*BLURB 
This is the second part of the Yacas function reference.
This reference contains functions that might be useful
for programming in Yacas.
				Yacas programmer's function reference

			Programming

*INTRO This chapter describes functions useful for writing Yacas scripts.

*CMD	/*, */, // --- comments
*CORE
*CALL
	/* comment */
	// comment

*DESC

Introduce a comment block in a source file, similar to C++ comments.
{//} makes everything until the end of the line a comment, while {/*} and {*/} may delimit a multi-line comment.

*E.G.

	a+b; // get result
	a + /* add them */ b;

*CMD Prog, [, ] --- block of statements
*CORE
*CALL
	Prog(statement1, statement2, ...)
	[ statement1; statement2; ... ]

*PARMS

{statement1}, {statement2} -- expressions

*DESC

The {Prog} and the {[ ... ]} construct have the same effect: they evaluate all
arguments in order and return the result of the last evaluated expression.

{Prog(a,b);} is the same as typing {[a;b;];} and is very useful for writing out
function bodies. The {[ ... ]} construct is a syntactically nicer version of the
{Prog} call; it is converted into {Prog(...)} during the parsing stage.



*CMD Bodied, Infix, Postfix, Prefix --- define function syntax
*CORE
*CALL
	Bodied("op", precedence)
	Infix("op")
	Infix("op", precedence)
	Postfix("op")
	Postfix("op", precedence)
	Prefix("op")
	Prefix("op", precedence)

*PARMS

{"op"} -- string, the name of a function

{precedence} -- nonnegative integer (evaluated)

*DESC

Declares a special syntax for the function to be parsed as a bodied, infix, postfix,
or prefix operator.

"Bodied" functions have all arguments except the first one inside parentheses and the last argument outside, for example:
	For(pre, condition, post) statement;
Here the function {For} has 4 arguments and the last argument is placed outside the parentheses.
The {precedence} of a "bodied" function refers to how tightly the last argument is bound to the parentheses.
This makes a difference when the last argument contains other operators.
For example, when taking the derivative
	D(x) Sin(x)+Cos(x)
both {Sin} and {Cos} are under the derivative because the bodied function {D} binds less tightly than the infix operator "{+}".

"Infix" functions must have two arguments and are syntactically placed between their arguments.
Names of infix functions can be arbitrary, although for reasons of readability they are usually made of non-alphabetic characters.

"Prefix" functions must have one argument and are syntactically placed before their argument.
"Postfix" functions must have one argument and are syntactically placed after their argument.

Function name can be any string but meaningful usage and readability would
require it to be either made up entirely of letters or entirely of non-letter
characters (such as "+", ":" etc.).
Precedence is optional (will be set to 0 by default).

*E.G.
	In> YY x := x+1;
	CommandLine(1) : Error parsing expression
	
	In> Prefix("YY", 2)
	Out> True;
	In> YY x := x+1;
	Out> True;
	In> YY YY 2*3
	Out> 12;
	In> Infix("##", 5)
	Out> True;
	In> a ## b ## c
	Out> a##b##c;

Note that, due to a current parser limitation, a function atom that is declared prefix cannot be used by itself as an argument.

	In> YY
	CommandLine(1) : Error parsing expression

*SEE IsBodied, OpPrecedence



*CMD IsBodied, IsInfix, IsPostfix, IsPrefix --- check for function syntax
*CORE
*CALL
	IsBodied("op")
	IsInfix("op")
	IsPostfix("op")
	IsPrefix("op")

*PARMS

{"op"} -- string, the name of a function

*DESC

Check whether the function with given name {"op"} has been declared as a
"bodied", infix, postfix, or prefix operator, and  return {True} or {False}.

*E.G.

	In> IsInfix("+");
	Out> True;
	In> IsBodied("While");
	Out> True;
	In> IsBodied("Sin");
	Out> False;
	In> IsPostfix("!");
	Out> True;

*SEE Bodied, OpPrecedence

*CMD OpPrecedence, OpLeftPrecedence, OpRightPrecedence --- get operator precedence
*CORE
*CALL
	OpPrecedence("op")
	OpLeftPrecedence("op")
	OpRightPrecedence("op")

*PARMS

{"op"} -- string, the name of a function

*DESC

Returns the precedence of the function named "op" which should have been declared as a bodied function or an infix, postfix, or prefix operator. Generates an error message if the string str does not represent a type of function that can have precedence.

For infix operators, right precedence can differ from left precedence. Bodied functions and prefix operators cannot have left precedence, while postfix operators cannot have right precedence; for these operators, there is only one value of precedence.

*E.G.
	In> OpPrecedence("+")
	Out> 6;
	In> OpLeftPrecedence("!")
	Out> 0;



*CMD RightAssociative --- declare associativity
*CORE
*CALL
	RightAssociative("op")

*PARMS

{"op"} -- string, the name of a function

*DESC
This makes the operator right-associative. For example:
	RightAssociative("*")
would make multiplication right-associative. Take care not to abuse this
function, because the reverse, making an infix operator left-associative, is
not implemented. (All infix operators are by default left-associative until
they are declared to be right-associative.)

*SEE OpPrecedence


*CMD LeftPrecedence, RightPrecedence --- set operator precedence
*CORE
*CALL
	LeftPrecedence("op",precedence)
	RightPrecedence("op",precedence)

*PARMS

{"op"} -- string, the name of a function

{precedence} -- nonnegative integer

*DESC

{"op"} should be an infix operator. This function call tells the
infix expression printer to bracket the left or right hand side of
the expression if its precedence is larger than precedence.

This functionality was required in order to display expressions like {a-(b-c)}
correctly. Thus, {a+b+c} is the same as {a+(b+c)}, but {a-(b-c)} is not
the same as {a-b-c}.

Note that the left and right precedence of an infix operator does not affect the way Yacas interprets expressions typed by the user. You cannot make Yacas parse {a-b-c} as {a-(b-c)} unless you declare the operator "{-}" to be right-associative.

*SEE OpPrecedence, OpLeftPrecedence, OpRightPrecedence, RightAssociative

*CMD RuleBase --- define function with a fixed number of arguments
*CORE
*CALL
	RuleBase(name,params)

*PARMS

{name} -- string, name of function

{params} -- list of arguments to function

*DESC
Define a new rules table entry for a
function "name", with {params} as the parameter list. Name can be
either a string or simple atom.

In the context of the transformation rule declaration facilities
this is a useful function in that it allows the stating of argument
names that can he used with HoldArg.

Functions can be overloaded: the same function can be defined
with different number of arguments.


*SEE MacroRuleBase, RuleBaseListed, MacroRuleBaseListed, HoldArg, Retract



*CMD RuleBaseListed --- define function with variable number of arguments
*CORE
*CALL
	RuleBaseListed("name", params)

*PARMS

{"name"} -- string, name of function

{params} -- list of arguments to function

*DESC

The command {RuleBaseListed} defines a new function. It essentially works the
same way as {RuleBase}, except that it declares a new function with a variable
number of arguments. The list of parameters {params} determines the smallest
number of arguments that the new function will accept. If the number of
arguments passed to the new function is larger than the number of parameters in
{params}, then the last argument actually passed to the new function will be a
list containing all the remaining arguments.

