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 \documentstyle[11pt,reduce]{article}
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 \newcommand{\bq}{\begin{quotation}}
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 \newcommand{\bi}{\begin{itemize}}
 \newcommand{\ei}{\end{itemize}}
 \date{}
 \title{{\bf ASSIST}\ :\\[2pt]
 A General Purpose Facility~for~\REDUCE \\[5pt]
 \mbox{\hfill Version 2.0\hfil}}
 \author{Hubert Caprasse \\
 D\'epartement d'Astronomie et d'Astrophysique \\
 Institut de Physique, B5, Sart Tilman \\
 B4000 LIEGE 1 \\[3pt]
 Email: caprasse@vm1.ulg.ac.be}
 \begin{document}
 \maketitle
 \index{ASSIST package}
 \section{Introduction}
 ASSIST contains
 an appreciable number of additional general purpose functions which allow
 to better adapt \REDUCE\ to various calculational strategies,
 to make the programming task more straightforward and more efficient.
 Contrary to all other packages, ASSIST does not aim to provide neither a new
 facility to compute a definite class of mathematical objects nor to extend
 the base of mathematical knowledge of \REDUCE\ .
 The functions it contains should be
 useful independently of the nature of the application which is considered.
 They were initially written while doing specific applications of
 \REDUCE\ to problems in theoretical physics. Most them were designed
 in such a way that their applicability range is broad. Though it was not
 the primary goal, efficiency has been sought whenever possible.
 The source code in ASSIST contains many comments concerning
 the meaaning and the use of the supplementary functions available
 in the algebraic mode. These comments, hopefully, makes the code transparent
 and allow a thorough exploitation of the package. The present documentation
 contains a nontechnical description of it and describes the
 various new facilities it provides.
 \section{ Survey of the Available New Facilities}
 The package contains several modules. Their content reflects closely
 the various categories of facilities quoted below. Some functions do
 already exist inside the KERNEL of \REDUCE\ . However, their range
 of applicability is {\em extended}.
 \begin{itemize}
 \item{Control of Switches:}
 \begin{quotation}
 \noindent
 \f{SWITCHES SWITCHORG}
 \end{quotation}
 \item{Operations on Lists and Bags:}
 \begin{quotation}
 \noindent
 \f{MKLIST KERNLIST FREQUENCY SEQUENCES \nl
 INSERT INSERT\_KEEP\_ORDER MERGE\_LIST \nl
 REMOVE DELETE DELETE\_ALL DELPAIR \nl
 FIRST SECOND THIRD REST REVERSE LAST \nl
 BELAST CONS ( . ) APPEND APPENDN \nl
 LENGTH REMOVE DELETE DELPAIR \nl
 MEMBER ELMULT PAIR DEPTH POSITION \nl
 REPFIRST REPREST ASFIRST ASLAST ASREST \nl
 ASFLIST ASSLIST RESTASLIST SUBSTITUTE \nl
 BAGPROP PUTBAG CLEARBAG BAGP BAGLISTP \nl
 ALISTP ABAGLISTP LISTBAG }
 \end{quotation}
 \item{Operations on Sets:}
 \begin{quotation}
 \noindent
 \f{MKSET SETP UNION INTERSECT DIFFSET SYMDIFF}
 \end{quotation}
 \item{General Purpose Utility Functions:}
 \begin{quotation}
 \noindent
 \f{LIST\_TO\_IDS MKIDN MKIDNEW DELLASTDIGIT DETIDNUM \\
 ODDP FOLLOWLINE == RANDOMLIST \\
 PERMUTATIONS CYCLICPERMLIST COMBNUM COMBINATIONS \\
 SYMMETRIZE REMSYM SORTNUMLIST SORTLIST EXTREMUM \\
 DEPATOM FUNCVAR IMPLICIT EXPLICIT KORDERLIST \\
 SIMPLIFY CHECKPROPLIST EXTRACTLIST}
 \end{quotation}
 \item{ Properties and Flags:}
 \begin{quotation}
 \noindent
 \f{PUTFLAG PUTPROP DISPLAYPROP DISPLAYFLAG \\
 CLEARFLAG CLEARPROP }
 \end{quotation}
 \item{ Control Statements, Control of Environment:}
 \begin{quotation}
 \noindent
 \f{NORDP DEPVARP ALATOMP ALKERNP PRECP \\
 SHOW SUPPRESS CLEAROP CLEARFUNCTIONS }
 \end{quotation}
 \item{Handling of Polynomials:}
 \begin{quotation}
 \noindent
 \f{ALG\_TO\_SYMB SYMB\_TO\_ALG \\
 DISTRIBUTE LEADTERM REDEXPR MONOM\\
 LOWESTDEG DIVPOL SPLITTERMS SPLITPLUSMINUS}
 \end{quotation}
 % \vfill\pagebreak
 \item{Handling of Transcendental Functions:}
 \begin{quotation}
 \noindent
 \f{TRIGEXPAND HYPEXPAND TRIGREDUCE HYPREDUCE \\
 PLUSLOG CONCSUMLOG}
 \end{quotation}
 \item{Handling of ndimensional Vectors:}
 \begin{quotation}
 \noindent
 \f{SUMVECT MINVECT SCALVECT CROSSVECT MPVECT }
 \end{quotation}
 {\item Handling of Grassmann Operators:}
 \begin{quotation}
 \noindent
 \f{PUTGRASS REMGRASS GRASSP GRASSPARITY GHOSTFACTOR }
 \end{quotation}
 \item{Handling of Matrices:}
 \begin{quotation}
 \noindent
 \f{UNITMAT MKIDM BAGLMAT COERCEMAT \\
 SUBMAT MATSUBR MATSUBC RMATEXTR RMATEXTC \\
 HCONCMAT VCONCMAT TPMAT HERMAT \\
 SETELTMAT GETELTMAT}
 \end{quotation}
 \end{itemize}
 In the following each group of functions is, successively, described.
 \section{Control of Switches}
 The two available functions i.e. \f{SWITCHES, SWITCHORG} have
 no argument and are called as if they were mere identifiers.
 \f{SWITCHES} displays the actual status of the most often used switches
 when manipulating rational functions. The chosen switches are
 \begin{quotation}
 \noindent
 {\tt EXP, DIV, MCD, GCD, ALLFAC, INTSTR,\\ RAT, RATIONAL, FACTOR }
 \end{quotation}
 The switch {\tt DISTRIBUTE} which controls the handling
 of distributed polynomials is added to them (see below the description of
 the new functions for manipulating polynomials ).
