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alphabetic is a property type recognized by declare.
The expression declare(s, alphabetic) tells Maxima to recognize
as alphabetic all of the characters in s, which must be a string.
See also Identifiers.
Example:
(%i1) xx\~yy\`\@ : 1729;
(%o1) 1729
(%i2) declare ("~`@", alphabetic);
(%o2) done
(%i3) xx~yy`@ + @yy`xx + `xx@@yy~;
(%o3) `xx@@yy~ + @yy`xx + 1729
(%i4) listofvars (%);
(%o4) [@yy`xx, `xx@@yy~]
The command declare(x, bindtest) tells Maxima to trigger an error
when the symbol x is evaluated unbound.
(%i1) aa + bb; (%o1) bb + aa (%i2) declare (aa, bindtest); (%o2) done (%i3) aa + bb; aa unbound variable -- an error. Quitting. To debug this try debugmode(true); (%i4) aa : 1234; (%o4) 1234 (%i5) aa + bb; (%o5) bb + 1234
declare(a, constant) declares a to be a constant. The
declaration of a symbol to be constant does not prevent the assignment of a
nonconstant value to the symbol.
Example:
(%i1) declare(c, constant); (%o1) done (%i2) constantp(c); (%o2) true (%i3) c : x; (%o3) x (%i4) constantp(c); (%o4) false
Returns true if expr is a constant expression, otherwise returns
false.
An expression is considered a constant expression if its arguments are
numbers (including rational numbers, as displayed with /R/),
symbolic constants such as %pi, %e, and %i,
variables bound to a constant or declared constant by declare,
or functions whose arguments are constant.
constantp evaluates its arguments.
See the property constant which declares a symbol to be constant.
Examples:
(%i1) constantp (7 * sin(2)); (%o1) true (%i2) constantp (rat (17/29)); (%o2) true (%i3) constantp (%pi * sin(%e)); (%o3) true (%i4) constantp (exp (x)); (%o4) false (%i5) declare (x, constant); (%o5) done (%i6) constantp (exp (x)); (%o6) true (%i7) constantp (foo (x) + bar (%e) + baz (2)); (%o7) false (%i8)
Assigns the atom or list of atoms a_i the property or list of properties p_i. When a_i and/or p_i are lists, each of the atoms gets all of the properties.
declare quotes its arguments. declare always returns done.
As noted in the description for each declaration flag, for some flags
featurep(object, feature) returns true if object
has been declared to have feature.
For more information about the features system, see features. To
remove a property from an atom, use remove.
declare recognizes the following properties:
additiveTells Maxima to simplify a_i expressions by the substitution
a_i(x + y + z + ...) -->
a_i(x) + a_i(y) + a_i(z) + ....
The substitution is carried out on the first argument only.
alphabeticTells Maxima to recognize all characters in a_i (which must be a string) as alphabetic characters.
antisymmetric, commutative, symmetricTells Maxima to recognize a_i as a symmetric or antisymmetric
function. commutative is the same as symmetric.
bindtestTells Maxima to trigger an error when a_i is evaluated unbound.
constantTells Maxima to consider a_i a symbolic constant.
even, oddTells Maxima to recognize a_i as an even or odd integer variable.
evenfun, oddfunTells Maxima to recognize a_i as an odd or even function.
evflagMakes a_i known to the ev function so that a_i is bound
to true during the execution of ev when a_i appears as
a flag argument of ev.
evfunMakes a_i known to ev so that the function named by a_i
is applied when a_i appears as a flag argument of ev.
featureTells Maxima to recognize a_i as the name of a feature. Other atoms may then be declared to have the a_i property.
increasing, decreasingTells Maxima to recognize a_i as an increasing or decreasing function.
integer, nonintegerTells Maxima to recognize a_i as an integer or noninteger variable.
integervaluedTells Maxima to recognize a_i as an integer-valued function.
lassociative, rassociativeTells Maxima to recognize a_i as a right-associative or left-associative function.
linearEquivalent to declaring a_i both outative and
additive.
mainvarTells Maxima to consider a_i a "main variable". A main variable
succeeds all other constants and variables in the canonical ordering of
Maxima expressions, as determined by ordergreatp.
multiplicativeTells Maxima to simplify a_i expressions by the substitution
a_i(x * y * z * ...) -->
a_i(x) * a_i(y) * a_i(z) * ....
