {\rm prec}, {\sl int32}\, {\rm sign}, {\sl int32}\, {\rm exp},
   {\sl int32}\, \mbox{a[1]}, \ldots , {\sl int32}\, \mbox{a[k]}
   \end{center}
   
   /*&jp
   p=prec 膕上墾, s=sign 膃(1 罩ｃ,  -1 (2茖域；)莢) ,
   E=exp 育, 筝若帥
   $$ s (a[k]/B + a[k-1]/B^2 + ... + a[1]/B^k) 2^E  $$
   茵.
    $B=2^{32}$,
   $k = \lceil p/32 \rceil$
   с.
   */
   /*&eg
   p=prec is the precision, s=sign is the sign(1 means positive,  -1 (expressed by two's complement) is nevative),
   E=exp is the exponent, and the data above expresses the number
   $$ s (a[k]/B + a[k-1]/B^2 + ... + a[1]/B^k) 2^E.  $$
   Here, $B=2^{32}$,
   $k = \lceil p/32 \rceil$.
   */
   
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