reduce-historical/r36/help/DECLARE.TEX

\section{Declarations}

\begin{Command}{ALGEBRAIC}
\index{evaluation}
The \name{algebraic} command changes REDUCE's mode of operation to
algebraic.  When \name{algebraic} is used as an operator (with an
argument inside parentheses) that argument is evaluated in algebraic
mode, but REDUCE's mode is not changed.

\begin{Examples}
algebraic; \\
symbolic;                  &      NIL \\
algebraic(x**2);           &      X^{2} \\
x**2;                      &      \begin{multilineoutput}{6cm}
  ***** The symbol X has no value.
\end{multilineoutput}
\end{Examples}

\begin{Comments}
REDUCE's symbolic mode does not know about most algebraic commands.
Error messages in this mode may also depend on the particular Lisp
used for the REDUCE implementation.
% You can tell that you are in algebraic mode if the numbered prompt
% contains a colon rather than an asterisk.
\end{Comments}
\end{Command}


\begin{Declaration}{ANTISYMMETRIC}
When an operator is declared \name{antisymmetric}, its arguments are
reordered to conform to the internal ordering of the system.  If an odd
number of argument interchanges are required to do this ordering,
the sign of the expression is changed.

\begin{Syntax}
\name{antisymmetric} \meta{identifier}\{\name{,}\meta{identifier}\}\optional
\end{Syntax}

\meta{identifier} is an identifier that has been declared as an operator.

\begin{Examples}
operator m,n; \\
antisymmetric m,n; \\
m(x,n(1,2));               &         - M( - N(2,1),X) \\
operator p; \\
antisymmetric p; \\
p(a,b,c);                  &           P(A,B,C) \\
p(b,a,c);                  &         - P(A,B,C)
\end{Examples}

\begin{Comments}
If \meta{identifier} has not been declared an operator, the flag
\name{antisymmetric} is still attached to it.  When \meta{identifier} is
subsequently used as an operator, the message \name{Declare} \meta{identifier}
 \name{operator? (Y or N)} is printed.  If the user replies \name{y}, the
antisymmetric property of the operator is used.

Note in the first example, identifiers are customarily ordered
alphabetically, while numbers are ordered from largest to smallest.
The operators may have any desired number of arguments (less than 128).
\end{Comments}
\end{Declaration}


\begin{Declaration}{ARRAY}
The \name{array} declaration declares a list of identifiers to be of type
\name{array}, and sets all their entries to 0.
\begin{Syntax}
\name{array}
\meta{identifier}\(\meta{dimensions}\)
              \{\name{,}\meta{identifier}\(\meta{dimensions}\)\}\optional
\end{Syntax}

\meta{identifier} may be any valid REDUCE identifier.  If the identifier
was already an array, a warning message is given that the array has been
redefined.  \meta{dimensions} are of form
 \meta{integer}\{,\meta{integer}\}\optional.

\begin{Examples}
array a(2,5),b(3,3,3),c(200);  \\
array a(3,5);              &     *** ARRAY A REDEFINED \\
a(3,4);                    &     0 \\
length a;                  &     {4,6}
\end{Examples}

\begin{Comments}
Arrays are always global, even if defined inside a procedure or block
statement.  Their status as an array remains until the variable is
reset by \nameref{clear}.  Arrays may not have the same names as operators,
procedures or scalar variables.

Array elements are referred to by the usual notation: \name{a(i,j)}
returns the jth element of the ith row.  The \nameref{assign}ment operator
\name{:=} is used to put values into the array.  Arrays as a whole
cannot be subject to assignment by \nameref{let} or \name{:=} ; the
assignment operator \name{:=} is only valid for individual elements.

When you use \nameref{let} on an array element, the contents of that
element become the argument to \name{let}.  Thus, if the element
contains a number or some other expression that is not a valid argument
for this command, you get an error message.  If the element contains an
identifier, the identifier has the substitution rule attached to it
globally.  The same behavior occurs with \nameref{clear}.  If the array
element contains an identifier or simple\_expression, it is cleared.  Do
\meta{not} use \name{clear} to try to set an array element to 0.  Because
of the side effects of either \name{let} or \name{clear}, it is unwise
to apply either of these to array elements.

Array indices always start with 0, so that the declaration \name{array a(5)}
sets aside 6 units of space, indexed from 0 through 5, and initializes
them to 0.  The \nameref{length} command returns a list of the true number of
elements in each dimension.
\end{Comments}
\end{Declaration}


\begin{Command}{CLEAR}
The \name{clear} command is used to remove assignments or remove substitution
rules from any expression.

\begin{Syntax}
\name{clear} \meta{identifier}\{,\meta{identifier}\}\repeated \ or \\
\meta{let-type statement} \name{clear} \meta{identifier}
\end{Syntax}

\meta{identifier} can be any \name{scalar}, \nameref{matrix}, 
or \nameref{array} variable or
\nameref{procedure} name.  \meta{let-type statement} can be any general 
or specific \nameref{let} statement (see below in Comments).

\begin{Examples}
array a(2,3); \\
a(2,2) := 15;                &             A(2,2) := 15 \\
clear a;                     \\
a(2,2);                      &             Declare A operator? (Y or N) \\
let x = y + z; \\
sin(x);                      &             SIN(Y + Z) \\
clear x; \\
sin(x);                      &             SIN(X) \\
let x**5 = 7; \\
clear x; \\
x**5;                        &             7 \\
clear x**5; \\
x**5;                        &             X^{5}
\end{Examples}
\begin{Comments}

Although it is not a good idea, operators of the same name but taking
different numbers of arguments can be defined.  Using a \name{clear} statement
on any of these operators clears every one with the same name, even if the
number of arguments is different.

The \name{clear} command is used to ``forget" matrices, arrays, operators
and scalar variables, returning their identifiers to the pristine state
to be used for other purposes.  When \name{clear} is applied to array
elements, the contents of the array element becomes the argument for
\name{clear}.  Thus, you get an error message if the element contains a
number, or some other expression that is not a legal argument to
\name{clear}.  If the element contains an identifier, it is cleared.
When clear is applied to matrix elements, an error message is returned
if the element evaluates to a number, otherwise there is no effect.  Do
{\em not} try to use \name{clear} to set array or matrix elements to 0.
You will not be pleased with the results.

If you are trying to clear power or product substitution rules made with
either \nameref{let} or \nameref{forall}\ldots\name{let}, you must
reproduce the rule, exactly as you typed it with the same arguments, up to
but not including the equal sign, using the word \name{clear} instead of
the word \name{let}.  This is shown in the last example.  Any other type of
\name{let} or \name{forall}\ldots\name{let} substitution can be cleared
with just the variable or operator name.  \nameref{match} behaves the same as
\nameref{let} in this situation.  There is a more complicated example under
\nameref{forall}.