The substitution is carried out on the first argument only.
naryTells Maxima to recognize a_i as an n-ary function.
The nary declaration is not the same as calling the nary
function. The sole effect of declare(foo, nary) is to instruct the
Maxima simplifier to flatten nested expressions, for example, to simplify
foo(x, foo(y, z)) to foo(x, y, z).
nonarrayTells Maxima to consider a_i not an array. This declaration prevents multiple evaluation of a subscripted variable name.
nonscalarTells Maxima to consider a_i a nonscalar variable. The usual application is to declare a variable as a symbolic vector or matrix.
nounTells Maxima to parse a_i as a noun. The effect of this is to
replace instances of a_i with 'a_i or
nounify(a_i), depending on the context.
outativeTells Maxima to simplify a_i expressions by pulling constant factors out of the first argument.
When a_i has one argument, a factor is considered constant if it is a literal or declared constant.
When a_i has two or more arguments, a factor is considered constant if the second argument is a symbol and the factor is free of the second argument.
posfunTells Maxima to recognize a_i as a positive function.
rational, irrationalTells Maxima to recognize a_i as a rational or irrational real variable.
real, imaginary, complexTells Maxima to recognize a_i as a real, pure imaginary, or complex variable.
scalarTells Maxima to consider a_i a scalar variable.
Examples of the usage of the properties are available in the documentation for each separate description of a property.
The commands declare(f, decreasing) or
declare(f, increasing) tell Maxima to recognize the function
f as an decreasing or increasing function.
See also declare for more properties.
Example:
(%i1) assume(a > b); (%o1) [a > b] (%i2) is(f(a) > f(b)); (%o2) unknown (%i3) declare(f, increasing); (%o3) done (%i4) is(f(a) > f(b)); (%o4) true
declare(a, even) or declare(a, odd) tells Maxima to
recognize the symbol a as an even or odd integer variable. The
properties even and odd are not recognized by the functions
evenp, oddp, and integerp.
See also declare and askinteger.
Example:
(%i1) declare(n, even); (%o1) done (%i2) askinteger(n, even); (%o2) yes (%i3) askinteger(n); (%o3) yes (%i4) evenp(n); (%o4) false
Maxima understands two distinct types of features, system features and features
which apply to mathematical expressions. See also status for information
about system features. See also features and featurep for
information about mathematical features.
feature itself is not the name of a function or variable.
Attempts to determine whether the object a has the feature f on the
basis of the facts in the current database. If so, it returns true,
else false.
Note that featurep returns false when neither f
nor the negation of f can be established.
featurep evaluates its argument.
See also declare and features.
(%i1) declare (j, even)$ (%i2) featurep (j, integer); (%o2) true
Maxima recognizes certain mathematical properties of functions and variables. These are called "features".
declare (x, foo) gives the property foo
to the function or variable x.
declare (foo, feature) declares a new feature foo.
For example,
declare ([red, green, blue], feature)
declares three new features, red, green, and blue.
The predicate featurep (x, foo)
returns true if x has the foo property,
and false otherwise.
The infolist features is a list of known features. These are
integer noninteger even odd rational irrational real imaginary complex analytic increasing decreasing oddfun evenfun posfun constant commutative lassociative rassociative symmetric antisymmetric integervalued
plus any user-defined features.
features is a list of mathematical features. There is also a list of
non-mathematical, system-dependent features. See status.
Example:
(%i1) declare (FOO, feature); (%o1) done (%i2) declare (x, FOO); (%o2) done (%i3) featurep (x, FOO); (%o3) true
Retrieves the user property indicated by i associated with
atom a or returns false if a doesn’t have property i.
get evaluates its arguments.
(%i1) put (%e, 'transcendental, 'type);
(%o1) transcendental
(%i2) put (%pi, 'transcendental, 'type)$
(%i3) put (%i, 'algebraic, 'type)$
(%i4) typeof (expr) := block ([q],
if numberp (expr)
then return ('algebraic),
if not atom (expr)
then return (maplist ('typeof, expr)),
q: get (expr, 'type),
if q=false
then errcatch (error(expr,"is not numeric.")) else q)$
(%i5) typeof (2*%e + x*%pi);
x is not numeric.
(%o5) [[transcendental, []], [algebraic, transcendental]]
(%i6) typeof (2*%e + %pi);
(%o6) [transcendental, [algebraic, transcendental]]
declare(a, integer) or declare(a, noninteger) tells
Maxima to recognize a as an integer or noninteger variable.