A function defined using {RuleBaseListed} will appear to have the arity equal
to the number of parameters in the {param} list, and it can accept any number
of arguments greater or equal than that. As a consequence, it will be impossible to define a new function with the same name and with a greater arity.

The function body will know that the function is passed more arguments than the
length of the {param} list, because the last argument will then be a list. The
rest then works like a {RuleBase}-defined function with a fixed number of
arguments. Transformation rules can be defined for the new function as usual.


*E.G.

The definitions

	RuleBaseListed("f",{a,b,c})
	10 # f(_a,_b,{_c,_d}) <--
	  Echo({"four args",a,b,c,d});
	20 # f(_a,_b,c_IsList) <--
	  Echo({"more than four args",a,b,c});
	30 # f(_a,_b,_c) <-- Echo({"three args",a,b,c});
give the following interaction:

	In> f(A)
	Out> f(A);
	In> f(A,B)
	Out> f(A,B);
	In> f(A,B,C)
	three args A B C 
	Out> True;
	In> f(A,B,C,D)
	four args A B C D 
	Out> True;
	In> f(A,B,C,D,E)
	more than four args A B {C,D,E} 
	Out> True;
	In> f(A,B,C,D,E,E)
	more than four args A B {C,D,E,E} 
	Out> True;

The function {f} now appears to occupy all arities greater than 3:

	In> RuleBase("f", {x,y,z,t});
	CommandLine(1) : Rule base with this arity
	  already defined


*SEE RuleBase, Retract, Echo


*CMD Rule --- define a rewrite rule
*CORE
*CALL
	Rule("operator", arity,
	  precedence, predicate) body
*PARMS

{"operator"} -- string, name of function

{arity}, {precedence} -- integers

{predicate} -- function returning boolean

{body} -- expression, body of rule

*DESC

Define a rule for the function "operator" with
"arity", "precedence", "predicate" and
"body". The "precedence" goes from low to high: rules with low precedence will be applied first.

The arity for a rules database equals the number of arguments. Different
rules data bases can be built for functions with the same name but with
a different number of arguments.

Rules with a low precedence value will be tried before rules with a high value, so
a rule with precedence 0 will be tried before a rule with precedence 1.

*CMD HoldArg --- mark argument as not evaluated
*CORE
*CALL
	HoldArg("operator",parameter)

*PARMS

{"operator"} -- string, name of a function

{parameter} -- atom, symbolic name of parameter

*DESC
Specify that parameter should
not be evaluated before used. This will be
declared for all arities of "operator", at the moment
this function is called, so it is best called
after all {RuleBase} calls for this operator.
"operator" can be a string or atom specifying the 
function name.

The {parameter} must be an atom from the list of symbolic 
arguments used when calling {RuleBase}.

*SEE RuleBase, HoldArgNr, RuleBaseArgList

*CMD Retract --- erase rules for a function
*CORE
*CALL
	Retract("function",arity)

*PARMS
{"function"} -- string, name of function

{arity} -- positive integer

*DESC

Remove a rulebase for the function named {"function"} with the specific {arity}, if it exists at all. This will make
Yacas forget all rules defined for a given function. Rules for functions with
the same name but different arities are not affected.

Assignment {:=} of a function does this to the function being (re)defined.

*SEE RuleBaseArgList, RuleBase, :=

*CMD UnFence --- change local variable scope for a function
*CORE
*CALL
	UnFence("operator",arity)

*PARMS
{"operator"} -- string, name of function

{arity} -- positive integers

*DESC

When applied to a user function, the bodies
defined for the rules for "operator" with given
arity can see the local variables from the calling
function. This is useful for defining macro-like
procedures (looping and such).

The standard library functions {For} and {ForEach} use {UnFence}.

*CMD HoldArgNr --- specify argument as not evaluated
*STD
*CALL
	HoldArgNr("function", arity, argNum)

*PARMS
{"function"} -- string, function name

{arity}, {argNum} -- positive integers

*DESC

Declares the argument numbered {argNum} of the function named {"function"} with
specified {arity} to be unevaluated ("held"). Useful if you don't know symbolic
names of parameters, for instance, when the function was not declared using an
explicit {RuleBase} call. Otherwise you could use {HoldArg}.

*SEE HoldArg, RuleBase


*CMD RuleBaseArgList --- obtain list of arguments
*CORE
*CALL
	RuleBaseArgList("operator", arity)

*PARMS
{"operator"} -- string, name of function

{arity} -- integer

*DESC

Returns a list of atoms, symbolic parameters specified in the {RuleBase} call
for the function named {"operator"} with the specific {arity}.

*SEE RuleBase, HoldArgNr, HoldArg


*CMD MacroSet, MacroClear, MacroLocal, MacroRuleBase, MacroRuleBaseListed, MacroRule --- define rules in functions
*CORE
*DESC

These functions have the same effect as their non-macro counterparts, except
that their arguments are evaluated before the required action is performed.
This is useful in macro-like procedures or in functions that need to define new
rules based on parameters.

Make sure that the arguments of {Macro}... commands evaluate to expressions that would normally be used in the non-macro versions!

*SEE Set, Clear, Local, RuleBase, Rule, Backquoting

*A {`}
*CMD Backquoting --- macro expansion (LISP-style backquoting)
*CORE
*CALL
	`(expression)

*PARMS

{expression} -- expression containing "{@var}" combinations to substitute the value of variable "{var}"

*DESC

Backquoting is a macro substitution mechanism. A backquoted {expression}
is evaluated in two stages: first, variables prefixed by {@} are evaluated
inside an expression, and second, the new expression is evaluated.

To invoke this functionality, a backquote {`} needs to be placed in front of
an expression. Parentheses around the expression are needed because the
backquote binds tighter than other operators.

The expression should contain some variables (assigned atoms) with the special
prefix operator {@}. Variables prefixed by {@} will be evaluated even if they
are inside function arguments that are normally not evaluated (e.g. functions
declared with {HoldArg}). If the {@var} pair is in place of a function name,
e.g. "{@f(x)}", then at the first stage of evaluation the function name itself
is replaced, not the return value of the function (see example); so at the
second stage of evaluation, a new function may be called.

One way to view backquoting is to view it as a parametric expression
generator. {@var} pairs get substituted with the value of the variable {var}
even in contexts where nothing would be evaluated. This effect can be also
achieved using {UnList} and {Hold} but the resulting code is much more
difficult to read and maintain.

This operation is relatively slow since a new expression is built
before it is evaluated, but nonetheless backquoting is a powerful mechanism
that sometimes allows to greatly simplify code.

*E.G.

This example defines a function that automatically evaluates to a number as
soon as the argument is a number (a lot of functions  do this only when inside
a {N(...)} section).

	In> Decl(f1,f2) := \
	In>   `(@f1(x_IsNumber) <-- N(@f2(x)));
	Out> True;
	In> Decl(nSin,Sin)
	Out> True;
	In> Sin(1)
	Out> Sin(1);
	In> nSin(1)
	Out> 0.8414709848;

This example assigns the expression {func(value)} to variable {var}. Normally
the first argument of {Set} would be unevaluated.