 The selection is somewhat arbitrary but it may be changed in a trivial
 fashion by the user.
 Most of the symbolic variables {\tt !*EXP, !*DIV, $\ldots$}
 which have either the value T or the value NIL are made available in the
 algebraic mode so that it becomes possible to write conditional
 statements of the kind
 \begin{verbatim}
 IF !*EXP THEN DO ......
 IF !*GCD THEN OFF GCD;
 \end{verbatim}
 \f{SWITCHORG} resets (almost) {\em all} switches in the status
 they have when {\bf entering} into \REDUCE\ .
 The new switch {\tt DISTRIBUTE} allows to put polynomials in a
 distributed form.
 \section{Manipulation of the List Structure}
 Additional functions for list manipulations are provided and some already
 defined functions in the kernel of \REDUCE\ are modified to properly
 generalize them to the available new structure {\tt BAG}.
 \begin{itemize}
 \item[i.]
 Generation of a list of length n with all its elements initialized to 0
 and possibility to append to a list $l$ a certain number of zero's to
 make it of length $n$:
 \begin{verbatim}
 MKLIST n ; n is an INTEGER
 MKLIST(l,n); l is Listlike, n is an INTEGER
 \end{verbatim}
 \item[ii.]
 Generation of a list of sublists of length n containing p elements
 equal to 0 and q elements equal to 1 such that $$p+q=n .$$
 The function \f{SEQUENCES} works both in the algebraic and
 symbolic modes. Here is an example:
 \begin{verbatim}
 SEQUENCES 2 ; ==> {{0,0},{0,1},{1,0},{1,1}}
 \end{verbatim}
 The function \f{KERNLIST} transforms any prefix of a kernel into the
 {\bf \verb+list+} prefix. The output list is a copy:
 \begin{verbatim}
 KERNLIST (<kernel>); ==> {<kernel arguments>}
 \end{verbatim}
 Four functions to delete elements are \f{DELETE, REMOVE, DELETE\_ALL} and
 \f{DELPAIR}. The first two act as in symbolic mode, the third
 eliminates from a given list {\em all}
 elements equal to its first argument. The fourth act on list of pairs
 and eliminates from it the {\em first} pair whose first element is equal to
 its first argument :
 \begin{verbatim}
 DELETE(x,{a,b,x,f,x}); ==> {a,b,f,x}
 REMOVE({a,b,x,f,x},3); ==> {a,b,f,x}
 DELETE_ALL(x,{a,b,x,f,x}); ==> {a,b,f}
 DELPAIR(a,{{a,1},{b,2},{c,3}}; ==> {{b,2},{c,3}}
 \end{verbatim}
 \item[iv.]
 The function \f{ELMULT} returns an {\em integer} which is the
 {\em multiplicity} of its first argument inside the list which is its
 second argument.
 The function \f{FREQUENCY} gives a list of pairs
 whose second element indicates the number of times the first element
 appears inside the original list:
 \begin{verbatim}
 ELMULT(x,{a,b,x,f,x}) ==> 2
 FREQUENCY({a,b,c,a}); ==> {{a,2},{b,1},{c,1}}
 \end{verbatim}
 \item[v.]
 The function \f{INSERT} allows to insert a given object into a list
 at the wanted position.
 The functions \f{INSERT\_KEEP\_ORDER} and \f{MERGE\_LIST} allow to
 keep a given ordering when inserting one element inside a list or
 when merging two lists. Both have 3 arguments. The last one is
 the name of a binary boolean ordering function:
 \begin{verbatim}
 ll:={1,2,3}$
 INSERT(x,ll,3); ==> {1,2,x,3}
 INSERT_KEEP_ORDER(5,ll,lessp); ==> {1,2,3,5}
 MERGE_LIST(ll,ll,lessp); ==> {1,1,2,2,3,3}
 \end{verbatim}
 \item[vi.]
 Algebraic lists can be read from right to left or left to right.
 They {\em look} symmetrical. One would like to dispose of manipulation
 functions which reflect this.
 So, to the already defined functions \f{FIRST} and \f{REST} are
 added the functions \f{LAST} and \f{BELAST}. \f{LAST} gives the last
 element of the list while \f{BELAST} gives the list {\em without} its
 last element. \\
 Various additional functions are provided. They are:
 \bq
 \f{ CONS (.), POSITION, DEPTH, PAIR, \\
 APPENDN, REPFIRST, REPLAST}
 \eq
 The token ``dot'' needs a special comment. It corresponds to
 several different operations.
 \begin{enumerate}
 \item If one applies it on the left of a list, it acts as the \f{CONS}
 function. Be careful, blank spaces are required around the dot:
 \begin{verbatim}
 4 . {a,b}; ==> {4,a,b}
 \end{verbatim}
 \item If one applies it on the right of a list, it has the same
 effect as the \f{PART} operator:
 \begin{verbatim}
 {a,b,c}.2; ==> b
 \end{verbatim}
 \item If one applies it on 4dimensional vectors, it acts as in the
 HEPHYS packge.
 \end{enumerate}
 \f{POSITION} returns the POSITION of the first occurrence of x in
 a list or a message if x is not present in it.
 \f{DEPTH} returns an {\em integer} equal to the number of levels where a
 list is found if and only if this number is the {\em same} for each
 element of the list otherwise it returns a message telling the user
 that list is of {\em unequal depth}.
 \f{PAIR} has two arguments which must be lists. It returns a list
 whose elements are {\em lists of two elements.}
 The $n^{th}$ sublist contains the $n^{th}$ element of the first list
 and the $n^{th}$ element of the second list. These types of lists are called
 {\em association lists} or ALISTS in the following.
 To test for these type of lists a boolean function \f{ABAGLISTP}
 is provided. It will be discussed below.\\
 \f{APPENDN} has {\em any} fixed number of lists as arguments. It
 generalizes the already existing function \f{APPEND} which accepts
 only two lists as arguments. \\
 \f{REPFIRST} has two arguments. The first one is any object, the second one
 is a list. It replaces the first element of the list by tho object. It
 works like the symbolic function \f{REPLACA} except that the
 original list is not destroyed.\\
 \f{REPREST} has also two arguments. It replaces the rest of the list by
 its first argument and returns the new list without destroying the
 original list. It is analogous to the symbolic function \f{REPLACD}.