See also declare.
Example:
(%i1) declare(n, integer, x, noninteger); (%o1) done (%i2) askinteger(n); (%o2) yes (%i3) askinteger(x); (%o3) no
declare(f, integervalued) tells Maxima to recognize f as an
integer-valued function.
See also declare.
Example:
(%i1) exp(%i)^f(x);
%i f(x)
(%o1) (%e )
(%i2) declare(f, integervalued);
(%o2) done
(%i3) exp(%i)^f(x);
%i f(x)
(%o3) %e
The command declare(a, nonarray) tells Maxima to consider a not
an array. This declaration prevents multiple evaluation, if a is a
subscripted variable.
See also declare.
Example:
(%i1) a:'b$ b:'c$ c:'d$
(%i4) a[x];
(%o4) d
x
(%i5) declare(a, nonarray);
(%o5) done
(%i6) a[x];
(%o6) a
x
Makes atoms behave as does a list or matrix with respect to the dot operator.
See also declare.
Returns true if expr is a non-scalar, i.e., it contains
atoms declared as non-scalars, lists, or matrices.
declare (f, posfun) declares f to be a positive function.
is (f(x) > 0) yields true.
See also declare.
Displays the property with the indicator i associated with the atom
a. a may also be a list of atoms or the atom all in which
case all of the atoms with the given property will be used. For example,
printprops ([f, g], atvalue). printprops is for properties that
cannot otherwise be displayed, i.e. for atvalue,
atomgrad, gradef, and matchdeclare.
Returns a list of the names of all the properties associated with the atom a.
Default value: []
props are atoms which have any property other than those explicitly
mentioned in infolists, such as specified by atvalue,
matchdeclare, etc., as well as properties specified in the
declare function.
Returns a list of those atoms on the props list which
have the property indicated by prop. Thus propvars (atvalue)
returns a list of atoms which have atvalues.
Assigns value to the property (specified by indicator) of atom. indicator may be the name of any property, not just a system-defined property.
rem reverses the effect of put.
put evaluates its arguments.
put returns value.
Examples:
(%i1) put (foo, (a+b)^5, expr);
5
(%o1) (b + a)
(%i2) put (foo, "Hello", str);
(%o2) Hello
(%i3) properties (foo);
(%o3) [[user properties, str, expr]]
(%i4) get (foo, expr);
5
(%o4) (b + a)
(%i5) get (foo, str);
(%o5) Hello
Assigns value to the property (specified by indicator) of
atom. This is the same as put, except that the arguments are
quoted.
See also get.
Example:
(%i1) foo: aa$
(%i2) bar: bb$
(%i3) baz: cc$
(%i4) put (foo, bar, baz);
(%o4) bb
(%i5) properties (aa);
(%o5) [[user properties, cc]]
(%i6) get (aa, cc);
(%o6) bb
(%i7) qput (foo, bar, baz);
(%o7) bar
(%i8) properties (foo);
(%o8) [value, [user properties, baz]]
(%i9) get ('foo, 'baz);
(%o9) bar
declare(a, rational) or declare(a, irrational) tells
Maxima to recognize a as a rational or irrational real variable.
See also declare.
declare(a, real), declare(a, imaginary), or
declare(a, complex) tells Maxima to recognize a as a real,
pure imaginary, or complex variable.
See also declare.
Removes the property indicated by indicator from atom.
rem reverses the effect of put.
rem returns done if atom had an indicator property
when rem was called, or false if it had no such property.
Removes properties associated with atoms.
remove (a_1, p_1, ..., a_n, p_n)
removes property p_k from atom a_k.
remove ([a_1, ..., a_m], [p_1, ..., p_n], ...)
removes properties p_1, ..., p_n
from atoms a_1, …, a_m.
There may be more than one pair of lists.
remove (all, p) removes the property p from all atoms which
have it.
The removed properties may be system-defined properties such as
function, macro, or mode_declare.
remove does not remove properties defined by put.
A property may be transfun to remove
the translated Lisp version of a function.
After executing this, the Maxima version of the function is executed
rather than the translated version.
remove ("a", operator) or, equivalently,
remove ("a", op) removes from a the operator properties
declared by prefix, infix,
nary, postfix, matchfix, or
nofix. Note that the name of the operator must be written as a quoted
string.
remove always returns done whether or not an atom has a specified
property. This behavior is unlike the more specific remove functions
remvalue, remarray, remfunction, and
remrule.
remove quotes its arguments.
declare(a, scalar) tells Maxima to consider a a scalar
variable.