	In> SetF(var,func,value) := \
	In>     `(Set(@var,@func(@value)));
	Out> True;
	In> SetF(a,Sin,x)
	Out> True;
	In> a
	Out> Sin(x);


*SEE MacroSet, MacroLocal, MacroRuleBase, Hold, HoldArg, DefMacroRuleBase



*CMD DefMacroRuleBase --- define a function as a macro
*STD
*CALL
	DefMacroRuleBase(name,params)

*PARMS

{name} -- string, name of a function

{params} -- list of arguments

*DESC

{DefMacroRuleBase} is similar to {RuleBase}, with the difference that it declares a macro,
instead of a function.
After this call, rules can be defined for the function "{name}", but their interpretation will be different.

With the usual functions, the evaluation model is that of the <i>applicative-order model of 
substitution</i>, meaning that first the arguments are evaluated, and then the function 
is applied to the result of evaluating these arguments. The function is entered, and the
code inside the function can not access local variables outside of its own local variables.

With macros, the evaluation model is that of the <i>normal-order model of substitution</i>,
meaning that all occurrences of variables in an expression are first substituted into the 
body of the macro, and only then is the resulting expression evaluated <i>in its
calling environment</i>. This is important, because then in principle a macro body
can access the local variables from the calling environment, whereas functions can not do that.

As an example, suppose there is a function {square}, which squares its argument, and a function
{add}, which adds its arguments. Suppose the definitions of these functions are:

	add(x,y) <-- x+y;
and
	square(x) <-- x*x;
In applicative-order mode (the usual way functions are evaluated), in the following expression
	add(square(2),square(3))
first the arguments to {add} get evaluated. So, first {square(2)} is evaluated.
To evaluate this, first {2} is evaluated, but this evaluates to itself. Then
the {square} function is applied to it, {2*2}, which returns 4. The same
is done for {square(3)}, resulting in {9}. Only then, after evaluating these two
arguments, {add} is applied to them, which is equivalent to
	add(4,9)
resulting in calling {4+9}, which in turn results in {13}.

In contrast, when {add} is a macro, the arguments to {add} are first
expanded. So 
	add(square(2),square(3))
first expands to 
	square(2) + square(3)
and then this expression is evaluated, as if the user had written it directly.
In other words, {square(2)} is not evaluated before the macro has been fully expanded.


*REM One more difference between macros and 

Macros are useful for customizing syntax, and compilers can potentially
greatly optimize macros, as they can be inlined in the calling environment,
and optimized accordingly. 

There are disadvantages, however. In interpreted mode, macros are slower,
as the requirement for substitution means that a new expression to be evaluated
has to be created on the fly. Also, when one of the parameters to the macro
occur more than once in the body of the macro, it is evaluated multiple times.

When defining transformation rules for macros, the variables to be substituted
need to be preceded by the {@} operator, similar to the back-quoting mechanism.
Apart from that, the two are similar, and all transformation rules can also be
applied to macros.

Macros can co-exist with functions with the same name but different arity.
For instance, one can have a function {foo(a,b)}
with two arguments, and a macro {foo(a,b,c)} with three arguments.


*EG

The following example defines a function {myfor}, and shows one use, referencing
a variable {a} from the calling environment.

	In> DefMacroRuleBase("myfor",{init,pred,inc,body})
	Out> True;
	In> myfor(_init,_pred,_inc,_body)<--[@init;While(@pred)[@body;@inc;];True;];
	Out> True;
	In> a:=10
	Out> 10;
	In> myfor(i:=1,i<10,i++,Echo(a*i))
	10 
	20 
	30 
	40 
	50 
	60 
	70 
	80 
	90 
	Out> True;
	In> i
	Out> 10;

*SEE RuleBase, Backquoting, DefMacroRuleBaseListed

*CMD DefMacroRuleBaseListed --- define macro with variable number of arguments
*CORE
*CALL
	DefMacroRuleBaseListed("name", params)

*PARMS

{"name"} -- string, name of function

{params} -- list of arguments to function

*DESC

This does the same as {DefMacroRuleBase} (define a macro), but with a variable
number of arguments, similar to {RuleBaseListed}.

*SEE RuleBase, RuleBaseListed, Backquoting, DefMacroRuleBase


*A object properties
*CMD SetExtraInfo, GetExtraInfo --- annotate objects with additional information
*CORE
*CALL
	SetExtraInfo(expr,tag)
	GetExtraInfo(expr)

*PARMS

{expr} -- any expression

{tag} -- tag information (any other expression)

*DESC

Sometimes it is useful to be able to add extra tag information to "annotate"
objects or to label them as having certain "properties". The functions
{SetExtraInfo} and {GetExtraInfo} enable this.

The function {SetExtraInfo} returns the tagged expression, leaving
the original expression alone. This means there is a common pitfall:
be sure to assign the returned value to a variable, or the tagged
expression is lost when the temporary object is destroyed.

The original expression is left unmodified, and the tagged expression
returned, in order to keep the atomic objects small. To tag an
object, a new type of object is created from the old object, with
one added property (the tag). The tag can be any expression whatsoever.

The function {GetExtraInfo(x)} retrieves this tag expression from an object
{x}. If an object has no tag, it looks the same as if it had a tag with value
{False}.

No part of the Yacas core uses tags in a way that is visible to the outside
world, so for specific purposes a programmer can devise a format to use for tag
information. Association lists (hashes) are a natural fit for this, although it
is not required and a tag can be any object (except the atom {False} because it
is indistinguishable from having no tag information). Using association lists
is highly advised since it is most likely to be the format used by other parts
of the library, and one needs to avoid clashes with other library code.
Typically, an object will either have no tag or a tag which is an associative
list (perhaps empty). A script that uses tagged objects will check whether an
object has a tag and if so, will add or modify certain entries of the
association list, preserving any other tag information.

Note that {FlatCopy} currently does <i>not</i> copy the tag information (see
examples).

*E.G.

	In> a:=2*b
	Out> 2*b;
	In> a:=SetExtraInfo(a,{{"type","integer"}})
	Out> 2*b;
	In> a
	Out> 2*b;
	In> GetExtraInfo(a)
	Out> {{"type","integer"}};
	In> GetExtraInfo(a)["type"]
	Out> "integer";
	In> c:=a
	Out> 2*b;
	In> GetExtraInfo(c)
	Out> {{"type","integer"}};
	In> c
	Out> 2*b;
	In> d:=FlatCopy(a);
	Out> 2*b;
	In> GetExtraInfo(d)
	Out> False;

*SEE Assoc, :=

*CMD GarbageCollect --- do garbage collection on unused memory
*CORE
*CALL
	GarbageCollect()

*DESC

{GarbageCollect} garbage-collects unused memory. The Yacas system
uses a reference counting system for most objects, so this call
is usually not necessary. 

Reference counting refers to bookkeeping where in each object a 
counter is held, keeping track of the number of parts in the system 
using that object. When this count drops to zero, the object is 
automatically removed. Reference counting is not the fastest way
of doing garbage collection, but it can be implemented in a very
clean way with very little code.