 Here are examples:
 \begin{verbatim}
 ll:={{a,b}}$
 ll1:=ll.1; ==> {a,b}
 ll.0; ==> list
 0 . ll; ==> {0,{a,b}}
 DEPTH ll; ==> 2
 PAIR(ll1,ll1); ==> {{a,a},{b,b}}
 REPFIRST{new,ll); ==> {new}
 ll3:=APPENDN(ll1,ll1,ll1); ==> {a,b,a,b,a,b}
 POSITION(b,ll3); ==> 2
 REPREST(new,ll3); ==> {a,new}
 \end{verbatim}
 \item[vii.]
 The functions \f{ASFIRST, ASLAST, ASREST, ASFLIST, ASSLIST, \\RESTASLIST}
 act on ALISTS or on list of lists of well defined depths
 and have two arguments. The first is the key object
 which one seeks to associate in some way to an element of the association
 list which is the second argument.\\
 \f{ASFIRST} returns the pair whose first element is equal to the
 first argument.\\
 \f{ASLAST} returns the pair whose last element is equal to the first
 argument.\\
 \f{ASREST} needs a {\em list} as its first argument. The function
 seeks the first sublist of a list of lists (which is its second argument)
 equal to its first argument and returns it.\\
 \f{RESTASLIST} has a {\em list of keys} as its first arguments. It
 returns the collection of pairs which meet the criterium of \f{ASREST}.\\
 \f{ASFLIST} returns a list containing {\em all pairs} which
 satisfy to the criteria of the function \f{ASFIRST}. So the output
 is also an ALIST or a list of lists.\\
 \f{ASSLIST} returns a list which contains {\em all pairs} which have
 their second element equal to the first argument.\\
 Here are a few examples:
 \begin{verbatim}
 lp:={{a,1},{b,2},{c,3}}$
 ASFIRST(a,lp); ==> {a,1}
 ASLAST(1,lp); ==> {a,1}
 ASREST({1},lp); ==> {a,1}
 RESTASLIST({a,b},lp); ==> {{1},{2}}
 lpp:=APPEND(lp,lp)$
 ASFLIST(a,lpp); ==> {{a,1},{a,1}}
 ASSLIST(1,lpp); ==> {{a,1},{a,1}}
 \end{verbatim}
 \item[vii.] The function \f{SUBSTITUTE} has three arguments. The first
 is the object to substitute, the second is the object which must be
 replaced by the first, the third is a list. Substitution is made to
 all levels. It is a more elementary function than \f{SUB} but its
 capabilities are less. When dealing with algebraic quantities, it is
 important to make sure that {\em all} objects involved in the function
 have either the prefix lisp or the standard quotient representation
 otherwise it will not properly work.
 \end{itemize}
 \section{ The Bag Structure and its Associated Functions}
 The LIST structure of \REDUCE\ is very convenient to manipulate
 groups of objects which are, a priori, unknown. This structure is
 endowed with other properties such as ``mapping'' i.e. the fact that
 if \verb+OP+ is an operator one gets, by default,
 \begin{verbatim}
 OP({x,y}); ==> {OP(x),OP(y)}
 \end{verbatim}
 It is not permitted to submit lists to the operations valid on rings
 so that lists cannot be indeterminates of polynomials.\\
 Very frequently too, procedure arguments cannot be lists.
 At the other extreme, so to say, one has the \verb+KERNEL+
 structure associated
 to the algebraic declaration \verb+operator+ . This structure behaves as
 an ``unbrokable'' one and, for that reason, behaves
 like an ordinary identifier.
 It may generally be bound to all nonnumeric procedure parameters
 and it may appear
 as an ordinary indeterminate inside polynomials. \\
 The \verb+BAG+ structure is intermediate between a list and an operator.
 From the operator it borrows the property to be a \verb+KERNEL+ and,
 therefore, may be an indeterminate of a polynomial. From the list structure
 it borrows the property to be a {\em composite} object.\\[5pt]
 \mbox{\underline{{\bf Definition}:\hfill}}\\[4pt]
 A bag is an object endowed with the following properties:
 \begin{enumerate}
 \item It is a \verb+KERNEL+ composed of an atomic prefix (its
 envelope) and
 its content (miscellaneous objects).
 \item Its content may be changed in an analogous way as the content of a
 list. During these manipulations the name of the bag is {\em conserved}.
 \item Properties may be given to the envelope. For instance, one may
 declare it \verb+NONCOM+ or \verb+SYMMETRIC+ etc.\ $\ldots$
 \end{enumerate}
 \vspace{5pt}
 \mbox{\underline{{\bf Available Functions}:\hfill}}
 \bi
 \item[i.] A default bag envelope \verb+BAG+ is defined.
 It is a reserved identifier.
 An identifier other than \verb+LIST+ or one which is already associated
 with a boolean function may be defined as a bag envelope through the
 command \f{PUTBAG}. In particular, any operator may also be declared
 to be a bag. {\bf When and only when} the identifier is not an already defined
 function does \f{PUTBAG} puts on it the property of an OPERATOR PREFIX.
 The command:
 \begin{verbatim}
 PUTBAG id1,id2,....idn;
 \end{verbatim}
 declares \verb+id1,.....,idn+ as bag envelopes.
 Analoguously, the command
 \begin{verbatim}
 CLEARBAG id1,...idn;
 \end{verbatim}
 eliminates the bag property on \verb+id1,...,idn+.
 \item[ii.] The boolean function \f{BAGP} detects the bag property.
 Here is an example:
 \begin{verbatim}
 aa:=bag(x,y,z)$
 if BAGP aa then "ok"; ==> ok
 \end{verbatim}
 \item[iii.] Almost all functions defined above for lists
 do also work for bags.
 Moreover, functions subsequently defined for SETS do also work.