See also declare.
Returns true if expr is a number, constant, or variable declared
scalar with declare, or composed entirely of numbers,
constants, and such variables, but not containing matrices or lists.
See also the predicate function nonscalarp.
Next: Functions and Variables for Predicates, Previous: Functions and Variables for Properties, Up: Maxima’s Database [Contents][Index]
Activates the contexts context_1, …, context_n.
The facts in these contexts are then available to
make deductions and retrieve information.
The facts in these contexts are not listed by facts ().
The variable activecontexts is the list
of contexts which are active by way of the activate function.
Default value: []
activecontexts is a list of the contexts which are active
by way of the activate function, as opposed to being active because
they are subcontexts of the current context.
askequal(expr1, expr2) attempts to determine from the
assume database whether expr1 is equal to expr2,
and prompts the user if it cannot tell.
If the user provides the answer,
the answer is stored in the assume database
for the duration of the evaluation of the expression currently in progress.
When the evaluation is completed,
the user-provided answer is removed from the database.
askequal returns yes or no,
whether the answer was determined from the assume database
or provided by the user.
See also equal.
askinteger (expr, integer) attempts to determine from the
assume database whether expr is an integer.
askinteger prompts the user if it cannot tell otherwise,
and attempt to install the information in the database if possible.
askinteger (expr) is equivalent to
askinteger (expr, integer).
askinteger (expr, even) and askinteger (expr, odd)
likewise attempt to determine if expr is an even integer or odd integer,
respectively.
First attempts to determine whether the specified
expression is positive, negative, or zero. If it cannot, it asks the
user the necessary questions to complete its deduction. The user’s
answer is recorded in the data base for the duration of the current
computation. The return value of asksign is one of pos,
neg, or zero.
Adds predicates pred_1, …, pred_n to the current context.
If a predicate is inconsistent or redundant with the predicates in the current
context, it is not added to the context. The context accumulates predicates
from each call to assume.
assume returns a list whose elements are the predicates added to the
context or the atoms redundant or inconsistent where applicable.
The predicates pred_1, …, pred_n can only be expressions
with the relational operators < <= equal notequal >= and >.
Predicates cannot be literal equality = or literal inequality #
expressions, nor can they be predicate functions such as integerp.
Compound predicates of the form pred_1 and ... and pred_n
are recognized, but not pred_1 or ... or pred_n.
not pred_k is recognized if pred_k is a relational predicate.
Expressions of the form not (pred_1 and pred_2)
and not (pred_1 or pred_2) are not recognized.
Maxima’s deduction mechanism is not very strong;
there are many obvious consequences which cannot be determined by is.
This is a known weakness.
assume does not handle predicates with complex numbers. If a predicate
contains a complex number assume returns inconsistent or
redundant.
assume evaluates its arguments.
See also is, facts, forget,
context, and declare.
Examples:
(%i1) assume (xx > 0, yy < -1, zz >= 0); (%o1) [xx > 0, yy < - 1, zz >= 0] (%i2) assume (aa < bb and bb < cc); (%o2) [bb > aa, cc > bb] (%i3) facts (); (%o3) [xx > 0, - 1 > yy, zz >= 0, bb > aa, cc > bb] (%i4) is (xx > yy); (%o4) true (%i5) is (yy < -yy); (%o5) true (%i6) is (sinh (bb - aa) > 0); (%o6) true (%i7) forget (bb > aa); (%o7) [bb > aa] (%i8) prederror : false; (%o8) false (%i9) is (sinh (bb - aa) > 0); (%o9) unknown (%i10) is (bb^2 < cc^2); (%o10) unknown
Default value: true
assumescalar helps govern whether expressions expr
for which nonscalarp (expr) is false
are assumed to behave like scalars for certain transformations.
Let expr represent any expression other than a list or a matrix,
and let [1, 2, 3] represent any list or matrix.
Then expr . [1, 2, 3] yields [expr, 2 expr, 3 expr]
if assumescalar is true, or scalarp (expr) is
true, or constantp (expr) is true.
If assumescalar is true, such
expressions will behave like scalars only for commutative
operators, but not for noncommutative multiplication ..