Among the most important objects that are not reference counted are
the strings. {GarbageCollect} collects these and disposes of them
when they are not used any more. 

{GarbageCollect} is useful when doing a lot of text processing,
to clean up the text buffers. It is not highly needed, but it keeps
memory use low.


*CMD FindFunction --- find the library file where a function is defined
*CORE
*CALL
	FindFunction(function)

*PARMS

{function} -- string, the name of a function

*DESC

This function is useful for quickly finding the file where a standard library
function is defined. It is likely to only be useful for developers. The
function {FindFunction} scans the {.def} files that were loaded at start-up.
This means that functions that are not listed in {.def} files will not be found with {FindFunction}.

*E.G.

	In> FindFunction("Sum")
	Out> "sums.rep/code.ys";
	In> FindFunction("Integrate")
	Out> "integrate.rep/code.ys";

*SEE Vi

*CMD Secure --- guard the host OS
*CORE
*CALL
	Secure(body)

*PARMS

{body} -- expression

*DESC

{Secure} evaluates {body} in a "safe" environment, where files cannot be opened
and system calls are not allowed. This can help protect the system
when e.g. a script is sent over the
Internet to be evaluated on a remote computer, which is potentially unsafe.

*SEE SystemCall

*REM functions for arbitrary-precision numerical programming
*INCLUDE numerics.chapt

			Error reporting

*INTRO
This chapter contains commands useful for reporting errors to the user.

*CMD Check --- report "hard" errors
*CMD TrapError --- trap "hard" errors
*CMD GetCoreError --- get "hard" error string
*CORE
*CALL
	Check(predicate,"error text")
	TrapError(expression,errorHandler)
	GetCoreError()

*PARMS

{predicate} -- expression returning {True} or {False}

{"error text"} -- string to print on error

{expression} -- expression to evaluate (causing potential error)

{errorHandler} -- expression to be called to handle error

*DESC
If {predicate} does not evaluate to {True},
the current operation will be stopped, the string {"error text"} will be printed, and control will be returned immediately to the command line. This facility can be used to assure that some condition
is satisfied during evaluation of expressions (guarding
against critical internal errors).

A "soft" error reporting facility that does not stop the execution is provided by the function {Assert}.

*EG

	In> [Check(1=0,"bad value"); Echo(OK);]
	In function "Check" : 
	CommandLine(1) : "bad value"

Note that {OK} is not printed.

TrapError evaluates its argument {expression}, returning the
result of evaluating {expression}. If an error occurs,
{errorHandler} is evaluated, returning its return value in stead.

GetCoreError returns a string describing the core error.
TrapError and GetCoreError can be used in combination to write
a custom error handler. 


*SEE Assert

*CMD Assert --- signal "soft" custom error
*STD
*CALL
	Assert("str", expr) pred
	Assert("str") pred
	Assert() pred
Precedence:
*EVAL OpPrecedence("Assert")

*PARMS

{pred} -- predicate to check

{"str"} -- string to classify the error

{expr} -- expression, error object

*DESC

{Assert} is a global error reporting mechanism. It can be used to check for
errors and report them. An error is considered to occur when the predicate
{pred} evaluates to anything except {True}. In this case, the function returns
{False} and an error object is created and posted to the global error tableau.
Otherwise the function returns {True}.

Unlike the "hard" error function {Check}, the function {Assert} does not stop
the execution of the program.

The error object consists of the string {"str"} and an arbitrary
expression {expr}. The string should be used to classify the kind of error that
has occurred, for example "domain" or "format". The error object can be any expression that might be useful for handling the error later;
for example, a list of erroneous values and explanations.
The association list of error objects is currently the global
variable {ErrorTableau}.

If the parameter {expr} is missing, {Assert} substitutes {True}. If both optional parameters {"str"} and {expr} are missing, {Assert} creates an error of class {"generic"}.

Errors can be handled by a
custom error handler in the portion of the code that is able to handle a certain class of
errors. The functions {IsError}, {GetError} and {ClearError} can be used.

Normally, all errors posted to the error tableau during evaluation of an expression should
be eventually printed to the screen. This is the behavior of prettyprinters
{DefaultPrint}, {Print}, {PrettyForm} and {TeXForm} (but not of the
inline prettyprinter, which is enabled by default); they call
{DumpErrors} after evaluating the expression. 

*E.G.

	In> Assert("bad value", "must be zero") 1=0
	Out> False;
	In> Assert("bad value", "must be one") 1=1
	Out> True;
	In> IsError()
	Out> True;
	In> IsError("bad value")
	Out> True;
	In> IsError("bad file")
	Out> False;
	In> GetError("bad value");
	Out> "must be zero";
	In> DumpErrors()
	Error: bad value: must be zero
	Out> True;
No more errors left:
	In> IsError()
	Out> False;
	In> DumpErrors()
	Out> True;

*SEE IsError, DumpErrors, Check, GetError, ClearError, ClearErrors

*CMD DumpErrors, ClearErrors --- simple error handlers
*STD
*CALL
	DumpErrors()
	ClearErrors()

*DESC

{DumpErrors} is a simple error handler for the global error reporting mechanism. It prints all errors posted using {Assert} and clears the error tableau.

{ClearErrors} is a trivial error handler that does nothing except it clears the tableau.

*SEE Assert, IsError

*CMD IsError --- check for custom error
*STD
*CALL
	IsError()
	IsError("str")

*PARMS

{"str"} -- string to classify the error

*DESC

{IsError()} returns {True} if any custom errors have been reported using {Assert}.
The second form takes a parameter {"str"} that designates the class of the
error we are interested in. It returns {True} if any errors of the given class
{"str"} have been reported.

*SEE GetError, ClearError, Assert, Check


*CMD GetError, ClearError --- custom errors handlers
*STD
*CALL
	GetError("str")
	ClearError("str")

*PARMS

{"str"} -- string to classify the error

*DESC

These functions can be used to create a custom error handler.

{GetError} returns the error object if a custom error of class {"str"} has been
reported using {Assert}, or {False} if no errors of this class have been
reported.

{ClearError("str")} deletes the same error object that is returned by
{GetError("str")}. It deletes at most one error object. It returns {True} if an
object was found and deleted, and {False} otherwise.


*E.G.

	In> x:=1
	Out> 1;
	In> Assert("bad value", {x,"must be zero"}) x=0
	Out> False;
	In> GetError("bad value")
	Out> {1, "must be zero"};
	In> ClearError("bad value");
	Out> True;
	In> IsError()
	Out> False;

*SEE IsError, Assert, Check, ClearErrors

*CMD CurrentFile, CurrentLine --- show current file and line of input
*CORE
*CALL
	CurrentFile()
	CurrentLine()

*DESC

The functions {CurrentFile} and {CurrentLine} return a string
with the file name of the current file and the current line 
of input respectively.

These functions are most useful in batch file calculations, where
there is a need to determine at which line an error occurred.
One can define a function 

	tst() := Echo({CurrentFile(),CurrentLine()});
which can then be inserted into the input file at various places,
to see how far the interpreter reaches before an error occurs.