\end{Comments}
\end{Command}


\begin{Command}{CLEARRULES}
\index{rule}
\begin{Syntax}
\name{clearrules} \meta{list}\{,\meta{list}\}\repeated
\end{Syntax}

The operator \name{clearrules} is used to remove previously defined
\nameref{rule} lists from the system.  \meta{list} can be an explicit rule
list, or evaluate to a rule list.

\begin{Examples}
trig1 := {cos(~x)*cos(~y) => (cos(x+y)+cos(x-y))/2,
	  cos(~x)*sin(~y) => (sin(x+y)-sin(x-y))/2,
	  sin(~x)*sin(~y) => (cos(x-y)-cos(x+y))/2,
	  cos(~x)^2       => (1+cos(2*x))/2,
	  sin(~x)^2       => (1-cos(2*x))/2}$ \\
let trig1;
cos(a)*cos(b); &
\rfrac{COS(A - B) + COS(A + B)}{2} \\
clearrules trig1;
cos(a)*cos(b); &  COS(A)*COS(B)
\end{Examples}

\end{Command}


\begin{Command}{DEFINE}
The command \name{define} allows you to supply a new name for an identifier
or replace it by any valid REDUCE expression.

\begin{Syntax}
\name{define} \meta{identifier}\name{=}\meta{substitution}
      \{\name{,}\meta{identifier}\name{=}\meta{substitution}\}\optional
\end{Syntax}


\meta{identifier} is any valid REDUCE identifier, \meta{substitution} can be a
number, an identifier, an operator, a reserved word, or an expression.

\begin{Examples}

define is= :=, xx=y+z; \\

a is 10;                     &            A := 10 \\

xx**2;                       &            Y^{2}  + 2*Y*Z + Z^{2} \\

xx := 10;                    &            Y + Z := 10
\end{Examples}

\begin{Comments}
The renaming is done at the input level, and therefore takes precedence
over any other replacement or substitution declared for the same identifier.
It remains in effect until the end of the REDUCE session.  Be careful with
it, since you cannot easily undo it without ending the session.
\end{Comments}
\end{Command}


\begin{Declaration}{DEPEND}
\index{dependency}
\name{depend} declares that its first argument depends on the rest of its
arguments.

\begin{Syntax}
\name{depend} \meta{kernel}\{\name{,}\meta{kernel}\}\repeated
\end{Syntax}

\meta{kernel} must be a legal variable name or a prefix operator (see
\nameref{kernel}).

\begin{Examples}

depend y,x; \\

df(y**2,x);                 &            2*DF(Y,X)*Y \\

depend z,cos(x),y; \\

df(sin(z),cos(x));          &            COS(Z)*DF(Z,COS(X)) \\

df(z**2,x);                 &            2*DF(Z,X)*Z \\

nodepend z,y; \\

df(z**2,x);                 &            2*DF(Z,X)*Z \\

cc := df(y**2,x);           &            CC := 2*DF(Y,X)*Y \\

y := tan x;                 &            Y := TAN(X); \\

cc;                         &            2*TAN(X)*(TAN(X)^{2}  + 1)
\end{Examples}
\begin{Comments}
Dependencies can be removed by using the declaration \nameref{nodepend}.
The differentiation operator uses this information, as shown in the
examples above.  Linear operators also use knowledge of dependencies
(see \nameref{linear}).  Note that dependencies can be nested:  Having
declared \IFTEX{$y$}{y} to depend on \IFTEX{$x$}{x}, and \IFTEX{$z$}{z}
to depend on \IFTEX{$y$}{y}, we
see that the chain rule was applied to the derivative of a function of
\IFTEX{$z$}{z} with respect to \IFTEX{$x$}{x}.   If the explicit function of the
dependency is later entered into the system, terms with \name{DF(Y,X)},
for example, are expanded when they are displayed again, as shown in the
last example.  The boolean operator \nameref{freeof} allows you to
check the dependency between two algebraic objects.
\end{Comments}
\end{Declaration}


\begin{Declaration}{EVEN}
\begin{Syntax}
\name{even} \meta{identifier}\{,\meta{identifier}\}\optional
\end{Syntax}
This declaration is used to declare an operator {\em even} in its first
argument.  Expressions involving an operator declared in this manner are
transformed if the first argument contains a minus sign.  Any other
arguments are not affected.
\begin{Examples}
	even f; \\
	f(-a)    &    F(A) \\
	f(-a,-b) &    F(A,-B)
\end{Examples}
\end{Declaration}


\begin{Declaration}{FACTOR}
\index{output}
When a kernel is declared by \name{factor}, all terms involving fixed
powers of that kernel are printed as a product of the fixed powers and
the rest of the terms. 
\begin{Syntax}
\name{factor} \meta{kernel} \{\name{,}\meta{kernel}\}\optional
\end{Syntax}

\meta{kernel} must be a \nameref{kernel} or a \nameref{list} of
\name{kernel}s.

\begin{Examples}
a := (x + y + z)**2;         &
     A := X^{2} + 2*X*Y + 2*X*Z + Y^{2} + 2*Y*Z + Z^{2} \\
factor y; \\
a;                           &
     Y^{2} + 2*Y*(X + Z) + X^{2} + 2*X*Z + Z^{2} \\
factor sin(x); \\
c := df(sin(x)**4*x**2*z,x); &
     C := 2*SIN(X)^{4}*X*Z + 4*SIN(X)^{3}*COS(X)*X^{2}*Z \\
remfac sin(x); \\
c;                           &
     2*SIN(X)^{3}*X*Z*(2*COS(X)*X + SIN(X))
\end{Examples}

\begin{Comments}
Use the \name{factor} declaration to display variables of interest so that
you can see their powers more clearly, as shown in the example.  Remove
this special treatment with the declaration \nameref{remfac}.  The
\name{factor} declaration is only effective when the switch \nameref{pri} 
is on.

The \name{factor} declaration is not a factoring command; to factor
expressions use the \nameref{factor} switch or the \nameref{factorize} command.

The \name{factor} declaration is helpful in such cases as Taylor polynomials
where the explicit powers of the variable are expected at the top level, not
buried in various factored forms.
\end{Comments}
\end{Declaration}


\begin{Command}{FORALL}
\index{substitution}
The \name{forall} or (preferably) \name{for all} command is used as a
modifier for \nameref{let} statements, indicating the universal applicability
of the rule, with possible qualifications.
\begin{Syntax}
\name{for all} \meta{identifier}\{,\meta{identifier}\}\optional\ \name{let}
\meta{let statement}

{\em or}

\name{for all} \meta{identifier}\{,\meta{identifier}\}\optional
\ \name{such that} \meta{condition} \name{let} \meta{let statement}
\end{Syntax}


\meta{identifier} may be any valid REDUCE identifier, \meta{let statement}
can be an operator, a product or power, or a group or block statement.
\meta{condition} must be a logical or comparison operator returning true or
false.

\begin{Examples}
for all x let f(x) = sin(x**2);
			     &     Declare F operator ? (Y or N) \\
y \\
f(a);                        &     SIN(A^{2}) \\
operator pos; \\
for all x such that x>=0 let pos(x) = sqrt(x + 1); \\
pos(5);                      &     SQRT(6) \\
pos(-5);                     &     POS(-5) \\
clear pos; \\
pos(5);                      &     Declare POS operator ? (Y or N) \\
for all a such that numberp a let x**a = 1; \\
x**4;                        &     1 \\
clear x**a;                  &     *** X**A not found \\
for all a  clear x**a; \\
x**4;                        &     1 \\
for all a such that numberp a clear x**a; \\
x**4;                        &     X^{4}
\end{Examples}

\begin{Comments}
Substitution rules defined by \name{for all} or \name{for
all}\ldots\name{such that} commands that involve products or powers are
cleared by reproducing the command, with exactly the same variable names
used, up to but not including the equal sign, with \nameref{clear}
replacing \name{let}, as shown in the last example.  Other substitutions
involving variables or operator names can be cleared with just the name,
like any other variable.

The \nameref{match} command can also be used in product and power substitutions.
The syntax of its use and clearing is exactly like \name{let}.  A \name{match}
substitution only replaces the term if it is exactly like the pattern, for
example \name{match x**5 = 1} replaces only terms of \name{x**5} and not
terms of higher powers.