 Here is a list of the main ones:
 \begin{quotation}
 \f{FIRST, SECOND, LAST, REST, BELAST, DEPTH,\\
 LENGTH, APPEND, CONS (.), REPFIRST, REPREST} $\ldots$
 \end{quotation}
 However, because of the conservation of the envelope, they act
 somewhat differently. Here are a few examples (more examples are
 given inside the test file):
 \begin{verbatim}
 PUTBAG op; ==> T
 aa:=op(x,y,z)$
 FIRST op(x,y,z); ==> op(x)
 REST op(x,y,z); ==> op(y,z)
 BELAST op(x,y,z); ==> op(x,y)
 APPEND(aa,aa); ==> op(x,y,z,x,y,z)
 LENGTH aa; ==> 3
 DEPTH aa; ==> 1
 \end{verbatim}
 When ``appending'' two bags with {\em different} envelopes, the resulting bag
 gets the name of the one bound to the first parameter of \f{APPEND}.\\
 The function \f{LENGTH} gives the actual number of variables on which
 the operator (or the function) depends.
 \vspace{5pt}
 \begin{center}
 The NAME of the ENVELOPE is KEPT by the functions \\[3pt]
 \f{FIRST, SECOND, LAST, BELAST }.
 \end{center}
 \vspace{5pt}
 \item[iv.]
 The connection between the list and the bag structures is made easy
 thanks to \f{KERNLIST} which transforms a bag into a list and thanks to
 the coercion function \f{LISTBAG}. This function has 2 arguments
 and is used as follows:
 \begin{verbatim}
 LISTBAG(<list>,<id>); ==> <id>(<arg_list>)
 \end{verbatim}
 The identifier \verb+<id>+, if allowed, is automatically declared as a bag
 envelope or an error message is generated. \\[3pt]
 Finally, two boolean functions which work both for bags and lists are
 provided. They are \f{BAGLISTP} and \f{ABAGLISTP}.
 They return t or nil (in a conditional statement) if their argument
 is a bag or a list for the first one, if their argument is a list of
 sublists or a bag containing bags for the second one .
 \end{itemize}
 \section{Sets and their Manipulation Functions}
 Functions for sets do exist on the level of the symbolic mode. The
 package make them available in the algebraic mode but also {\em generalizes}
 them so that they can be applied on baglike objects as well.
 \bi
 \item[i.]
 The constructor \f{MKSET} transforms a list or bag into a set by eliminating
 duplicates.
 \begin{verbatim}
 MKSET({1,a,a1}); ==> {1,a}
 MKSET bag(1,a,a1); ==> bag(1,a)
 \end{verbatim}
 \f{SETP} is a boolean function which recognizes setlike objects.
 \begin{verbatim}
 if SETP {1,2,3} then ... ;
 \end{verbatim}
 \item[ii.]
 The available functions are
 \begin{center}
 \f{UNION, INTERSECT, DIFFSET, SYMDIFF}.
 \end{center}
 They have two arguments which must be sets otherwise an error message
 is issued.
 Their meaning is transparent from their name. They respectively give the
 union, the intersection, the difference and the symmetric difference of two
 sets.
 \ei
 \section{General Purpose Utility Functions}
 Functions in this sections have various purposes. They have all been used
 many times in applications under some form or another. The form given
 to them in this package is adjusted to maximize their range of applications.
 \bi
 \item[i.]
 The functions \f{MKIDNEW DELLASTDIGIT DETIDNUM LIST\_TO\_IDS}
 handle identifiers. \f{MKIDNEW} is a variant of \f{MKID}.
 \f{MKIDNEW} has either 0 or 1 argument. It generates an identifier which
 has not yet been used before.
 \begin{verbatim}
 MKIDNEW(); ==> g0001
 MKIDNEW(a); ==> ag0002
 \end{verbatim}
 \f{DELLASTDIGIT} takes an integer as argument, it strips it from its last
 digit.
 \begin{verbatim}
 DELLASTDIGIT 45; ==> 4
 \end{verbatim}
 \f{DETIDNUM}, deletes the last digit from an
 identifier. It is a very convenient when one wants to make a do loop
 starting from a set of indices $ a_1, \ldots , a_{n} $.
 \begin{verbatim}
 DETIDNUM a23; ==> 23
 \end{verbatim}
 \f{LIST\_to\_IDS} generalizes the function \f{MKID} to a list of
 atoms. It creates and intern an identifier from the concatenation of
 the atoms. The first atom cannot be an integer.
 \begin{verbatim}
 LIST_TO_IDS {a,1,id,10}; ==> a1id10
 \end{verbatim}
 The function \f{ODDP} detects odd integers.
 The function \f{FOLLOWLINE} is convenient when using the function \f{PRIN2}.
 It allows to format an output text in a much more flexible way than with
 the function \f{WRITE}. \\
 Try the following examples :
 \begin{verbatim}
 <<prin2 2; prin2 5>>$ ==> ?
 <<prin2 2; followline(5); prin2 5;>>; ==> ?
 \end{verbatim}
 The function \f{==} is a short and convenient notation for the \f{SET}
 function. In fact it is a {\em generalization} of it to allow to
 deal also with KERNELS:
 \begin{verbatim}
 operator op;
 op(x):=abs(x)$
 op(x) == x; ==> x
 op(x); ==> x
 abs(x); ==> x
 \end{verbatim}
 The function \f{RANDOMLIST} generates a list of random numbers. It takes
 two arguments which are both integers. The first one indicates the range
 inside which the random numbers are chosen. The second one indicates how
 many numbers are to be generated. It is also the length of the list which
 is generated.
 \begin{verbatim}
 RANDOMLIST(10,5); ==> {2,1,3,9,6}
 \end{verbatim}
 \f{COMBNUM} gives the number of combinations of $n$ objects
 taken $p$ to $p$. It has the two integer arguments $n$ and $p$.
 \f{PERMUTATIONS} gives the list of permutations on $n$ objects.
 Each permutation is itself a list. \f{CYCLICPERMLIST} gives the list of
 {\em cyclic} permutations. For both functions, the argument may
 also be a {\tt bag}.
 \begin{verbatim}
 PERMUTATIONS {1,2} ==> {{1,2},{2,1}}
 CYCLICPERMLIST {1,2,3} ==>
 {{1,2,3},{2,3,1},{3,1,2}}
 \end{verbatim}
 \f{COMBINATIONS} gives a list of combinations on $n$ objects taken $p$
 to $p$. It has two arguments. The first one is a list (or a bag) and
 the second one is the integer $p$.
 \begin{verbatim}
 COMBINATIONS({1,2,3},2) ==> {{2,3},{1,3},{1,2}}
 \end{verbatim}
 \f{REMSYM} is a command that erases the \REDUCE\ commands
 \verb+symmetric+ or \verb+antisymmetric+ .
 \f{SYMMETRIZE} is a powerful function which generate a symmetric expression.