*SEE Echo


			Built-in (core) functions

*INTRO
Yacas comes with a small core of built-in functions and a large library of
user-defined functions. Some of these core functions are documented in this
chapter.




It is important for a developer to know which functions are built-in and cannot
be redefined or {Retract}-ed. Also, core functions may be somewhat faster to
execute than functions defined in the script library. All core functions are
listed in the file {corefunctions.h} in the {src/} subdirectory of the Yacas
source tree. The declarations typically look like this:

	SetCommand(LispSubtract, "MathSubtract");
Here {LispSubtract} is the Yacas internal name for the function and {MathSubtract} is the name visible to the Yacas language.
Built-in bodied functions and infix operators are declared in the same file.

*CMD MathNot --- built-in logical "not"
*CORE
*CALL
	MathNot(expression)

*DESC

Returns "False" if "expression" evaluates
to "True", and vice versa.

*CMD MathAnd --- built-in logical "and"

*CALL
	MathAnd(...)

*DESC
Lazy logical {And}: returns {True} if all args evaluate to
{True}, and does this by looking at first, and then at the
second argument, until one is {False}.
If one of the arguments is {False}, {And} immediately returns {False} without
evaluating the rest. This is faster, but also means that none of the
arguments should cause side effects when they are evaluated.

*CMD MathOr --- built-in logical "or"
*CORE
*CALL
	MathOr(...)

{MathOr} is the basic logical "or" function. Similarly to {And}, it is
lazy-evaluated. {And(...)} and {Or(...)} do also exist, defined in the script
library. You can redefine them as infix operators yourself, so you have the
choice of precedence. In the standard scripts they are in fact declared as
infix operators, so you can write {expr1 And expr}.

*CMD BitAnd, BitOr, BitXor --- bitwise arithmetic
*CORE
*CALL
	BitAnd(n,m)
	BitOr(n,m)
	BitXor(n,m)

*DESC
These functions return bitwise "and", "or" and "xor"
of two numbers.

*CMD Equals --- check equality
*CORE
*CALL
	Equals(a,b)

*DESC
Compares evaluated {a} and {b} recursively
(stepping into expressions). So "Equals(a,b)" returns
"True" if the expressions would be printed exactly
the same, and "False" otherwise.

*CMD GreaterThan, LessThan --- comparison predicates
*CORE
*CALL
	GreaterThan(a,b)
	LessThan(a,b)

*PARMS
{a}, {b} -- numbers or strings
*DESC
Comparing numbers or strings (lexicographically).

*EG
	In> LessThan(1,1)
	Out> False;
	In> LessThan("a","b")
	Out> True;


*A {MathExp}
*A {MathLog}
*A {MathPower}
*A {MathSin}
*A {MathCos}
*A {MathTan}
*A {MathArcSin}
*A {MathArcCos}
*A {MathArcTan}
*A {MathSinh}
*A {MathCosh}
*A {MathTanh}
*A {MathArcSinh}
*A {MathArcCosh}
*A {MathArcTanh}
*A {MathGcd}
*A {MathAdd}
*A {MathSubtract}
*A {MathMultiply}
*A {MathDivide}
*A {MathSqrt}
*A {MathFloor}
*A {MathCeil}
*A {MathAbs}
*A {MathMod}
*A {MathDiv}
*CMD Math... --- arbitrary-precision math functions
*CORE
*REM these are not made into code examples to save space
*CALL
	MathGcd(n,m)      (Greatest Common Divisor)
	MathAdd(x,y)      (add two numbers)
	MathSubtract(x,y) (subtract two numbers)
	MathMultiply(x,y) (multiply two numbers)
	MathDivide(x,y)   (divide two numbers)
	MathSqrt(x)    (square root, must be x>=0)
	MathFloor(x)   (largest integer not larger than x)
	MathCeil(x)    (smallest integer not smaller than x)
	MathAbs(x)     (absolute value of x, or |x| )
	MathExp(x)     (exponential, base 2.718...)
	MathLog(x)     (natural logarithm, for x>0)
	MathPower(x,y) (power, x ^ y)
	MathSin(x)     (sine)
	MathCos(x)     (cosine)
	MathTan(x)     (tangent)
	MathSinh(x)     (hyperbolic sine)
	MathCosh(x)     (hyperbolic cosine)
	MathTanh(x)     (hyperbolic tangent)
	MathArcSin(x)   (inverse sine)
	MathArcCos(x)   (inverse cosine)
	MathArcTan(x)   (inverse tangent)
	MathArcSinh(x)  (inverse hyperbolic sine)
	MathArcCosh(x)  (inverse hyperbolic cosine)
	MathArcTanh(x)  (inverse hyperbolic tangent)
	MathDiv(x,y)    (integer division, result is an integer)
	MathMod(x,y)    (remainder of division, or x mod y)

*DESC

These commands perform the calculation of elementary mathematical functions.
The arguments <i>must</i> be numbers.
The reason for the prefix {Math} is that
the library needs to define equivalent
non-numerical functions for symbolic computations, such as {Exp}, {Sin} and so on.

Note that all functions, such as the {MathPower}, {MathSqrt}, {MathAdd} etc., accept integers as well as floating-point numbers.
The resulting values may be integers or floats.
If the mathematical result is an exact integer, then the integer is returned.
For example, {MathSqrt(25)} returns the integer {5}, and {MathPower(2,3)} returns the integer {8}.
In such cases, the integer result is returned even if the calculation requires more digits than set by {Precision}.
However, when the result is mathematically not an integer, the functions return a floating-point result which is correct only to the current precision.

*EG
	In> Precision(10)
	Out> True
	In> Sqrt(10)
	Out> Sqrt(10)
	In> MathSqrt(10)
	Out> 3.16227766
	In> MathSqrt(490000*2^150)
	Out> 26445252304070013196697600
	In> MathSqrt(490000*2^150+1)
	Out> 0.264452523e26
	In> MathPower(2,3)
	Out> 8
	In> MathPower(2,-3)
	Out> 0.125


*A {FastExp}
*A {FastLog}
*A {FastPower}
*A {FastSin}
*A {FastCos}
*A {FastTan}
*A {FastArcSin}
*A {FastArcCos}
*A {FastArcTan}
*CMD Fast... --- double-precision math functions
*CORE
*CALL
*REM these are not made into code examples to save space

FastExp(x), FastLog(x) (natural logarithm),
FastPower(x,y),
FastSin(x), FastCos(x), FastTan(x),
FastArcSin(x), FastArcCos(x), FastArcTan(x)

*DESC
Versions of these functions using the C++ library. These
should then at least be faster than the arbitrary precision versions.

*CMD ShiftLeft, ShiftRight --- built-in bit shifts
*CORE
*CALL
	ShiftLeft(expr,bits)
	ShiftRight(expr,bits)

*DESC

Shift bits to the left or to the right.



*CMD IsPromptShown --- test for the Yacas prompt option
*CORE
*CALL
	IsPromptShown()
*DESC
Returns {False} if Yacas has been started with the option to suppress the prompt, and {True} otherwise.

*CMD MathLibrary --- obtain current math library name
*CORE
*CALL
	MathLibrary()
*DESC
Returns a string that describes the currently used arbitrary-precision arithmetic library.