It is easier to declare your potential operator before defining the 
\name{for all} rule, since the system will ask you to declare it an
operator anyway.  Names of declared arrays or matrices or scalar
variables are invalid as operator names, to avoid ambiguity.  Either
\name{for all}\ldots\name{let} statements or procedures are often used to define
operators.  One difference is that procedures implement ``call by value"
meaning that assignments involving their formal parameters do not change
the calling variables that replace them.  If you use assignment statements
on the formal parameters in a \name{for all}\ldots\name{let} statement, the
effects are seen in the calling variables.  Be careful not to redefine a
system operator unless you mean it: the statement \name{for all x let
sin(x)=0;} has exactly that effect, and the usual definition for sin(x) has
been lost for the remainder of the REDUCE session. \end{Comments}
\end{Command}

\begin{Declaration}{INFIX}
\index{operator}
\name{infix} declares identifiers to be infix operators.
\begin{Syntax}
\name{infix} \meta{identifier}\{,\meta{identifier}\}\optional
\end{Syntax}

\meta{identifier} can be any valid REDUCE identifier, which has not already
been declared an operator, array or matrix, and is not reserved by the
system.

\begin{Examples}
infix aa; \\
for all x,y let aa(x,y) = cos(x)*cos(y) - sin(x)*sin(y); \\
x aa y;                      &       COS(X)*COS(Y) - SIN(X)*SIN(Y) \\
pi/3 aa pi/2;                &        - \rfrac{SQRT(3)}{2} \\
aa(pi,pi);                   &           1
\end{Examples}

\begin{Comments}
A \nameref{let} statement must be used to attach functionality to 
the operator. Note that the operator is defined in prefix form in 
the \name{let} statement.
After its definition, the operator may be used in either prefix or infix
mode.  The above operator \IFTEX{$aa$}{aa} finds the cosine of the sum 
of two angles by the formula 
\begin{TEX}
\begin{displaymath}
\cos(x+y) = \cos x \cos y  - \sin x \sin y.
\end{displaymath}
\end{TEX}
\begin{INFO}
cos(x+y) = cos(x)*cos(y) - sin(x)*sin(y).
\end{INFO}
Precedence may be attached to infix operators with the
\nameref{precedence} declaration.

User-defined infix operators may be used in prefix form.  If they are used
in infix form, a space must be left on each side of the operator to avoid
ambiguity.  Infix operators are always binary.
\end{Comments}
\end{Declaration}


\begin{Declaration}{INTEGER}
The \name{integer} declaration must be made immediately after a
\nameref{begin} (or other variable declaration such as \nameref{real}
and \nameref{scalar}) and declares local integer variables.  They are
initialized to 0.
\begin{Syntax}
\name{integer} \meta{identifier}\{,\meta{identifier}\}\optional
\end{Syntax}

\meta{identifier} may be any valid REDUCE identifier, except
\name{t} or \name{nil}.

\begin{Comments}
Integer variables remain local, and do not share values with variables of
the same name outside the \nameref{begin}\ldots\name{end} block.  When the
block is finished, the variables are removed.  You may use the words
\nameref{real} or \nameref{scalar} in the place of \name{integer}.
\name{integer} does not indicate typechecking by the
current REDUCE; it is only for your own information.  Declaration
statements must immediately follow the \name{begin}, without a semicolon
between \name{begin} and the first variable declaration.

Any variables used inside \name{begin}\ldots\name{end} blocks that were not
declared \name{scalar}, \name{real} or \name{integer} are global, and any
change made to them inside the block affects their global value.  Any
\ref{array} or \ref{matrix} declared inside a block is always global.
\end{Comments}
\end{Declaration}


\begin{Declaration}{KORDER}
\index{kernel order}\index{variable order}\index{order}
The \name{korder} declaration changes the internal canonical ordering of
kernels.
\begin{Syntax}
\name{korder} \meta{kernel}\{\name{,}\meta{kernel}\}\optional
\end{Syntax}

\meta{kernel} must be a REDUCE \nameref{kernel} or a \nameref{list} of
\name{kernel}s.

\begin{Comments}
The declaration \name{korder} changes the internal ordering, but not the print
ordering, so the effects cannot be seen on output.  However, in some 
calculations, the order of the variables can have significant effects on the 
time and space demands of a calculation.  If you are doing a demanding 
calculation with several kernels, you can experiment with changing the 
canonical ordering to improve behavior.

The first kernel in the argument list is given the highest priority, the
second gets the next highest, and so on.  Kernels not named in a
\name{korder} ordering otherwise.  A new \name{korder} declaration replaces 
the previous one.  To return to canonical ordering, use the command 
\name{korder nil}.

To change the print ordering, use the declaration \nameref{order}.
\end{Comments}
\end{Declaration}


\begin{Command}{LET}
\index{substitution}\index{rule}
The \name{let} command defines general or specific substitution rules.
\begin{Syntax}
\name{let} \meta{identifier} \name{=} \meta{expression}\{,\meta{identifier}
\name{=} \meta{expression}\}\optional
\end{Syntax}


\meta{identifier} can be any valid REDUCE identifier except an array, and in
some cases can be an expression; \meta{expression} can be any valid REDUCE
expression.   

\begin{Examples}
let a = sin(x); \\
b := a;                      &          B := SIN X; \\
let c = a;  \\
exp(a);                      &          E^{SIN(X)} \\
a := x**2;                   &          A := X^{2} \\
exp(a);                      &          E^{X^{2}} \\
exp(b);                      &          E^{SIN(X)} \\
exp(c);                      &          E^{X^{2}} \\
let m + n = p; \\
(m + n)**5;                  &          P^{5} \\
%%%let z**6 = 0; \\
%%%z**3*(z + 1)**4;             &          Z^{3}*(6*Z^{2}  + 4*Z + 1) \\
operator h; \\
let h(u,v) = u - v; \\
h(u,v);                      &          U - V \\
h(x,y);                      &          H(X,Y) \\
array q(10); \\
let q(1) = 15;               &   ***** Substitution for 0 not allowed
\end{Examples}

The \name{let} command is also used to activate a \name{rule sets}.
\begin{Syntax}
\name{let} \meta{list}\{,\meta{list}\}\repeated
\end{Syntax}

\meta{list} can be an explicit \nameref{rule} \name{list}, or evaluate 
to a rule list.

\begin{Examples}
trig1 := {cos(~x)*cos(~y) => (cos(x+y)+cos(x-y))/2,
	  cos(~x)*sin(~y) => (sin(x+y)-sin(x-y))/2,
	  sin(~x)*sin(~y) => (cos(x-y)-cos(x+y))/2,
	  cos(~x)^2       => (1+cos(2*x))/2,
	  sin(~x)^2       => (1-cos(2*x))/2}$ \\
let trig1;
cos(a)*cos(b); &
\rfrac{COS(A - B) + COS(A + B)}{2}
\end{Examples}

\begin{Comments}
A \name{let} command returns no value, though the substitution rule is
entered.  Assignment rules made by \nameref{assign} and \name{let} 
rules are at the
same level, and cancel each other.  There is a difference in their
operation, however, as shown in the first example:  a \name{let} assignment
tracks the changes in what it is assigned to, while a \name{:=} assignment
is fixed at the value it originally had.  

The use of expressions as left-hand sides of \name{let} statements is a 
little complicated.  The rules of operation are:
\begin{itemize}

\item[(i)]
Expressions of the form A*B = C do not change A, B or C, but set A*B to C.

\item[(ii)]
Expressions of the form A+B = C substitute C - B for A, but do not change
B or C.

\item[(iii)]
Expressions of the form A-B = C substitute B + C for A, but do not change
B or C.

\item[(iv)]
Expressions of the form A/B = C substitute B*C for A, but do not change B or
C.