When assumescalar is false, such
expressions will behave like non-scalars.
When assumescalar is all, such expressions will behave like
scalars for all the operators listed above.
Default value: false
When assume_pos is true and the sign of a parameter x
cannot be determined from the current context
or other considerations,
sign and asksign (x) return true.
This may forestall some automatically-generated asksign queries,
such as may arise from integrate or other computations.
By default, a parameter is x such that symbolp (x)
or subvarp (x).
The class of expressions considered parameters can be modified to some extent
via the variable assume_pos_pred.
sign and asksign attempt to deduce the sign of expressions
from the sign of operands within the expression.
For example, if a and b are both positive,
then a + b is also positive.
However, there is no way to bypass all asksign queries.
In particular, when the asksign argument is a
difference x - y or a logarithm log(x),
asksign always requests an input from the user,
even when assume_pos is true and assume_pos_pred is
a function which returns true for all arguments.
Default value: false
When assume_pos_pred is assigned the name of a function
or a lambda expression of one argument x,
that function is called to determine
whether x is considered a parameter for the purpose of assume_pos.
assume_pos_pred is ignored when assume_pos is false.
The assume_pos_pred function is called by sign and asksign
with an argument x
which is either an atom, a subscripted variable, or a function call expression.
If the assume_pos_pred function returns true,
x is considered a parameter for the purpose of assume_pos.
By default, a parameter is x such that symbolp (x)
or subvarp (x).
See also assume and assume_pos.
Examples:
(%i1) assume_pos: true$
(%i2) assume_pos_pred: symbolp$
(%i3) sign (a);
(%o3) pos
(%i4) sign (a[1]);
(%o4) pnz
(%i5) assume_pos_pred: lambda ([x], display (x), true)$
(%i6) asksign (a);
x = a
(%o6) pos
(%i7) asksign (a[1]);
x = a
1
(%o7) pos
(%i8) asksign (foo (a));
x = foo(a)
(%o8) pos
(%i9) asksign (foo (a) + bar (b));
x = foo(a)
x = bar(b)
(%o9) pos
(%i10) asksign (log (a));
x = a
Is a - 1 positive, negative, or zero?
p;
(%o10) pos
(%i11) asksign (a - b);
x = a
x = b
x = a
x = b
Is b - a positive, negative, or zero?
p;
(%o11) neg
Default value: initial
context names the collection of facts maintained by assume and
forget. assume adds facts to the collection named by
context, while forget removes facts.
Binding context to a name foo changes the current context to
foo. If the specified context foo does not yet exist,
it is created automatically by a call to newcontext.
The specified context is activated automatically.
See contexts for a general description of the context mechanism.
Default value: [initial, global]
contexts is a list of the contexts which
currently exist, including the currently active context.
The context mechanism makes it possible for a user to bind together and name a collection of facts, called a context. Once this is done, the user can have Maxima assume or forget large numbers of facts merely by activating or deactivating their context.
Any symbolic atom can be a context, and the facts contained in that
context will be retained in storage until destroyed one by one
by calling forget or destroyed as a whole by calling kill
to destroy the context to which they belong.
Contexts exist in a hierarchy, with the root always being
the context global, which contains information about Maxima that some
functions need. When in a given context, all the facts in that
context are "active" (meaning that they are used in deductions and
retrievals) as are all the facts in any context which is a subcontext
of the active context.
When a fresh Maxima is started up, the user is in a
context called initial, which has global as a subcontext.
See also facts, newcontext, supcontext,
killcontext, activate, deactivate,
assume, and forget.
Deactivates the specified contexts context_1, …, context_n.
If item is the name of a context, facts (item) returns a
list of the facts in the specified context.
If item is not the name of a context, facts (item) returns a
list of the facts known about item in the current context. Facts that
are active, but in a different context, are not listed.
facts () (i.e., without an argument) lists the current context.
Removes predicates established by assume.
The predicates may be expressions equivalent to (but not necessarily identical
to) those previously assumed.
forget (L), where L is a list of predicates,
forgets each item on the list.
Attempts to determine whether the predicate expr is provable from the
facts in the assume database.
If the predicate is provably true or false, is returns
true or false, respectively. Otherwise, the return value is
governed by the global flag prederror. When prederror is
true, is complains with an error message. Otherwise, is
returns unknown.
ev(expr, pred) (which can be written expr, pred at
the interactive prompt) is equivalent to is(expr).