Possible names supported at the moment are {"Internal"} and {"Gmp"}, indicating the internal math library {libyacasnumbers} and the GNU Multiple Precision library {libgmp}.

*EG
	In>  MathLibrary()
	Out> "Internal";


*CMD GetTime --- measure the time taken by an evaluation
*CORE
*CALL
	GetTime(expr)
*PARMS
{expr} -- any expression
*DESC
The function {GetTime(expr)} evaluates the expression {expr} and returns the time needed for the evaluation.
The result is returned as a floating-point number of seconds.
The value of the expression {expr} is lost.

The result is the "user time" as reported by the OS, not the real ("wall clock") time.
Therefore, any CPU-intensive processes running alongside Yacas will not significantly affect the result of {GetTime}.

*EG
	In> GetTime(Simplify((a*b)/(b*a)))
	Out> 0.09;

*SEE Time

		Full listing of core functions

The following Yacas functions are currently declared in {corefunctions.h}
as core functions.
The list indicates whether a function is a real function (evaluating its arguments) or a macro (not evaluating its arguments).
Also a function can either take a fixed number of arguments or a variable number (say, 2 or more).
Some kernel functions are additionally declared as operators (bodied, prefix, infix, postfix) with the given precedence.

*A core functions, listing of
*REM yacasapi.chapt is generated from corefunctions.h

*INCLUDE yacasapi.chapt




			The Yacas plugin structure

*INTRO
Yacas supports dynamically loading libraries at runtime. This chapter describes functions for working with plugins.

The plugin feature allows Yacas to interface with other libraries that support
additional functionality. For example, there could be a plugin enabling the
user to script a user interface from within Yacas, or a specific powerful
library to do numeric calculations.

The plugin feature is currently in an experimental stage, and it will
not work on each platform. To be precise, the {libltdl} library is
used to load the plugins. This will only work on platforms which are
supported by {libltdl}, which includes Linux, Mac OS X, Solaris, and
HP-UX.

The remainder of the section is only of interest to users who want to
write plugins themselves. It is assumed that the reader is comfortable
programming in C++.

In addition to the plugin structure in the Yacas engine, there is
a module {cstubgen} (currently still under development) that allows
a rapid creation of a plugin that interfaces to an external library.
Essentially all that is required is
to write a file that looks like the header file of the original
library, but written in Yacas syntax. The module {cstubgen} is then
able to write out a C++ file that can be compiled and linked with
the original library, and then loaded from within Yacas.
To include
a function in the plugin typically takes one line of
Yacas code. There are a few examples in the {plugins/}
directory (the files ending with api.stub). The build system
converts these automatically to the required C++ files. 

In addition to the C++ stub file, {cstubgen} also automatically generates
some documentation on the functions included in the stub. This
documentation is put in a file with extension '.man.txt'.

An example describing the use of {cstubgen} can be found in the essay
<*"Creating plugins for Yacas"|yacasdoc://essays/3/3/*>. 
See <*yacasdoc://essays/5/#build system!plugins*> for a description of
the integration of plugins in the build system.


*CMD DllLoad --- load a plugin
*CORE
*CALL
	DllLoad(file)

*PARMS
{file} -- file name of the plugin

*DESC

{DllLoad} loads the plugin {file} into Yacas. Neither the directory
nor the extension of {file} need to be specified. The result is {True}
if the plugin is loaded successfully, and {False} otherwise.

If the argument {file} does not specify a directory, all directories
in the path for Yacas plugins are searched. This path is built up by
calls to {DllDirectory}. In addition, it contains the default plugin
directory which is specified by the command line option {--dlldir} 
(see <*yacasdoc://ref/1/#command-line options!summary*>).
If this option is not present, the default plugin directory is as
specified at compile time (the default value is
{/usr/local/lib/yacas}).

If the argument {file} does not specify the extension, then first the
extension {.la} is appended. This is the standard extension for
Libtool libraries. If no plugin with this name can be found, a
platform-specific extension is tried (as an example, this is {.so} on
platforms with ELF libraries).

*EG

The following commands load the example plugin, and then call the
function {AddTwoIntegers} which is defined in this plugin.

	 In> DllLoad("example");
	 Out> True
	 In> AddTwoIntegers(2,3);
	 Out> 5

Note that it suffices to specify the string {example}. Yacas knows
that the plugin resides in the file {/usr/local/lib/yacas/example.la}
(if plugins are installed in the default locations).

*SEE DllDirectory, DllUnload, DllEnumerate


*CMD DllUnload --- unload a plugin
*CORE
*CALL
	DllUnload(file)

*PARMS
{file} -- file name of the plugin

*DESC

{DllUnload} unloads a plugin previously loaded with {DllLoad}.  The
argument {file} has to be exactly the same as specified in the call to
{DllLoad}, or the system will not be able to determine which plugin to
unload.  It will scan all plugins which are currently loaded, and
delete the first one found to exactly match.

{DllUnload} always returns {True}, even if no plugin with the name
{file} is found.

*EG notest

First, we load the example plugin, and call the function
{AddTwoIntegers} which is defined in this plugin.

	In> DllLoad("example");
	Out> True
	In> AddTwoIntegers(2,3);
	Out> 5

Then, we unload the plugin, and we check that the function
{AddTwoIntegers} is indeed no longer defined.

	In> DllUnload("example");
	Out> True
	In> AddTwoIntegers(2,3);
	Out> AddTwoIntegers(2,3)

*SEE DllLoad, DllEnumerate


*CMD DllEnumerate --- enumerate all loaded plugins
*CORE
*CALL
	DllEnumerate()

*DESC

{DllEnumerate} returns a list with the names of all the plugins which
are currently loaded.

*EG notest

At startup, no plugins are loaded, so {DllEnumerate()} evaluates to
the empty list.

	In> DllEnumerate();
	Out> {}

This changes after we load the example plugin.

	In> DllLoad("example");
	Out> True
	In> DllEnumerate();
	Out> {"example"}

*SEE DllLoad, DllUnload



*CMD StubApiCStart --- begin a C++ plugin API description
*STD
*CALL
	StubApiCStart(name)

*PARMS
{name} -- string, name of the plugin

*DESC

This function must be called to begin generating a stub file for linking a C or C++ library with Yacas.
A stub specification file needs to start with this
function call, to reset the internal state of Yacas for emitting
a stub C++ file. The parameter {name} should identify the plugin
uniquely.

*SEE StubApiCShortIntegerConstant, StubApiCInclude, StubApiCFunction, StubApiCFile, StubApiCSetEnv, StubApiCPrettyName


*CMD StubApiCPrettyName --- give a descriptive name for documentation
*STD
*CALL
	StubApiCPrettyName(name)

*PARMS
{name} -- descriptive name of the plugin

*DESC

This function sets a descriptive name for a plugin. It should be very short, for example, "plotting" or "XYZ library", and it should not contain the word "plugin".

This name is used for producing documentation. It will appear in the title of the documentation section describing the plugin.

*EG notest
	StubApiCPrettyName("multithreaded GUI")
The documentation section will be titled "The multithreaded GUI plugin".