\item[(v)]
Expressions of the form A**N = C substitute C for A**N in every expression of
a power of A to N or greater.  An asymptotic command such as A**N = 0 sets
all terms involving A to powers greater than or equal to N to 0.  Finite
fields may be generated by requiring modular arithmetic (the \nameref{modular}
switch) and defining the primitive polynomial via a \name{let} statement.

\end{itemize}
\name{let} substitutions involving expressions are cleared by using
the \nameref{clear} command with exactly the same expression.

Note when a simple \name{let} statement is used to assign functionality to an
operator, it is valid only for the exact identifiers used.  For the use of the 
\name{let} command to attach more general functionality to an operator,
see \nameref{forall}.

Arrays as a whole cannot be arguments to \name{let} statements, but
matrices as a whole can be legal arguments, provided both arguments are
matrices.  However, it is important to note that the two matrices are then
linked.  Any change to an element of one matrix changes the corresponding
value in the other.  Unless you want this behavior, you should not use
\name{let} for matrices.  The assignment operator \nameref{assign} can be used
for non-tracking assignments, avoiding the side effects.  Matrices are
redimensioned as needed in \name{let} statements.

When array or matrix elements are used as the left-hand side of \name{let}
statements, the contents of that element is used as the argument.  When the 
contents is a number or some other expression that is not a valid left-hand 
side for \name{let}, you get an error message.  If the contents is an
identifier or simple expression, the \name{let} rule is globally attached 
to that identifier, and is in effect not only inside the array or matrix, 
but everywhere.  Because of such unwanted side effects, you should not 
use \name{let} with array or matrix elements.  The assignment operator 
\name{:=} can be used to put values into array or matrix elements without 
the side effects.

Local variables declared inside \name{begin}\ldots\name{end} blocks cannot
be used as the left-hand side of \name{let} statements.  However,
\nameref{begin}\ldots\name{end} blocks themselves can be used as the
right-hand side of \name{let} statements.  The construction:
\begin{Syntax}
        \name{for all} \meta{vars}
        \name{let}\meta{operator}\(\meta{vars}\)\name{=}\meta{block}
\end{Syntax}
is an alternative to the 
\begin{Syntax}
        \name{procedure} \meta{name}\(\meta{vars}\)\name{;}\meta{block}
\end{Syntax}
construction.  One important difference between the two constructions is that
the \meta{vars} as formal parameters to a procedure have their global values
protected against change by the procedure, while the \meta{vars} of a 
\name{let} statement are changed globally by its actions. 

Be careful in using a construction such as \name{let x = x + 1} except inside
a controlled loop statement.  The process of resubstitution continues until 
a stack overflow message is given.  

The \name{let} statement may be used to make global changes to variables from
inside procedures.  If \name{x} is a formal parameter to a procedure, the 
command \name{let x = }\ldots makes the change to the calling variable.  
For example, if a procedure was defined by 
\begin{verbatim}
        procedure f(x,y);
        let x = 15;
\end{verbatim}
and the procedure was called as
\begin{verbatim}
        f(a,b);
\end{verbatim}
\name{a} would have its value changed to 15.  Be careful when using \name{let}
statements inside procedures to avoid unwanted side effects.

It is also important to be careful when replacing \name{let} statements with
other \name{let} statements.  The overlapping of these substitutions can be
unpredictable.  Ordinarily the latest-entered rule is the first to be applied.
Sometimes the previous rule is superseded completely; other times it stays
around as a special case.  The order of entering a set of related \name{let}
expressions is very important to their eventual behavior.  The best
approach is to assume that the rules will be applied in an arbitrary order.
%%%Be sure to research this subject carefully when you are programming
%%%mathematical operations using related \name{let} substitutions.
\end{Comments}
\end{Command}


\begin{Declaration}{LINEAR}
\index{operator}
An operator can be declared linear in its first argument over powers of
its second argument by the declaration \name{linear.}
\begin{Syntax}
 \name{linear} \meta{operator}\{\name{,}\meta{operator}\}\optional
\end{Syntax}
\meta{operator} must have been declared to be an operator.  Be careful not
to use a system operator name, because this command may change its definition.
The operator being declared must have at least two arguments, and the
second one must be a \nameref{kernel}.

\begin{Examples}
operator f; \\
linear f; \\
f(0,x);                      &        0 \\
f(-y,x);                     &        - F(1,X)*Y \\
f(y+z,x);                    &        F(1,X)*(Y + Z) \\
f(y*z,x);                    &        F(1,X)*Y*Z \\
depend z,x; \\
f(y*z,x);                    &        F(Z,X)*Y \\
f(y/z,x);                    &        F(\rfrac{1}{Z},X)*Y \\
depend y,x;  \\
f(y/z,x);                    &        F(\rfrac{Y}{Z},X) \\
nodepend z,x; \\
f(y/z,x);                    &        \rfrac{F(Y,X)}{Z} \\
f(2*e**sin(x),x);            &        2*F(E^{SIN(X)},X)
\end{Examples}

\begin{Comments}
Even when the operator has not had its functionality attached, it exhibits
linear properties as shown in the examples.  Notice the difference when
dependencies are added.  Dependencies are also in effect when the operator's
first argument contains its second, as in the last line above.

For a fully-developed example of the use of linear operators, refer to the
article in the \meta{Journal of Computational Physics}, Vol. 14 (1974), pp.
301-317, ``Analytic Computation of Some Integrals in Fourth Order Quantum
Electrodynamics," by J.A. Fox and A.C. Hearn.  The article includes the
complete listing of REDUCE procedures used for this work.
\end{Comments}
\end{Declaration}


\begin{Declaration}{LINELENGTH}
\index{output}
The \name{linelength} declaration sets the length of the output line. Default
is 80.
\begin{Syntax}
\name{linelength} \meta{expression}
\end{Syntax}

To change the linelength,
\meta{expression} must evaluate to a positive integer less than 128
(although this varies from system to system), and should not be less than
20 or so for proper operation.

\begin{Comments}
% If you want to use a thin window to set beside another window,
% reset the linelength shorter to avoid problems with raised exponents.
\name{linelength} returns the previous linelength.  If you want the current
linelength value, but not change it, say \name{linelength nil}.
\end{Comments}
\end{Declaration}


\begin{Command}{LISP}
The \name{lisp} command changes REDUCE's mode of operation to symbolic.  When
\name{lisp} is followed by an expression, that expression is evaluated in
symbolic mode, but REDUCE's mode is not changed.  This command is
equivalent to \nameref{symbolic}.

\begin{Examples}
lisp;                       &            NIL \\
car '(a b c d e);           &            A  \\
algebraic; \\
c := (lisp car '(first second))**2; \\
			    &           C := FIRST^{2}

\end{Examples}

\end{Command}


\begin{Declaration}{LISTARGP}
\index{list}\index{argument}
\begin{Syntax}
\name{listargp} \meta{operator}\{\name{,}\meta{operator}\}\optional
\end{Syntax}
If an operator other than those specifically defined for lists is given a
single argument that is a \nameref{list}, then the result of this 
operation will be a list in which that operator is applied to each element 
of the list.
This process can be inhibited for a specific operator, or list of operators,
by using the declaration \name{listargp}.

\begin{Examples}
log {a,b,c}; & {LOG(A),LOG(B),LOG(C)} \\
listargp log; \\
log {a,b,c}; & LOG({A,B,C})
\end{Examples}

\begin{Comments}
It is possible to inhibit such distribution globally by turning on the
switch \nameref{listargs}.  In addition, if an operator has more than one
argument, no such distribution occurs, so \name{listargp} has no effect.
\end{Comments}

\end{Declaration}


\begin{Declaration}{NODEPEND}
\index{depend}
The \name{nodepend} declaration removes the dependency declared with
\nameref{depend}.
\begin{Syntax}
\name{nodepend} \meta{dep-kernel}\{,\meta{kernel}\}\repeated