 It has 3 arguments. The first is a list (or a list of list) containing
 the expressions which will appear as variables for a kernel. The second
 argument is the kernelname and the third is a permutation function
 which either exist in the algebraic or in the symbolic mode. This
 function may have been constructed by the user. Within this package
 the two functions \f{PERMUTATIONS} and \f{CYCLICPERMLIST} may be used.
 Here are two examples:
 \begin{verbatim}
 ll:={a,b,c}$
 SYMMETRIZE(ll,op,cyclicpermlist); ==>
 OP(A,B,C) + OP(B,C,A) + OP(C,A,B)
 SYMMETRIZE(list ll,op,cyclicpermlist); ==>
 OP({A,B,C}) + OP({B,C,A}) + OP({C,A,B})
 \end{verbatim}
 Notice that, taking for the first argument a list of list gives rise to
 an expression where each kernel has a {\em list as argument}. Another
 peculiarity of this function is the fact that, unless a pattern maching is
 made on the operator \verb+OP+, it needs to be reevaluated. This peculiarity
 is induced by the need to maximize efficiency when \verb+OP+ is an abstract
 operator. Here is an illustration:
 \begin{verbatim}
 op(a,b,c):=a*b*c$
 SYMMETRIZE(ll,op,cyclicpermlist); ==>
 OP(A,B,C) + OP(B,C,A) + OP(C,A,B)
 for all x let op(x,a,b)=sin(x*a*b);
 SYMMETRIZE(ll,op,cyclicpermlist); ==>
 OP(B,C,A) + SIN(A*B*C) + OP(A,B,C)
 \end{verbatim}
 The functions \f{SORTNUMLIST} and \f{SORTLIST} are functions which sort
 lists. They use {\em bubblesort} and {\em quicksort} algorithms.
 \f{SORTNUMLIST} takes as argument a list of numbers. It sorts it in
 increasing order.
 \f{SORTLIST} is a generalization of the above function.
 It sorts the list according
 to any well defined ordering. Its first argument is the list and its
 second argument is the ordering function. The content of the list
 is not necessary numbers but must be such that the ordering function has
 a meaning.
 \begin{verbatim}
 l:={1,3,4,0}$ SORTNUMLIST l; ==> {0,1,3,4}
 ll:={1,a,tt,z}$ SORTLIST(ll,ordp); ==> {a,z,tt,1}
 \end{verbatim}
 One must know that using these functions for kernels or bags may be
 dangerous since they are destructive. If it is needed, it is recommended
 to first apply \f{KERNLIST} on them.
 The function \f{EXTREMUM} is a generalization of the already defined functions
 \f{MIN, MAX} to include general orderings. It is a 2 arguments function.
 The first is the list and the second is the ordering function.
 With the list \verb+ll+ defined in the last example, one gets
 \begin{verbatim}
 EXTREMUM(ll,ordp); ==> 1
 \end{verbatim}
 \item[iii.] There are four functions to identify dependencies.
 \f{FUNCVAR} takes any expression as argument and returns the set of
 variables of which it depends. Constants are eliminated.
 \begin{verbatim}
 FUNCVAR(e+pi+sin(log(y)); ==> {y}
 \end{verbatim}
 \f{DEPATOM} has an {\bf atom} as argument. It returns it if it is
 a number or if no dependency has previously been declared. Otherwise,
 it returns the list of variables which the various depend declarations
 imply.
 \begin{verbatim}
 depend a,x,y;
 DEPATOM a; ==> {x,y}
 \end{verbatim}
 The functions \f{EXPLICIT} and \f{IMPLICIT} make explicit or
 implicit the dependencies. This example show how they work:
 \begin{verbatim}
 depend a,x; depend x,y,z;
 EXPLICIT a; ==> a(x(y,z))
 IMPLICIT ws; ==> a
 \end{verbatim}
 These are useful when one does not know the names of the variables
 and (or) the nature of the dependencies.
 \f{KORDERLIST} is a zero argument function which display the actual
 ordering.
 \begin{verbatim}
 korder x,y,z;
 KORDERLIST; ==> (x,y,z)
 \end{verbatim}
 \item[iv.] A function \f{SIMPLIFY} which takes an arbitrary expression
 is available which {\em forces} downtothebotttom simplification of
 an expression. It is usefull with \f{SYMMETRIZE}. It has also proved
 usefull to simplify some output expressions of the package
 EXCALC\ (E. Schruefer). Here is an example:
 \begin{verbatim}
 l:=op(x,y,z)$
 op(x,y,z):=x*y*z$
 SYMMETRIZE(l,op,cyclicpermlist); ==>
 op(x,y,z)+op(y,z,x)+op(z,x,y)
 SIMPLIFY ws; ==> op(y,z,x)+op(z,x,y)+x*y*z
 \end{verbatim}
 \item[v.] Filtering functions for lists.
 \f{CHECKPROLIST} is a boolean function which checks if the
 elements of a list have a definite property. Its first argument
 is the list, its second argument is a boolean function
 (\f{FIXP NUMBERP $\ldots$}) or an ordering function (as \f{ORDP}).
 \f{EXTRACTLIST} extracts from the list given as its first argument
 the elements which satisfy the boolean function given as its second
 argument. For example:
 \begin{verbatim}
 l:={1,a,b,"st")$
 EXTRACTLIST(l,fixp); ==> {1}
 EXTRACTLIST(l,stringp); ==> {st}
 \end{verbatim}
 \ei
 \section{Properties and Flags}
 In spite of the fact that many facets of the handling of
 property lists is easily accessible in the algebraic mode, it is useful to
 provide analogous functions {\em genuine} to the algebraic mode. The reason is
 that, altering property lists of objects, may easily destroy the integrity
 of the system. The functions, which are here described, {\bf do ignore}
 the property list and flags already defined by the system itself. They
 generate and track the {\em addtional properties and flags} that the user
 issues using them. They offer him
 the possibility to work on property lists so
 that he can design a programming style of the ``conceptual'' type.
 \bi
 \item[i.] We first consider ``flags''. \\
 To a given identifier, one may
 associates another one linked to it ``in the background''. The three
 functions \f{PUTFLAG, DISPLAYFLAG} and \f{CLEARFLAG} handle them.
 \f{PUTFLAG} has 3 arguments. The first is the identifier or a list
 of identifiers, the second is the name of the flag,
 the third is T (true) or 0 (zero).