*SEE StubApiCSStart, StubApiCPluginPurpose




*CMD StubApiCPluginPurpose --- document the purpose of the plugin
*STD
*CALL
	StubApiCPluginPurpose(text)

*PARMS
{text} -- a short description of the plugin

*DESC
The text should be in the reference manual plain text format.
(See <*the documentation on the format|yacasdoc://essays/5/2/*>.)
It will be inserted after the section title in the plugin documentation.

*SEE StubApiCRemark, StubApiCPrettyName

*CMD StubApiCShortIntegerConstant --- declare integer constant in plugin
*STD
*CALL
	StubApiCShortIntegerConstant(const,value)

*PARMS

{const} -- string representing the global variable to be bound runtime

{value} -- integer value the global should be bound to

*DESC

define a constant 'const' to have value 'value'.  The value should
be short integer constant. This is useful for linking in
defines and enumerated values into Yacas.
If the library for instance has a define
	#define FOO 10
Then
	StubApiCShortIntegerConstant("FOO","FOO")
will bind the global variable FOO to the value for FOO defined in
the library header file.

*SEE StubApiCStart, StubApiCInclude, StubApiCFunction, StubApiCFile, StubApiCSetEnv



*CMD StubApiCInclude --- declare include file in plugin
*STD
*CALL
	StubApiCInclude(file)

*PARMS

{file} -- file to include from the library the plugin is based on

*DESC

Declare an include file (a header file for the library, for instance)
The delimiters need to be specified too. So, for a standard library
like the one needed for OpenGL, you need to specify
	StubApiCInclude("\<GL/gl.h\>")
and for user include file:
	StubApiCInclude("\"GL/gl.h\"")

*SEE StubApiCStart, StubApiCShortIntegerConstant, StubApiCFunction, StubApiCFile, StubApiCSetEnv

*CMD StubApiCFunction --- declare C++ function in plugin
*STD
*CALL
	StubApiCFunction(returntype,fname,args)
	StubApiCFunction(returntype,fname,
	  fname2,args)

*PARMS

{returntype} -- return type of new function

{fname} -- function of built-in function

{fname2} -- (optional) function name to be used from within Yacas

{args} -- list of arguments to the function

*DESC

This function declares a new library function, along with its
calling sequence. {cstubgen} will then generate the C++ code
required to call this function.

Return type, function name, and list of arguments should be
literal strings (surrounded by quotes).

If {fname2} is not supplied, it will be assumed to be the same as fname.

The return types currently supported are "{int}", "{double}" and "{void}".

The argument values that are currently supported
are "{int}", "{double}", and "{input_string}".

Argument types can be specified simply as a string referring to their
type, like {"int"}, or they can be lists with an additional element
stating the name of the variable: {{"int","n"}}. The variable
will then show up in the automatically generated documentation as
having the name "n".

*E.G.

To define an OpenGL function {glVertex3d} that accepts three
doubles and returns void:

	StubApiCFunction("void","glVertex3d",
	  {"double","double","double"});

*SEE StubApiCStart, StubApiCShortIntegerConstant, StubApiCInclude, StubApiCFile, StubApiCSetEnv

*CMD StubApiCRemark --- provide more documentation for plugin
*STD
*CALL
	StubApiCRemark(string)

*PARMS

{string} -- text to be added to the documentation

*DESC

{StubApiCRemark} adds the text to the stub documentation
file that gets generated automatically. The documentation is put in
a {.man.txt} file while the input file is being processed, so adding
a remark on a function just after a function declaration adds a remark
on that function.
The format of the text must be that of the reference manual source in plaintext, see <*documentation on the format|yacasdoc://essays/5/2*>.


*SEE StubApiCPrettyName, StubApiCPluginPurpose, StubApiCStart, StubApiCSetEnv, StubApiCFile

*CMD StubApiCSetEnv --- access Yacas environment in plugin
*STD
*CALL
	StubApiCSetEnv(func)

*PARMS

{func} -- name of the function to call to set the environment variable

*DESC

This function forces the plugin to call the function func, with as
argument {LispEnvironment&} {aEnvironment}. This lets the plugin store
the environment class (which is needed for almost any thing to do with
Yacas), somewhere in a global variable. aEnvironment can then be used
from within a callback function in the plugin that doesn't take the
extra argument by design.

There needs to ba a function in the plugin somewhere of the form

	static LispEnvironment* env = NULL;
	void GlutSetEnv(LispEnvironment& aEnv)
	{
	    env = &aEnv;
	}
Then calling
	StubApiCSetEnv("GlutSetEnv");
will force the plugin to call {GlutSetEnv} at load time. All functions
in the plugin will then have access to the Yacas environment.

*SEE StubApiCStart, StubApiCShortIntegerConstant, StubApiCInclude, StubApiCFunction, StubApiCFile

*CMD StubApiCFile --- set file name for plugin API
*STD
*CALL
	StubApiCFile(pluginname,basename)

*PARMS

{pluginname} -- name of the plugin, as passed to {DllLoad}

{basename} -- name for the generation of the stub file

*DESC

Generate the C++ stub file, "basename.cc", and a documentation file
named "basename.description". The descriptions are automatically
generated while adding functions and constants to the stub.

{pluginname} should be the name as passed to {DllLoad}. If the
plugin name is loaded with {DllLoad("example")}, then pluginname
should be {"example"}.

*SEE StubApiCStart, StubApiCShortIntegerConstant, StubApiCInclude, StubApiCFunction, StubApiCSetEnv

*CMD StubApiCStruct --- declare C struct in plugin
*STD
*CALL
	StubApiCStruct(name)
	StubApiCStruct(name,freefunction)

*PARMS

{name} -- name of structure

{freefunction} -- function that can be called to clean up the object

*DESC

{StubApiCStruct} declares a {struct} in a specific library. The name
should be followed by an asterisk (clearly showing that it is a pointer).
After that, in the stub API definition, this type can be used as
argument or return type to functions to the library.

By default the struct will be deleted from memory with a normal
call to {free(...)}. This can be overridden with a function given
as second argument, freefunction. This is needed in the case where
there are additional operations that need to be performed in order
to delete the object from memory.

*E.G.

In a library header file, define:

	typedef struct SomeStruct
	{
	  int a;
	  int b;
	} SomeStruct;
Then in the stub file you can declare this struct by calling:

	StubApiCStruct("SomeStruct*")

*SEE StubApiCFunction


			Generic objects

*INTRO Generic objects are objects that are implemented in C++, but
can be accessed through the Yacas interpreter.

*CMD IsGeneric --- check for generic object
*CORE
*CALL
	IsGeneric(object)

*DESC
Returns {True} if an object is of a generic object type.

*CMD GenericTypeName --- get type name
*CORE
*CALL
	GenericTypeName(object)

*DESC
Returns a string representation of
the name of a generic object.

EG

	In> GenericTypeName(ArrayCreate(10,1))
	Out> "Array";

*CMD ArrayCreate --- create array
*CORE
*CALL
	ArrayCreate(size,init)

*DESC
Creates an array with {size} elements, all initialized to the
value {init}.