\end{Syntax}


\meta{dep-kernel} must be a kernel that has had a dependency declared upon the 
one or more other kernels that are its other arguments.  

\begin{Examples}
depend y,x,z; \\
df(sin y,x);                 &        COS(Y)*DF(Y,X) \\
df(sin y,x,z);               &  COS(Y)*DF(Y,X,Z) - DF(Y,X)*DF(Y,Z)*SIN(Y) \\
nodepend y,z; \\
df(sin y,x);                 &        COS(Y)*DF(Y,X) \\
df(sin y,x,z);               &        0
\end{Examples}

\begin{Comments}
A warning message is printed if the dependency had not been declared by
\name{depend}.
\end{Comments}
\end{Declaration}


\begin{Command}{MATCH}
\index{substitution}
The \name{match} command is similar to the \nameref{let} command, except
that it matches only explicit powers in substitution.
\begin{Syntax}
\name{match} \meta{expr} \name{=} \meta{expression}\{,\meta{expr}
 \name{=}\meta{expression}\}\optional
\end{Syntax}

\meta{expr} is generally a term involving powers, and is limited by
the rules for the \nameref{let} command.  \meta{expression} may be
any valid REDUCE scalar expression.


\begin{Examples}
match c**2*a**2 = d;
(a+c)**4;                    &      
            A^{4}  + 4*A^{3}*C + 4*A*C^{3} + C^{4} + 6*D \\
match a+b = c; \\
a + 2*b;                     &       B + C \\
(a + b + c)**2;              &       
            A^{2}  - B^{2}  + 2*B*C + 3*C^{2} \\
clear a+b; \\
(a + b + c)**2;              &       
            A ^{2} + 2*A*B + 2*A*C + B^{2} + 2*B*C + C^{2} \\
let p*r = s; \\
match p*q = ss; \\
(a + p*r)**2;                &       A^{2} + 2*A*S + S^{2} \\
(a + p*q)**2;                &       A^{2}  + 2*A*SS + P^{2}*Q^{2}
\end{Examples}

\begin{Comments}
Note in the last example that \name{a + b} has been explicitly matched
after the squaring was done, replacing each single power of \name{a} by
\name{c - b}.  This kind of substitution, although following the rules, is
confusing and could lead to unrecognizable results.  It is better to use
\name{match} with explicit powers or products only.  \name{match} should
not be used inside procedures for the same reasons that \name{let} should
not be.

Unlike \nameref{let} substitutions, \name{match} substitutions are executed
after all other operations are complete.  The last example shows the
difference. \name{match} commands can be cleared by using \nameref{clear},
with exactly the expression that the original \name{match} took.
\name{match} commands can also be done more generally with \name{for all}
or \nameref{forall}\ldots\name{such that} commands.
\end{Comments}
\end{Command}


\begin{Declaration}{NONCOM}
\index{commutative}\index{non commutative}\index{operator}
\name{noncom} declares that already-declared operators are noncommutative
under multiplication.
\begin{Syntax}
\name{noncom} \meta{operator}\{,\meta{operator}\}\optional
\end{Syntax}

\meta{operator} must have been declared an \nameref{operator}, or a warning
message is given.

\begin{Examples}
operator f,h; \\
noncom f; \\
f(a)*f(b) - f(b)*f(a);       &         F(A)*F(B) - F(B)*F(A) \\
h(a)*h(b) - h(b)*h(a);       &         0 \\
operator comm; \\
\begin{multilineinput}
for all x,y such that x neq y and ordp(x,y)
        let f(x)*f(y) = f(y)*f(x) + comm(x,y);
\end{multilineinput}\\
f(1)*f(2);                   &         F(1)*F(2) \\
f(2)*f(1);                   &         COMM(2,1) + F(1)*F(2)
\end{Examples}

\begin{Comments}
The last example introduces the commutator of $f(x)$ and $f(y)$
for all x and y.  The equality check is to prevent an infinite loop.  The 
operator {\it f} can have other functionality attached to it if desired, or it 
can remain an indeterminate operator.
\end{Comments}
\end{Declaration}


\begin{Declaration}{NONZERO}
\index{operator}
\begin{Syntax}
\name{nonzero} \meta{identifier}\{,\meta{identifier}\}\optional
\end{Syntax}
If an \ref{operator} \name{f} is declared \nameref{odd}, then \name{f(0)}
is replaced by zero unless \name{f} is also declared {\em non zero} by the
declaration \name{nonzero}.
\begin{Examples}
	odd f; \\
	f(0)    &   0 \\
	nonzero f; \\
	f(0) &   F(0)
\end{Examples}
\end{Declaration}


\begin{Declaration}{ODD}
\index{operator}

\begin{Syntax}
\name{odd} \meta{identifier}\{,\meta{identifier}\}\optional
\end{Syntax}
This declaration is used to declare an operator {\em odd} in its first
argument.  Expressions involving an operator declared in this manner are
transformed if the first argument contains a minus sign.  Any other
arguments are not affected.
\begin{Examples}
	odd f; \\
	f(-a)    &   -F(A) \\
	f(-a,-b) &   -F(A,-B) \\
	f(a,-b)  &    F(A,-B)
\end{Examples}

\begin{Comments}

If say \name{f} is declared odd, then \name{f(0)} is replaced by zero
unless \name{f} is also declared {\em non zero} by the declaration
\nameref{nonzero}.
\end{Comments}
\end{Declaration}


\begin{Command}{OFF}
\index{switch}
The \name{off} command is used to turn switches off.
\begin{Syntax}
\name{off} \meta{switch}\{,\meta{switch}\}\optional
\end{Syntax}

\meta{switch} can be any \name{switch} name.  There is no problem if the
switch is already off.  If the switch name is mistyped, an error message is
given.
\end{Command}


\begin{Command}{ON}
\index{switch}
The \name{on} command is used to turn switches on.
\begin{Syntax}
\name{on} \meta{switch}\{,\meta{switch}\}\optional
\end{Syntax}

\meta{switch} can be any \name{switch} name.  There is no problem if the
switch is already on.  If the switch name is mistyped, an error message is
given.
\end{Command}


\begin{Declaration}{OPERATOR}
Use the \name{operator} declaration to declare your own operators.
\begin{Syntax}
\name{operator} \meta{identifier}\{,\meta{identifier}\}\optional
\end{Syntax}

\meta{identifier} can be any valid REDUCE identifier, which is not the name
of a \nameref{matrix}, \nameref{array}, scalar variable or previously-defined
operator.  

\begin{Examples}
operator dis,fac; \\
let dis(~x,~y) = sqrt(x^2 + y^2); \\
dis(1,2);                   &       SQRT(5) \\
dis(a,10);                  &       SQRT(A^{2} + 100) \\
on rounded; \\
dis(1.5,7.2);               &       7.35459040329\\
\begin{multilineinput}
let fac(~n) = if n=0 then 1
               else if not(fixp n and n>0)
                then rederr "choose non-negative integer"
               else for i := 1:n product i;
\end{multilineinput} \\
fac(5);                     &      120 \\
fac(-2);                    &      ***** choose non-negative integer
\end{Examples}
 
\begin{Comments}
The first operator is the Euclidean distance metric, the distance of point
$(x,y)$ from the origin.  The second operator is the factorial.

Operators can have various properties assigned to them; they can be
declared \nameref{infix}, \nameref{linear}, \nameref{symmetric}, 
\nameref{antisymmetric}, or \nameref{noncom}\name{mutative}.
The default operator is prefix, nonlinear, and commutative.
Precedence can also be assigned to operators using the declaration
\nameref{precedence}.