 When the third argument is T, it creates the flag, when it is 0 it
 destroys it.
 \begin{verbatim}
 PUTFLAG(z1,flag_name,t); ==> flag_name
 PUTFLAG({z1,z2},flag1_name,t); ==> t
 PUTFLAG(z2,flag1_name,0) ==>
 \end{verbatim}
 \f{DISPLAYFLAG} allows to extract flags. The previous actions give:
 \begin{verbatim}
 DISPLAYFLAG z1; ==>{flag_name,flag1_name}
 DISPLAYFLAG z2 ; ==> {}
 \end{verbatim}
 \f{CLEARFLAG} is a command which clears {\em all} flags associated to
 the identifiers $id_1, \ldots , id_n .$
 \item[ii.] Properties are handled by similar functions.
 \f{PUTPROP} has four arguments. The second argument is, here, the
 {\em indicator} of the property. The third argument may be {\em any
 valid expression}. The fourth one is also T or 0.
 \begin{verbatim}
 PUTPROP(z1,property,x^2,t); ==> z1
 \end{verbatim}
 In general, one enter
 \begin{verbatim}
 PUTPROP(LIST(idp1,idp2,..),<propname>,<value>,T);
 \end{verbatim}
 To display a specific property, one uses
 \f{DISPLAYPROP} which takes two arguments. The first is the name of the
 identifier, the second is the indicator of the property.
 \begin{verbatim}
 2
 DISPLAYPROP(z1,property); ==> {property,x }
 \end{verbatim}
 Finally, \f{CLEARPROP} is a nary commmand which clears {\em all}
 properties of the identifiers which appear as arguments.
 \ei
 \section{Control Functions}
 Here we describe additional functions which
 improve the user control on the environment.
 \bi
 \item[i.]
 The first set of functions is composed of unary and binary boolean functions.
 They are:
 \begin{verbatim}
 ALATOMP x; x is anything.
 ALKERNP x; x is anything.
 DEPVARP(x,v); x is anything.
 (v is an atom or a kernel)
 \end{verbatim}
 \f{ALATOMP} has the value T iff x is an integer or an identifier
 {\em after} it has been evaluated down to the bottom.
 \f{ALKERNP} has the value T iff x is a kernel {\em after}
 it has been evaluated down to the bottom.
 \f{DEPVARP} returns T iff the expression x depends on v at
 {\bf any level}.
 The above functions together with \f{PRECP} have
 been declared operator functions to ease the verification of
 their value.
 \f{NORDP} is essentially equivalent to \verb+not+\f{ ORDP}
 when inside a conditional statement. Otherwise, it can be used
 while \verb+not+\f{ ORDP} cannot.
 \item[ii.]
 The next functions allow one to {\em analyze} and to
 {\em clean} the environment
 of \REDUCE\ which is created by the user while he is working
 {\bf interactively}. To functions are provided:\\
 \f{SHOW} allows to get the various identifiers already
 assigned and to see their type. \f{SUPPRESS} selectively clears the
 used identifiers or clears them all. It is to be stressed that identifiers
 assigned from the input of files are {\bf ignored}.
 Both functions have one argument and the same options for this
 argument:
 \begin{verbatim}
 SHOW (SUPPRESS) all
 SHOW (SUPPRESS) scalars
 SHOW (SUPPRESS) lists
 SHOW (SUPPRESS) saveids (for saved expressions)
 SHOW (SUPPRESS) matrices
 SHOW (SUPPRESS) arrays
 SHOW (SUPPRESS) vectors
 (contains vector, index and tvector)
 SHOW (SUPPRESS) forms
 \end{verbatim}
 The option \verb+all+ is the most convenient for \f{SHOW} but,
 with it, it may
 takes time to get the answer after one has worked several hours.
 When entering \REDUCE\ the option \verb+all+ for \f{SHOW} gives:
 \begin{verbatim}
 SHOW all; ==>
 scalars are: NIL
 arrays are: NIL
 lists are: NIL
 matrices are: NIL
 vectors are: NIL
 forms are: NIL
 \end{verbatim}
 It is a convenient way to remember the various options. Here an example
 which is valid when one starts from a fresh environment:
 \begin{verbatim}
 a:=b:=1$
 SHOW scalars; ==> scalars are: (A B)
 SUPPRESS scalars; ==> t
 SHOW scalars; ==> scalars are: NIL
 \end{verbatim}
 \item[iii.]
 The \f{CLEAR} function of the system does not do a complete cleaning of
 \verb+OPERATORS+ and \verb+FUNCTIONS+ . The following two functions do a more
 complete cleaning and, also, takes automatically into account the
 {\em user} flag and properties that the functions
 \f{PUTFLAG} and \f{PUTPROP} may have introduced.
 Their names are \f{CLEAROP} and \f{CLEARFUNCTIONS}.
 \f{CLEAROP} takes one operator as its argument.\\
 \f{CLEARFUNCTIONS} is a nary command. If one issues
 \begin{verbatim}
 CLEARFUNCTIONS a1,a2, ... , an $
 \end{verbatim}
 The functions with names \verb+ a1,a2, ... ,an+ are cleared.
 One should be careful when using this facility since the
 only functions which cannot be erased are those which are
 protected with the \verb+lose+ flag.
 \ei
 \section{Handling of Polynomials}
 The module contains some utility functions to handle
 standard quotients and several new facilities to manipulate polynomials.
 \bi
 \item[i.] Two functions \f{ALG\_TO\_SYMB} and \f{SYMB\_TO\_ALG}
 allow to change an expression which is in the algebraic standard
 quotient form into a prefix lisp form and viceversa. This is made
 in such a way that the symbol \verb+list+ which appears in the
 algebraic mode desappear in the symbolic form (there it becomes
 a parenthesis ``()'' ) and it is reintroduced in the translation
 from a symbolic prefix lisp expression to an algebraic one.
 Here, is an example, showing how the wellknown lisp function
 \f{FLATTENS} can be trivially transposed inside the algebraic mode:
 \begin{verbatim}
 algebraic procedure ecrase x;
 lisp symb_to_alg flattens1 alg_to_symb algebraic x;
 symbolic procedure flattens1 x;
 % ll; ==> ((A B) ((C D) E))
 % flattens1 ll; (A B C D E)
 if atom x then list x else
 if cdr x then
 append(flattens1 car x, flattens1 cdr x)
 else flattens1 car x;
 \end{verbatim}
 gives, for instance,
 \begin{verbatim}
 ll:={a,{b,{c},d,e},{{{z}}}}$
 ECRASE ll; ==> {A, B, C, D, E, Z}
 \end{verbatim}
 \item[ii.]