*CMD ArraySize --- get array size
*CORE
*CALL
	ArraySize(array)

*DESC
Returns the size of an array (number of elements in the array).

*CMD ArrayGet --- fetch array element
*CORE
*CALL
	ArrayGet(array,index)

*DESC
Returns the element at position index in the array passed. Arrays are treated
as base-one, so {index} set to 1 would return the first element.

Arrays can also be accessed through the {[]} operators. So
{array[index]} would return the same as {ArrayGet(array, index)}.

*CMD ArraySet --- set array element
*CORE
*CALL
	ArraySet(array,index,element)

*DESC
Sets the element at position index in the array passed to the value
passed in as argument to element. Arrays are treated
as base-one, so {index} set to 1 would set first element.

Arrays can also be accessed through the {[]} operators. So
{array[index] := element} would do the same as {ArraySet(array, index,element)}.

*CMD ArrayCreateFromList --- convert list to array
*CORE
*CALL
	ArrayCreateFromList(list)

*DESC
Creates an array from the contents of the list passed in.

*CMD ListFromArray --- convert array to list
*CORE
*CALL
	ListFromArray(array)

*DESC
Creates a list from the contents of the array passed in.



			The Yacas test suite

*INTRO This chapter describes commands used for verifying correct performance
of Yacas.

Yacas comes with a test suite which can be found in
the directory {tests/}. Typing 
	make test
on the command line after Yacas was built will run the test.
This test can be run even before {make install}, as it only 
uses files in the local directory of the Yacas source tree.
The default extension for test scripts is {.yts} (Yacas test script).

The verification commands described in this chapter only  display the
expressions that do not evaluate correctly. Errors do not terminate the
execution of the Yacas script that uses these testing commands, since they are
meant to be used in test scripts.


*CMD Verify, TestYacas, LogicVerify --- verifying equivalence of two expressions
*STD
*CALL
	Verify(question,answer)
	TestYacas(question,answer)
	LogicVerify(question,answer)

*PARMS

{question} -- expression to check for

{answer} -- expected result after evaluation

*DESC

The commands {Verify}, {TestYacas}, {LogicVerify} can be used to verify that an
expression is <I>equivalent</I> to  a correct answer after evaluation. All
three commands return {True} or {False}.

For some calculations, the demand that two expressions
are <I>identical</I> syntactically is too stringent. The 
Yacas system might change at various places in the future,
but $ 1+x $ would still be equivalent, from a mathematical
point of view, to $ x+1 $.

The general problem of deciding that two expressions $ a $ and $ b $
are equivalent, which is the same as saying that $ a-b=0 $ , 
is generally hard to decide on. The following commands solve
this problem by having domain-specific comparisons.

The comparison commands do the following comparison types:

*	{Verify} -- verify for literal equality. 
This is the fastest and simplest comparison, and can be 
used, for example, to test that an expression evaluates to $ 2 $.
*	{TestYacas} -- compare two expressions after simplification as 
multivariate polynomials. If the two arguments are equivalent
multivariate polynomials, this test succeeds. {TestYacas} uses {Simplify}. Note: {TestYacas} currently should not be used to test equality of lists.
*	{LogicVerify} -- Perform a test by using {CanProve} to verify that from 
{question} the expression {answer} follows. This test command is 
used for testing the logic theorem prover in Yacas.

*E.G.

	In> Verify(1+2,3)
	Out> True;
	In> Verify(x*(1+x),x^2+x)
	******************
	x*(x+1) evaluates to x*(x+1) which differs
	  from x^2+x
	******************
	Out> False;
	In> TestYacas(x*(1+x),x^2+x)
	Out> True;
	In> Verify(a And c Or b And Not c,a Or b)
	******************
	 a And c Or b And Not c evaluates to  a And c
	  Or b And Not c which differs from  a Or b
	******************
	Out> False;
	In> LogicVerify(a And c Or b And Not c,a Or b)
	Out> True;
	In> LogicVerify(a And c Or b And Not c,b Or a)
	Out> True;

*SEE Simplify, CanProve, KnownFailure


*CMD KnownFailure --- Mark a test as a known failure
*STD
*CALL
	KnownFailure(test)

*PARMS

{test} -- expression that should return {False} on failure

*DESC

The command {KnownFailure} marks a test as known to fail
by displaying a message to that effect on screen. 

This might be used by developers when they have no time
to fix the defect, but do not wish to alarm users who download
Yacas and type {make test}.

*E.G.

	In> KnownFailure(Verify(1,2))
	Known failure:
	******************
	 1 evaluates to  1 which differs from  2
	******************
	Out> False;
	In> KnownFailure(Verify(1,1))
	Known failure:
	Failure resolved!
	Out> True;

*SEE Verify, TestYacas, LogicVerify

*CMD RoundTo --- Round a real-valued result to a set number of digits
*STD
*CALL
	RoundTo(number,precision)

*PARMS

{number} -- number to round off

{precision} -- precision to use for round-off

*DESC

The function {RoundTo} rounds a floating point number to a
specified precision, allowing for testing for correctness
using the {Verify} command.

*E.G.

	In> N(RoundTo(Exp(1),30),30)
	Out> 2.71828182110230114951959786552;
	In> N(RoundTo(Exp(1),20),20)
	Out> 2.71828182796964237096;

*SEE Verify, VerifyArithmetic, VerifyDiv



*CMD VerifyArithmetic, RandVerifyArithmetic, VerifyDiv --- Special purpose arithmetic verifiers
*STD
*CALL
	VerifyArithmetic(x,n,m)
	RandVerifyArithmetic(n)
	VerifyDiv(u,v)

*PARMS

{x}, {n}, {m}, {u}, {v} -- integer arguments

*DESC

The commands {VerifyArithmetic} and {VerifyDiv} test a 
mathematic equality which should hold, testing that the
result returned by the system is mathematically correct
according to a mathematically provable theorem.

{VerifyArithmetic} verifies for an arbitrary set of numbers
$ x $, $ n $ and $ m $ that
$$ (x^n-1)*(x^m-1) = x^(n+m)-(x^n)-(x^m)+1 $$.

The left and right side represent two ways to arrive at the
same result, and so an arithmetic module actually doing the
calculation does the calculation in two different ways. 
The results should be exactly equal.

{RandVerifyArithmetic(n)} calls {VerifyArithmetic} with
random values, {n} times.

{VerifyDiv(u,v)} checks that 
$$ u = v*Div(u,v) + Mod(u,v) $$.


*E.G.

	In> VerifyArithmetic(100,50,60)
	Out> True;
	In> RandVerifyArithmetic(4)
	Out> True;
	In> VerifyDiv(x^2+2*x+3,x+1)
	Out> True;
	In> VerifyDiv(3,2)
	Out> True;

*SEE Verify

*INCLUDE glossary.chapt


*REM GNU General Public License

			GNU General Public License
*INTRO This chapter contains the GNU General Public License (GPL).
This information is important for everyone using or modifying the Yacas source code or binaries.

*A license
*A licence

*INCLUDE GPL.chapt


*REM GNU Free Documentation License
*INCLUDE FDL.chapt