Functionality is assigned to an operator by a \nameref{let} statement or
a \nameref{forall}\ldots\name{let} statement, 
(or possibly by a procedure with the name
of the operator). Be careful not to redefine a system operator by
accident. REDUCE permits you to redefine system operators, giving you a 
warning message that the operator was already defined.  This flexibility 
allows you to add mathematical rules that do what you want them to do, but 
can produce odd or erroneous behavior if you are not careful. 

You can declare operators from inside \nameref{procedure}s, as long as they 
are not local variables.  Operators defined inside procedures are global. 
A formal parameter may be declared as an operator, and has the effect of 
declaring the calling variable as the operator.
\end{Comments}
\end{Declaration}


\begin{Declaration}{ORDER}
\index{order}\index{variable order}\index{output}
The \name{order} declaration changes the order of precedence of kernels for
display purposes only.
\begin{Syntax}
\name{order} \meta{identifier}\{,\meta{identifier}\}\optional
\end{Syntax}
\meta{kernel} must be a valid \nameref{kernel} or \nameref{operator} name
complete with argument or a \nameref{list} of such objects.

\begin{Examples}
x + y + z + cos(a);         &         COS(A) + X + Y + Z \\
order z,y,x,cos(a); \\
x + y + z + cos(a);         &         Z + Y + X + COS(A) \\
(x + y)**2;                 &         Y^{2} + 2*Y*X + X^{2} \\
order nil; \\
(z + cos(z))**2;            &         COS(Z)^{2} + 2*COS(Z)*Z + Z^{2}
\end{Examples}

\begin{Comments}
\name{order} affects the printing order of the identifiers only; internal
order is unchanged.  Change internal order of evaluation with the
declaration \nameref{korder}.  You can use \name{order} to feature variables
or functions you are particularly interested in.

Declarations made with \name{order} are cumulative:  kernels in new order
declarations are ordered behind those in previous declarations, and
previous declarations retain their relative order.  Of course, specific
kernels named in new declarations are removed from previous ones and given
the new priority.  Return to the standard canonical printing order with the 
statement \name{order nil}.

The print order specified by \name{order} commands is not in effect if the
switch \nameref{pri} is off.
\end{Comments}
\end{Declaration}


\begin{Declaration}{PRECEDENCE}
\index{operator}
The \name{precedence} declaration attaches a precedence to an infix operator.
\begin{Syntax}
\name{precedence} \meta{operator},\meta{known\_operator}
\end{Syntax}

\meta{operator} should have been declared an operator but may be a REDUCE
identifier that is not already an operator, array, or matrix.
\meta{known\_operator} must be a system infix operator or have had its
precedence already declared.

\begin{Examples}
operator f,h; \\
precedence f,+; \\
precedence h,*;  \\
a + f(1,2)*c;                &         (1 F 2)*C + A \\
a + h(1,2)*c;                &         1 H 2*C + A \\
a*1 f 2*c;                   &         A F 2*C \\
a*1 h 2*c;                   &         1 H 2*A*C
%%%symbolic;                    &         NIL \\
%%%preclis!*;                  &
%%%   (IREM IEQUAL OVER ADD MULT TO OR AND NOT MEMBER MEMQ EQUAL NEQ EQ GEQ
%%%   GREATERP IGREATERP LEQ LESSP ILESSP FREEOF PLUS F IPLUS DIFFERENCE
%%%   IDIFFERENCE TIMES H ITIMES QUOTIENT IQUOTIENT EXPT CONS)
\end{Examples}

\begin{Comments}
The operator whose precedence is being declared is inserted into the infix
operator precedence list at the next higher place than \meta{known\_operator}.

Attaching a precedence to an operator has the side effect of declaring the
operator to be infix.  If the identifier argument for \name{precedence} has
not been declared to be an operator, an attempt to use it causes an error
message.  After declaring it to be an operator, it becomes an infix operator
with the precedence previously given.  Infix operators may be used in prefix
form; if they are used in infix form, a space must be left on each side of
the operator to avoid ambiguity.  Declared infix operators are always binary.

To see the infix operator precedence list, enter symbolic mode and type
\name{preclis!*;}.  The lowest precedence operator is listed first.

All prefix operators have precedence higher than infix operators.
\end{Comments}
\end{Declaration}


\begin{Declaration}{PRECISION}
\index{rounded}\index{floating point}
The \name{precision} declaration sets the number of decimal places used when
\nameref{rounded} is on.  Default is system dependent, and normally about 12.
\begin{Syntax}
\name{precision}(\meta{integer}) or \name{precision} \meta{integer}
\end{Syntax}

\meta{integer} must be a positive integer.  When \meta{integer} is 0, the
current precision is displayed, but not changed.  There is no upper limit,
but precision of greater than several hundred causes unpleasantly slow
operation on numeric calculations.

\begin{Examples}
on rounded; \\
7/9;                         &       0.777777777778 \\
precision 20;                &       20 \\
7/9;                         &       0.77777777777777777778 \\
sin(pi/4);                   &       0.7071067811865475244
\end{Examples}

\begin{Comments}
Trailing zeroes are dropped, so sometimes fewer than 20 decimal places are
printed as in the last example.  Turn on the switch \nameref{fullprec} if
you want to print all significant digits.  The \nameref{rounded} mode
carries calculations to two more places than given by \name{precision}, and
rounds off.
\end{Comments}
\end{Declaration}


\begin{Declaration}{PRINT\_PRECISION}
\index{output}\index{floating point}\index{rounded}

\begin{Syntax}
\name{print\_precision}(\meta{integer})
   or \name{print\_precision} \meta{integer}
\end{Syntax}

In \nameref{rounded} mode, numbers are normally printed to the specified
precision.  If the user wishes to print such numbers with less precision,
the printing precision can be set by the declaration \name{print\_precision}.

\begin{Examples}
on rounded; \\
1/3; & 0.333333333333 \\
print_precision 5; \\
1/3 & 0.33333
\end{Examples}
\end{Declaration}


\begin{Declaration}{REAL}
The \name{real} declaration must be made immediately after a
\nameref{begin} (or other variable declaration such as \nameref{integer}
and \nameref{scalar}) and declares local integer variables.  They are
initialized to zero.
\begin{Syntax}
\name{real} \meta{identifier}\{,\meta{identifier}\}\optional
\end{Syntax}

\meta{identifier} may be any valid REDUCE identifier, except
\name{t} or \name{nil}.

\begin{Comments}
Real variables remain local, and do not share values with variables of the
same name outside the \nameref{begin}\ldots\name{end} block.  When the
block is finished, the variables are removed.  You may use the words
\nameref{integer} or \nameref{scalar} in the place of \name{real}.
\name{real} does not indicate typechecking by the current REDUCE; it is
only for your own information.  Declaration statements must immediately
follow the \name{begin}, without a semicolon between \name{begin} and the
first variable declaration.