 \f{LEADTERM} and \f{REDEXPR} are the algebraic equivalent of the
 symbolic functions \f{LT} and \f{RED}. They give, respectively, the
 {\em leading term} and the {\em reductum} of a polynomial. They also work
 for rational functions. Their interest lies in the fact that they do not
 require to extract the main variable. They work according to the current
 ordering of the system:
 \begin{verbatim}
 pol:=x++y+z$
 LEADTERM pol; ==> x
 korder y,x,z;
 LEADTERM pol; ==> y
 REDEXPR pol; ==> x + z
 \end{verbatim}
 By default, the representation of multivariate polynomials is recursive.
 It is justified since it is the one which takes the least of memory.
 With such a representation, the function \f{LEADTERM} does not necessarily
 extract a true monom. It extracts a monom in the leading indeterminate
 multiplied by a polynomial in the other indeterminates. However, very often,
 one needs to handle true monoms separately. In that case, one needs a
 polynomial in {\em distributive} form. Such a form is provided by the
 package GROEBNER (H. Melenk et al.). The facility there is, however,
 much too involved and the necessity to load the package makes interesting
 an elementary facility to write polynomials in a distributive form. So,
 a new switch is created to handle {\em distributed} polynomials. It is
 called {\tt DISTRIBUTE} and a new function \f{DISTRIBUTE} puts a
 polynomial in distributive form. With the switch put to {\bf on},
 \f{LEADTERM} gives {\bf true} monoms.
 \f{MONOM} transforms a polynomial into a list of monoms. It works
 {\em whatever the position of the switch} {\tt DISTRIBUTE}.
 \f{SPLITTERMS} is analoguous to \f{MONOM} except that it gives
 a list of two lists. The first sublist contains the positive terms
 while the second sublist contains the negative terms.
 \f{SPLITPLUSMINUS} gives a list whose first element is the positive
 part of the polynomial and its second element is its negative part.
 \item[iii.]
 Two complementary functions \f{LOWESTDEG} and \f{DIVPOL} are provided.
 The first takes a polynomial as its first argument and the name of an
 indeterminate as its second argument. It returns the {\em lowest degree}
 in that indeterminate. The second function takes two polynomials and
 returns both the quotient and its remainder.
 \ei
 \section{Handling of Transcendental Functions}
 \bi
 \item[i.]
 The functions \f{TRIGREDUCE} and \f{TRIGEXPAND} and the equivalent
 one for hyperbolic functions \f{HYPREDUCE} and \f{HYPEXPAND}
 make the transformations to multiple arguments and from
 multiple arguments to elementary arguments. Here, a simple example:
 \begin{verbatim}
 aa:=sin(x+y)$
 TRIGEXPAND aa; ==> SIN(X)*COS(Y) + SIN(Y)*COS(X)
 TRIGREDUCE ws; ==> SIN(Y + X)
 \end{verbatim}
 When a trigonometric or hyperbolic expression is symmetric with
 respect to the interchange of {\tt SIN (SINH)} and {\tt COS (COSH)},
 the application of \f{TRIG(HYP)REDUCE} may often lead to great
 simplifications. However, if it is highly assymetric, the repeated
 application of \f{TRIG(HYP)REDUCE} followed by the use of
 \f{TRIG(HYP)EXPAND} will lead to {\em more} complicated
 but more symmetric expressions:
 \begin{verbatim}
 aa:=(sin(x)^2+cos(x)^2)^3$
 TRIGREDUCE aa; ==> 1
 \end{verbatim}
 \pagebreak
 \begin{verbatim}
 bb:=1+sin(x)^3$
 TRIGREDUCE bb; ==>
  SIN(3*X) + 3*SIN(X) + 4
 
 4
 TRIGEXPAND ws; ==>
 3 2
 SIN(X)  3*SIN(X)*COS(X) + 3*SIN(X) + 4
 
 4
 \end{verbatim}
 \item[ii.]
 Two functions for the manipulation of logarithmic expression are
 available. They are \f{PLUSLOG} and \f{CONCSUMLOG}.
 Both function take an arbitrary expression as argument. The first one
 transforms products, ratios and exponentiations which appear under a
 \verb+log+ respectively into sums, differnces of log's and products
 of exponents with log's. The second one does the opposite job.
 \ei
 \section{Handling of ndimensional Vectors}
 Explicit vectors in {\tt EUCLIDEAN} space may be represented by
 listlike or baglike objects of depth 1.
 The components may be bags but may {\bf not} be lists.
 Functions are provided to do the sum, the difference and the
 scalar product. When, spacedimension is three, there are also functions
 for the cross and mixed products.
 \f{SUMVECT, MINVECT, SCALVECT, CROSSVECT} have two arguments.
 \f{MPVECT} has three arguments. The following example
 is sufficient to explain how they work:
 \begin{verbatim}
 l:={1,2,3}$
 ll:=list(a,b,c)$
 SUMVECT(l,ll); ==> {A + 1,B + 2,C + 3}
 MINVECT(l,ll); ==> {  A + 1,  B + 2,  C + 3}
 SCALVECT(l,ll); ==> A + 2*B + 3*C
 CROSSVECT(l,ll); ==> {  3*B + 2*C,3*A  C,  2*A + B}
 MPVECT(l,ll,l); ==> 0
 \end{verbatim}
 \section{Handling of Grassmann Operators}
 Grassman variables are often used in physics. For them the multiplication
 operation is associative, distributive but anticommutative. The
 {\tt KERNEL} of \REDUCE\ does not provide it. However, implementing
 it in full generality would almost
 certainly decrease the overall efficiency of the system. This small
 module together with the declaration of antisymmetry for operators is
 enough to deal with most calculations. The reason is, that a
 product of similar anticommuting kernels can easily be transformed
 into an antisymmetric operator with as many indices as the number of
 these kernels. Moreover, one may also issue pattern matching rules
 to implement the anticommutativity of the product.
 The functions in this module represent the minimum functionality
 required to identify them and to handle their specific features.
 \f{PUTGRASS} is a (nary) command which give identifiers the property
 to be the names of Grassmann kernels. \f{REMGRASS} removes this property.