See also assume, facts, and maybe.
Examples:
is causes evaluation of predicates.
(%i1) %pi > %e; (%o1) %pi > %e (%i2) is (%pi > %e); (%o2) true
is attempts to derive predicates from the assume database.
(%i1) assume (a > b); (%o1) [a > b] (%i2) assume (b > c); (%o2) [b > c] (%i3) is (a < b); (%o3) false (%i4) is (a > c); (%o4) true (%i5) is (equal (a, c)); (%o5) false
If is can neither prove nor disprove a predicate from the assume
database, the global flag prederror governs the behavior of is.
(%i1) assume (a > b); (%o1) [a > b] (%i2) prederror: true$ (%i3) is (a > 0); Maxima was unable to evaluate the predicate: a > 0 -- an error. Quitting. To debug this try debugmode(true); (%i4) prederror: false$ (%i5) is (a > 0); (%o5) unknown
Kills the contexts context_1, …, context_n.
If one of the contexts is the current context, the new current context will
become the first available subcontext of the current context which has not been
killed. If the first available unkilled context is global then
initial is used instead. If the initial context is killed, a
new, empty initial context is created.
killcontext refuses to kill a context which is
currently active, either because it is a subcontext of the current
context, or by use of the function activate.
killcontext evaluates its arguments.
killcontext returns done.
Attempts to determine whether the predicate expr is provable from the
facts in the assume database.
If the predicate is provably true or false, maybe returns
true or false, respectively. Otherwise, maybe returns
unknown.
maybe is functionally equivalent to is with
prederror: false, but the result is computed without actually assigning
a value to prederror.
See also assume, facts, and is.
Examples:
(%i1) maybe (x > 0); (%o1) unknown (%i2) assume (x > 1); (%o2) [x > 1] (%i3) maybe (x > 0); (%o3) true
Creates a new, empty context, called name, which
has global as its only subcontext. The newly-created context
becomes the currently active context.
If name is not specified, a new name is created (via gensym) and returned.
newcontext evaluates its argument.
newcontext returns name (if specified) or the new context name.
Attempts to determine the sign of expr on the basis of the facts in the
current data base. It returns one of the following answers: pos
(positive), neg (negative), zero, pz (positive or zero),
nz (negative or zero), pn (positive or negative), or pnz
(positive, negative, or zero, i.e. nothing known).
See also signum.
Creates a new context, called name, which has context as a subcontext. context must exist.
If context is not specified, the current context is assumed.
If name is not specified, a new name is created (via gensym) and returned.
supcontext evaluates its argument.
supcontext returns name (if specified) or the new context name.
Previous: Functions and Variables for Facts, Up: Maxima’s Database [Contents][Index]
Return 0 when the predicate p evaluates to false; return 1 when
the predicate evaluates to true. When the predicate evaluates to
something other than true or false (unknown), return a noun form.
Examples:
(%i1) charfun (x < 1);
(%o1) charfun(x < 1)
(%i2) subst (x = -1, %);
(%o2) 1
(%i3) e : charfun ('"and" (-1 < x, x < 1))$
(%i4) [subst (x = -1, e), subst (x = 0, e), subst (x = 1, e)];
(%o4) [0, 1, 0]
Return a comparison operator op (<, <=, >, >=,
=, or #) such that is (x op y) evaluates
to true; when either x or y depends on %i and
x # y, return notcomparable; when there is no such
operator or Maxima isn’t able to determine the operator, return unknown.
Examples:
(%i1) compare (1, 2); (%o1) < (%i2) compare (1, x); (%o2) unknown (%i3) compare (%i, %i); (%o3) = (%i4) compare (%i, %i + 1); (%o4) notcomparable (%i5) compare (1/x, 0); (%o5) # (%i6) compare (x, abs(x)); (%o6) <=
The function compare doesn’t try to determine whether the real domains of
its arguments are nonempty; thus
(%i1) compare (acos (x^2 + 1), acos (x^2 + 1) + 1); (%o1) <
The real domain of acos (x^2 + 1) is empty.
Represents equivalence, that is, equal value.
By itself, equal does not evaluate or simplify.
The function is attempts to evaluate equal to a Boolean value.
is(equal(a, b)) returns true (or false) if
and only if a and b are equal (or not equal) for all possible
values of their variables, as determined by evaluating
ratsimp(a - b); if ratsimp returns 0, the two
expressions are considered equivalent. Two expressions may be equivalent even
if they are not syntactically equal (i.e., identical).