Any variables used inside a \name{begin}\ldots\name{end} \nameref{block}
that were not declared \name{scalar}, \name{real} or \name{integer} are
global, and any change made to them inside the block affects their global
value.  Any \ref{array} or \ref{matrix} declared inside a block is always
global.
\end{Comments}
\end{Declaration}


\begin{Declaration}{REMFAC}
\index{factor}\index{output}
The \name{remfac} declaration removes the special factoring treatment of its
arguments that was declared with \nameref{factor}.
\begin{Syntax}
\name{remfac} \meta{kernel}\{,\meta{kernel}\}\repeated
\end{Syntax}

\meta{kernel} must be a \nameref{kernel} or \nameref{operator} name that
was declared as special with the \nameref{factor} declaration.
\end{Declaration}


\begin{Declaration}{SCALAR}
The \name{scalar} declaration must be made immediately after a
\nameref{begin} (or other variable declaration such as \nameref{integer}
and \nameref{real}) and declares local scalar variables.  They are
initialized to 0.
\begin{Syntax}
\name{scalar} \meta{identifier}\{,\meta{identifier}\}\optional
\end{Syntax}

\meta{identifier} may be any valid REDUCE identifier, except \name{t} or
\name{nil}.

\begin{Comments}
Scalar variables remain local, and do not share values with variables of
the same name outside the \nameref{begin}\ldots\name{end} \nameref{block}.
When the block is finished, the variables are removed.  You may use the
words \nameref{real} or \nameref{integer} in the place of \name{scalar}.
\name{real} and \name{integer} do not indicate typechecking by the current
REDUCE; they are only for your own information.  Declaration statements
must immediately follow the \name{begin}, without a semicolon between
\name{begin} and the first variable declaration.

Any variables used inside \name{begin}\ldots\name{end} blocks that were not
declared \name{scalar}, \name{real} or \name{integer} are global, and any
change made to them inside the block affects their global value.  Arrays
declared inside a block are always global.
\end{Comments}
\end{Declaration}


\begin{Declaration}{SCIENTIFIC\_NOTATION}
\index{output}\index{floating point}\index{rounded}
\begin{Syntax}
\name{scientific\_notation}(\meta{m}) or
\name{scientific\_notation}(\{\meta{m},\meta{n}\})
\end{Syntax}

\meta{m} and \meta{n} are positive integers.
\name{scientific\_notation} controls the output format of floating point
numbers.  At the default settings, any number with five or less digits
before the decimal point is printed in a fixed-point notation, e.g., 
{\tt 12345.6}.  Numbers with more than five digits are printed in scientific
notation, e.g., {\tt 1.234567E+5}.  Similarly, by default, any number with
eleven or more zeros after the decimal point is printed in scientific
notation.

When \name{scientific\_notation} is called with the numerical argument
{\em m} a number with more than {\em m} digits before the decimal point,
or {\em m} or more zeros after the decimal point, is printed in scientific
notation.  When \name{scientific\_notation} is called with a list
\{\meta{m},\meta{n}\}, a number with more than {\em m} digits before the
decimal point, or {\em n} or more zeros after the decimal point is
printed in scientific notation.

\begin{Examples}

on rounded;\\

12345.6;&

12345.6\\

123456.5;&1.234565e+5\\

0.00000000000000012;&

1.2e-16\\

scientific_notation 20;&

{5,11}\\

5: 123456.7;&

123456.7\\

0.00000000000000012;&

0.00000000000000012\\

\end{Examples}
\end{Declaration}


\begin{Declaration}{SHARE}
The \name{share} declaration allows access to its arguments by both
algebraic and symbolic modes.
\begin{Syntax}
\name{share} \meta{identifier}\{,\meta{identifier}\}\optional
\end{Syntax}

\meta{identifier} can be any valid REDUCE identifier.

\begin{Comments}
Programming in \nameref{symbolic} as well as algebraic mode allows 
you a wider range
of techniques than just algebraic mode alone.  Expressions do not cross the
boundary since they have different representations, unless the \name{share}
declaration is used.  For more information on using symbolic mode, see
the \meta{REDUCE User's Manual}, and the \meta{Standard Lisp Report}.

You should be aware that a previously-declared array is destroyed by the
\name{share} declaration.  Scalar variables retain their values.  You can
share a declared \nameref{matrix} that has not yet 
been dimensioned so that it can be
used by both modes.  Values that are later put into the matrix are
accessible from symbolic mode too, but not by the usual matrix reference
mechanism.  In symbolic mode, a matrix is stored as a list whose first
element is \nameref{MAT}, and whose next elements are the rows of the matrix
stored as lists of the individual elements.  Access in symbolic mode is by
the operators \nameref{first}, \nameref{second}, \nameref{third} and
\nameref{rest}.
\end{Comments}
\end{Declaration}


\begin{Command}{SYMBOLIC}
The \name{symbolic} command changes REDUCE's mode of operation to symbolic.
When \name{symbolic} is followed by an expression, that expression is
evaluated in symbolic mode, but REDUCE's mode is not changed.  It is
equivalent to the \nameref{lisp} command.

\begin{Examples}
symbolic;                    &       NIL \\
cdr '(a b c);                &       (B C) \\
algebraic; \\
x + symbolic car '(y z);     &       X + Y
\end{Examples}
% \begin{Comments}
% You can tell when you are in symbolic mode by noting that the numbered
% prompt contains an asterisk (\name{*}) rather than a colon.
% \end{Comments}
\end{Command}


\begin{Declaration}{SYMMETRIC}
\index{operator}
When an operator is declared \name{symmetric}, its arguments are reordered
to conform to the internal ordering of the system.  
\begin{Syntax}
\name{symmetric} \meta{identifier}\{,\meta{identifier}\}\optional
\end{Syntax}

\meta{identifier} is an identifier that has been declared an operator.

\begin{Examples}
operator m,n; \\
symmetric m,n; \\
m(y,a,sin(x));               &           M(SIN(X),A,Y) \\
n(z,m(b,a,q));               &           N(M(A,B,Q),Z)
\end{Examples}

\begin{Comments}
If \meta{identifier} has not been declared to be an operator, the flag
\name{symmetric} is still attached to it.  When \meta{identifier} is
subsequently used as an operator, the message \name{Declare}\meta{identifier}
 \name{operator ? (Y or N)} is printed.  If the user replies \name{y}, the
symmetric property of the operator is used.
\end{Comments}
\end{Declaration}


\begin{Declaration}{TR}
\index{trace}
The \name{tr} declaration is used to trace system or user-written procedures.
It is only useful to those with a good knowledge of both Lisp and the
internal formats used by REDUCE.

\begin{Syntax}
\name{tr} \meta{name}\{,\meta{name}\}\optional
\end{Syntax}

\meta{name} is the name of a REDUCE system procedure or one of your own
procedures.

\begin{Examples}
\explanation{The system procedure \name{prepsq} is traced,
	     which prepares REDUCE standard
forms for printing by converting them to a Lisp prefix form.} \\
tr prepsq;               &      (PREPSQ) \\
x**2 + y;                &
\begin{multilineoutput}{6cm}
PREPSQ entry:
  Arg 1: (((((X . 2) . 1) ((Y . 1) . 1)) . 1)
PREPSQ return value = (PLUS (EXPT X 2) Y)
PREPSQ entry:
  Arg 1: (1 . 1)
PREPSQ return value = 1
X^{2} + Y
\end{multilineoutput}\\
untr prepsq;             &       (PREPSQ)
\end{Examples}

\begin{Comments}
This example is for a PSL-based system; the above format will vary if
other Lisp systems are used.