 \f{GRASSP} is a boolean function which detects grassmann kernels.
 \f{GRASSPARITY} takes a {\bf monom} as argument and gives its parity.
 If the monom is a simple grassmann kernel it returns 1.
 \f{GHOSTFACTOR} has two arguments. Each one is a monom. It is equal to
 \begin{verbatim}
 (1)**(GRASSPARITY u * GRASSPARITY v)
 \end{verbatim}
 Here is an illustration to show how the above functions work:
 \begin{verbatim}
 PUTGRASS eta; ==> t
 if GRASSP eta(1) then "grassmann kernel"; ==>
 grassmann kernel
 aa:=eta(1)*eta(2)eta(2)*eta(1); ==>
 AA :=  ETA(2)*ETA(1) + ETA(1)*ETA(2)
 GRASSPARITY eta(1); ==> 1
 GRASSPARITY (eta(1)*eta(2)); ==> 0
 GHOSTFACTOR(eta(1),eta(2)); ==> 1
 grasskernel:=
 {eta(~x)*eta(~y) => eta y * eta x when nordp(x,y),
 (~x)*(~x) => 0 when grassp x};
 exp:=eta(1)^2$
 exp where grasskernel; ==> 0
 aa where grasskernel; ==>  2*ETA(2)*ETA(1)
 \end{verbatim}
 \section{Handling of Matrices}
 This module provides functions for handling matrices more comfortably.
 \bi
 \item[i.]
 Often, one needs to construct some {\tt UNIT} matrix of
 a given dimension. This construction is done by the system thanks
 to the function \f{UNITMAT}. it is a nary function. The command is
 \begin{verbatim}
 UNITMAT M1(n1), M2(n2), .....Mi(ni) ;
 \end{verbatim}
 where \verb+M1,...Mi+ are names of matrices and
 \verb+ n1, n2, ..., ni+ are integers .
 \f{MKIDM} is a generalization of \f{MKID}. It allows to connect
 two or several matrices. If \verb+u+ and \verb+u1+ are two matrices,
 one can go from one to the other:
 \begin{verbatim}
 matrix u(2,2);$ unitmat u1(2)$
 u1; ==>
 [1 0]
 [ ]
 [0 1]
 mkidm(u,1); ==>
 [1 0]
 [ ]
 [0 1]
 \end{verbatim}
 This function allows to make loops on matrices like in the following
 illustration. If \verb+U, U1, U2,.., U5+ are matrices:
 \begin{verbatim}
 FOR I:=1:5 DO U:=UMKIDM(U,I);
 \end{verbatim}
 can be issued.
 \item[ii.]
 The next functions map matrices on baglike or listlike objects
 and conversely they generate matrices from bags or lists.
 \f{COERCEMAT} transforms the matrix \verb+U+ into a list of lists.
 The entry is
 \begin{verbatim}
 COERCEMAT(U,id)
 \end{verbatim}
 where \verb+id+ is equal to \verb+list+ othewise it transforms it into
 a bag of bags whose envelope is equal to \verb+id+.
 \f{BAGLMAT} does the opposite job. The {\bf first} argument is the
 baglike or listlike object while the second argument is the matrix
 identifier. The entry is
 \begin{verbatim}
 BAGLMAT(bgl,U)
 \end{verbatim}
 \verb+bgl+ becomes the matrix \verb+U+ . The transformation is
 {\bf not} done if \verb+U+ is {\em already} the name of a
 previously defined matrix. This is to avoid ACCIDENTAL redefinition
 of that matrix.
 \item[ii.]
 The functions \f{SUBMAT, MATEXTR, MATEXTC} take parts of a given matrix.
 \f{SUBMAT} has three arguments. The entry is
 \begin{verbatim}
 SUBMAT(U,nr,nc)
 \end{verbatim}
 The first is the matrix name, and the other two are the row and column
 numbers . It gives the
 submatrix obtained from \verb+U+ deleting the row \verb+nr+ and
 the column \verb+nc+.
 When one of them is equal to zero only column \verb+nc+
 or row \verb+nr+ is deleted.
 \f{MATEXTR} and \f{MATEXTC} extract a row or a column and place it into
 a listlike or baglike object.
 The entry are
 \begin{verbatim}
 MATEXTR(U,VN,nr)
 MATEXTC(U,VN,nc)
 \end{verbatim}
 where \verb+U+ is the matrix, \verb+VN+ is the ``vector name'',
 \verb+nr+ and \verb+nc+ are integers. If \verb+VN+ is equal
 to {\tt list} the vector is given as a list otherwise it is
 given as a bag.
 \item[iii.]
 Functions which manipulate matrices. They are
 \f{MATSUBR, MATSUBC, HCONCMAT, VCONCMAT, TPMAT, HERMAT}
 \f{MATSUBR MATSUBC} substitute rows and columns. They have three arguments.
 Entries are:
 \begin{verbatim}
 MATSUBR(U,bgl,nr)
 MATSUBC(U,bgl,nc)
 \end{verbatim}
 The meaning of the variables \verb+U, nr, nc+ is the same as above
 while \verb+bgl+ is a listlike or baglike vector.
 Its length should be compatible with the dimensions of the matrix.
 \f{HCONCMAT VCONCMAT} concatenate two matrices. The entries are
 \begin{verbatim}
 HCONCMAT(U,V)
 VCONCMAT(U,V)
 \end{verbatim}
 The first function concatenates horizontally, the second one
 concatenates vertically. The dimensions must match.
 \f{TPMAT} makes the tensor product of two matrices. It is also an
 {\em infix} function. The entry is
 \begin{verbatim}
 TPMAT(U,V) or U TPMAT V
 \end{verbatim}
 \f{HERMAT} takes the hermitian conjuguate of a matrix
 The entry is
 \begin{verbatim}
 HERMAT(U,HU)
 \end{verbatim}
 where \verb+HU+ is the identifier for the hermitian matrix of \verb+U+.
 It should {\bf unassigned} for this function to work successfully.
 This is done on purpose to prevent accidental redefinition of an already
 used identifier .
 \item[iv.]
 \f{SETELMAT GETELMAT} are functions of two integers. The first one
 reset the element \verb+(i,j)+ while the second one extract an
 element identified by \verb+(i,j)+. They may be useful when
 dealing with matrices {\em inside procedures}.
 \ei
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