When is fails to reduce equal to true or false, the
result is governed by the global flag prederror. When prederror
is true, is complains with an error message. Otherwise, is
returns unknown.
In addition to is, some other operators evaluate equal and
notequal to true or false, namely if,
and, or, and not.
The negation of equal is notequal.
Examples:
By itself, equal does not evaluate or simplify.
(%i1) equal (x^2 - 1, (x + 1) * (x - 1));
2
(%o1) equal(x - 1, (x - 1) (x + 1))
(%i2) equal (x, x + 1);
(%o2) equal(x, x + 1)
(%i3) equal (x, y);
(%o3) equal(x, y)
The function is attempts to evaluate equal to a Boolean value.
is(equal(a, b)) returns true when
ratsimp(a - b) returns 0. Two expressions may be equivalent
even if they are not syntactically equal (i.e., identical).
(%i1) ratsimp (x^2 - 1 - (x + 1) * (x - 1)); (%o1) 0 (%i2) is (equal (x^2 - 1, (x + 1) * (x - 1))); (%o2) true (%i3) is (x^2 - 1 = (x + 1) * (x - 1)); (%o3) false (%i4) ratsimp (x - (x + 1)); (%o4) - 1 (%i5) is (equal (x, x + 1)); (%o5) false (%i6) is (x = x + 1); (%o6) false (%i7) ratsimp (x - y); (%o7) x - y (%i8) is (equal (x, y)); (%o8) unknown (%i9) is (x = y); (%o9) false
When is fails to reduce equal to true or false,
the result is governed by the global flag prederror.
(%i1) [aa : x^2 + 2*x + 1, bb : x^2 - 2*x - 1];
2 2
(%o1) [x + 2 x + 1, x - 2 x - 1]
(%i2) ratsimp (aa - bb);
(%o2) 4 x + 2
(%i3) prederror : true;
(%o3) true
(%i4) is (equal (aa, bb));
Maxima was unable to evaluate the predicate:
2 2
equal(x + 2 x + 1, x - 2 x - 1)
-- an error. Quitting. To debug this try debugmode(true);
(%i5) prederror : false;
(%o5) false
(%i6) is (equal (aa, bb));
(%o6) unknown
Some operators evaluate equal and notequal to true or
false.
(%i1) if equal (y, y - 1) then FOO else BAR;
(%o1) BAR
(%i2) eq_1 : equal (x, x + 1);
(%o2) equal(x, x + 1)
(%i3) eq_2 : equal (y^2 + 2*y + 1, (y + 1)^2);
2 2
(%o3) equal(y + 2 y + 1, (y + 1) )
(%i4) [eq_1 and eq_2, eq_1 or eq_2, not eq_1];
(%o4) [false, true, true]
Because not expr causes evaluation of expr,
not equal(a, b) is equivalent to
is(notequal(a, b)).
(%i1) [notequal (2*z, 2*z - 1), not equal (2*z, 2*z - 1)]; (%o1) [notequal(2 z, 2 z - 1), true] (%i2) is (notequal (2*z, 2*z - 1)); (%o2) true
Represents the negation of equal(a, b).
Examples:
(%i1) equal (a, b); (%o1) equal(a, b) (%i2) maybe (equal (a, b)); (%o2) unknown (%i3) notequal (a, b); (%o3) notequal(a, b) (%i4) not equal (a, b); (%o4) notequal(a, b) (%i5) maybe (notequal (a, b)); (%o5) unknown (%i6) assume (a > b); (%o6) [a > b] (%i7) equal (a, b); (%o7) equal(a, b) (%i8) maybe (equal (a, b)); (%o8) false (%i9) notequal (a, b); (%o9) notequal(a, b) (%i10) maybe (notequal (a, b)); (%o10) true
Returns true if and only if expr contains an operator or function
not recognized by the Maxima simplifier.
Tests whether the expression expr in the variable v is equivalent
to zero, returning true, false, or dontknow.
zeroequiv has these restrictions:
For example zeroequiv (sin(2 * x) - 2 * sin(x) * cos(x), x) returns
true and zeroequiv (%e^x + x, x) returns false.
On the other hand zeroequiv (log(a * b) - log(a) - log(b), a) returns
dontknow because of the presence of an extra parameter b.
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