When a procedure is traced, the first lines show entry to the procedure and
the arguments it is given.  The value returned by the procedure is printed
upon exit.  If you are tracing several procedures, with a call to one of
them inside the other, the inner trace will be indented showing procedure
nesting.  There are no trace options. However, the format of the trace
depends on the underlying Lisp system used.  The trace can be removed with
the command \nameref{untr}.  Note that \name{trace}, below, is a matrix
operator, while \name{tr} does procedure tracing.
\end{Comments}
\end{Declaration}


\begin{Declaration}{UNTR}
\index{trace}
The \name{untr} declaration is used to remove a trace from system or
user-written procedures declared with \nameref{tr}.  It is only useful to
those with a good knowledge of both Lisp and the internal formats used by
REDUCE.

\begin{Syntax}
\name{untr} \meta{name}\{,\meta{name}\}\optional
\end{Syntax}

\meta{name} is the name of a REDUCE system procedure or one of your own
procedures that has previously been the argument of a \name{tr}
declaration.
\end{Declaration}


\begin{Declaration}{VARNAME}
The declaration \name{varname} instructs REDUCE to use its argument as the
default Fortran (when \nameref{fort} is on) or \nameref{structr} identifier
and identifier stem, rather than using \name{ANS}.
\begin{Syntax}
\name{varname} \meta{identifier}
\end{Syntax}

\meta{identifier} can be any combination of one or more alphanumeric
characters.  Try to avoid REDUCE reserved words.

\begin{Examples}
varname ident;               &         IDENT \\
on fort; \\
x**2 + 1;                    &         IDENT=X**2+1. \\
off fort,exp; \\
structr(((x+y)**2 + z)**3);  &\begin{multilineoutput}{6cm}
IDENT2^{3}
    where
       IDENT2 := IDENT1^{2} + Z
IDENT1 := X + Y
\end{multilineoutput}
\end{Examples}
\begin{Comments}
\nameref{exp} was turned off so that \nameref{structr} could show the
structure.  If \name{exp} had been on, the expression would have been
expanded into a polynomial.
\end{Comments}
\end{Declaration}


\begin{Command}{WEIGHT}
The \name{weight} command is used to attach weights to kernels for asymptotic
constraints.
\begin{Syntax}
\name{weight} \meta{kernel} \name{=}\meta{number}
\end{Syntax}

\meta{kernel} must be a REDUCE \nameref{kernel}, \meta{number} must be a
positive integer, not 0.

\begin{Examples}
a := (x+y)**4;               & 
     A := X^{4} + 4*X^{3}*Y + 6*X^{2}*Y^{2} + 4*X*Y^{3} + Y^{4} \\
weight x=2,y=3; \\
wtlevel 8; \\
a;                           &     X^{4} \\
wtlevel 10; \\
a;                           &      X^{2}*(6*Y^{2} + 4*X*Y  + X^{2}) \\
int(x**2,x);                 &     ***** X invalid as KERNEL
\end{Examples}
\begin{Comments}
Weights and \nameref{wtlevel} are used for asymptotic constraints, where
higher-order terms are considered insignificant.  

Weights are originally equivalent to 0 until set by a \name{weight}
command.  To remove a weight from a kernel, use the \nameref{clear}
command.  Weights once assigned cannot be changed without clearing the
identifier.  Once a weight is assigned to a kernel, it is no longer a
kernel and cannot be used in any REDUCE commands or operators that require
kernels, until the weight is cleared.  Note that terms are ordered by
greatest weight.

The weight level of the system is set by \nameref{wtlevel}, initially at
2.  Since no kernels have weights, no effect from \name{wtlevel} can be
seen.  Once you assign weights to kernels, you must set \name{wtlevel}
correctly for the desired operation.  When weighted variables appear in a
term, their weights are summed for the total weight of the term (powers of
variables multiply their weights).  When a term exceeds the weight level
of the system, it is discarded from the result expression.
\end{Comments}
\end{Command}


\begin{Operator}{WHERE}
\index{substitution}
The \name{where} operator provides an infix notation for one-time
substitutions for kernels in expressions.
\begin{Syntax}
\meta{expression} \name{where} \meta{kernel} 
       \name{=}\meta{expression}\
	 \{,\meta{kernel} \name{=}\meta{expression}\}\optional
\end{Syntax}

\meta{expression} can be any REDUCE scalar expression, \meta{kernel} must
be a \nameref{kernel}. Alternatively a \nameref{rule} or a \name{rule list}
can be a member of the right-hand part of a \name{where} expression. 

\begin{Examples}
x**2 + 17*x*y + 4*y**2 where x=1,y=2;          &      51 \\
for i := 1:5 collect x**i*q where q= for j := 1:i product j;
			     & \{X,2*X^{2},6*X^{3},24*X^{4},120*X^{5}\} \\
x**2 + y + z where z=y**3,y=3;                 &      X^{2} + Y^{3} + 3
\end{Examples}

\begin{Comments}
Substitution inside a \name{where} expression has no effect upon the values
of the kernels outside the expression.  The \name{where} operator has the
lowest precedence of all the infix operators, which are lower than prefix
operators, so that the substitutions apply to the entire expression
preceding the \name{where} operator.  However, \name{where} is applied
before command keywords such as \name{then}, \name{repeat}, or \name{do}.

A \nameref{rule} or a \name{rule set} in the right-hand part of the
\name{where} expression act as if the rules were activated by \nameref{let}
immediately before the evaluation of the expression and deactivated
by \nameref{clearrules} immediately afterwards.

\name{where} gives you a natural notation for auxiliary variables in
expressions.  As the second example shows, the substitute expression can be
a command to be evaluated.  The substitute assignments are made in
parallel, rather than sequentially, as the last example shows.   The
expression resulting from the first round of substitutions is not
reexamined to see if any further such substitutions can be made.
\name{where} can also be used to define auxiliary variables in
\nameref{procedure} definitions.
\end{Comments}
\end{Operator}


\begin{Command}{WHILE}
\index{loop}
The \name{while} command causes a statement to be repeatedly executed until a
given condition is true.  If the condition is initially false, the statement
is not executed at all.
\begin{Syntax}
\name{while} \meta{condition} \name{do} \meta{statement}
\end{Syntax}

\meta{condition} is given by a logical operator, \meta{statement} must be a
single REDUCE statement, or a \nameref{group} (\name{<<}\ldots\name{>>}) or
\nameref{begin}\ldots\name{end} \nameref{block}.

\begin{Examples}
a := 10;                     &           A := 10 \\
while a <= 12 do <<write a; a := a + 1>>;
			     &           10 \\
                                          11 \\
                                          12 \\
while a < 5 do <<write a; a := a + 1>>;
			     &           {\it nothing is printed}
\end{Examples}

\end{Command}


\begin{Command}{WTLEVEL}
In conjunction with \nameref{weight}, \name{wtlevel} is used to implement
asymptotic constraints.  Its default value is 2.
\begin{Syntax}
\name{wtlevel} \meta{expression}
\end{Syntax}

To change the weight level, \meta{expression} must evaluate to a positive
integer that is the greatest weight term to be retained in expressions
involving kernels with weight assignments. \name{wtlevel} returns the
new weight level.  If you want the current weight level, but not
change it, say \name{wtlevel nil}.

\begin{Examples}
(x+y)**4;          
        & X^{4} + 4*X^{3}*Y + 6*X^{2}*Y^{2} + 4*X*Y^{3} + Y^{4} \\
weight x=2,y=3; \\
wtlevel 8; \\
(x+y)**4;          & X^{4} \\
wtlevel 10; \\
(x+y)**4;          & X^{2}*(6*Y^{2} + 4*X*Y + X^{2}) \\
int(x**2,x);       & ***** X invalid as KERNEL
\end{Examples}
\begin{Comments}
\name{wtlevel} is used in conjunction with the command \nameref{weight} to
enable asymptotic constraints.  Weight of a term is computed by multiplying
the weights of each variable in it by the power to which it has been
raised, and adding the resulting weights for each variable.  If the weight
of the term is greater than \name{wtlevel}, the term is dropped from the
expression, and not used in any further computation involving the
expression.

Once a weight has been attached to a \nameref{kernel}, it is no longer
recognized by the system as a kernel, though still a variable.  It cannot
be used in REDUCE commands and operators that need kernels.  The weight
attachment can be undone with a \nameref{clear} command. \name{wtlevel} can
be changed as desired.
\end{Comments}
\end{Command}