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11.1 Introduction to Maximas Database | ||
11.2 Functions and Variables for Properties | ||
11.3 Functions and Variables for Facts | ||
11.4 Functions and Variables for Predicates |
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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~]
Categories: Declarations and inferences
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
Categories: Declarations and inferences Constants
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
,
%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
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)
Categories: Predicate functions Constants
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
.
remove a property from an atom, use remove
.
declare
recognizes the following properties:
additive
Tells 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.
alphabetic
Tells Maxima to recognize all characters in a_i (which must be a string) as alphabetic characters.
antisymmetric, commutative, symmetric
Tells Maxima to recognize a_i as a symmetric or antisymmetric
function. commutative
bindtest
Tells Maxima to trigger an error when a_i is evaluated unbound.
constant
Tells Maxima to consider a_i a symbolic constant.
even, odd
Tells Maxima to recognize a_i as an even or odd integer variable.
evenfun, oddfun
Tells Maxima to recognize a_i as an odd or even function.
evflag
Makes 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
. See evflag
.
evfun
Makes 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
.
See evfun
.
feature
Tells Maxima to recognize a_i as the name of a feature. Other atoms may then be declared to have the a_i property.
increasing, decreasing
Tells Maxima to recognize a_i as an increasing or decreasing function.
integer, noninteger
Tells Maxima to recognize a_i as an integer or noninteger variable.
integervalued
Tells Maxima to recognize a_i as an integer-valued function.
lassociative, rassociative
Tells Maxima to recognize a_i as a right-associative or left-associative function.
linear
Equivalent to declaring a_i both outative
and
additive
.
mainvar
Tells 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
.
multiplicative
Tells 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.
nary
Tells 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)
.
nonarray
Tells Maxima to consider a_i not an array. This declaration prevents multiple evaluation of a subscripted variable name.
nonscalar
Tells Maxima to consider a_i a nonscalar variable. The usual application is to declare a variable as a symbolic vector or matrix.
noun
Tells 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.
outative
Tells 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.
posfun
Tells Maxima to recognize a_i as a positive function.
rational, irrational
Tells Maxima to recognize a_i as a rational or irrational real variable.
real, imaginary, complex
Tells Maxima to recognize a_i as a real, pure imaginary, or complex variable.
scalar
Tells 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.
Categories: Declarations and inferences
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
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
Categories: Declarations and inferences
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
,
integerp
.
See also declare
askinteger
.
Example:
(%i1) declare(n, even); (%o1) done (%i2) askinteger(n, even); (%o2) yes (%i3) askinteger(n); (%o3) yes (%i4) evenp(n); (%o4) false
Categories: Declarations and inferences
Maxima understands two distinct types of features, system features and features
which apply to mathematical expressions. See also status
about system features. See also features
featurep
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.
(%i1) declare (j, even)$ (%i2) featurep (j, integer); (%o2) true
Categories: Predicate functions Declarations and inferences
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 commutative lassociative rassociative symmetric antisymmetric
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
Categories: Declarations and inferences
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]]
Categories: Declarations and inferences
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
Categories: Declarations and inferences
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
Categories: Declarations and inferences
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
Categories: Expressions
Makes atoms behave as does a list or matrix with respect to the dot operator.
See also declare
.
Categories: Declarations and inferences Vectors Matrices
Returns true
if expr is a non-scalar, i.e., it contains
atoms declared as non-scalars, lists, or matrices.
See also the predicate function scalarp
declare
.
Categories: Predicate functions Vectors Matrices
declare (f, posfun)
declares f
to be a positive function.
is (f(x) > 0)
yields true
.
See also declare
.
Categories: Declarations and inferences Operators
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
,
matchdeclare
.
Categories: Declarations and inferences Display functions
Returns a list of the names of all the properties associated with the atom a.
Categories: Declarations and inferences
Default value: []
props
are atoms which have any property other than those explicitly
mentioned in infolists
,
atvalue
,
matchdeclare
,
declare
Categories: Declarations and inferences Global variables
Returns a list of those atoms on the props
have the property indicated by prop. Thus propvars (atvalue)
returns a list of atoms which have atvalues.
Categories: Declarations and inferences
Assigns value to the property (specified by indicator) of atom. indicator may be the name of any property, not just a system-defined property.
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
Categories: Declarations and inferences
Assigns value to the property (specified by indicator) of
atom. This is the same as put
,
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
Categories: Declarations and inferences
declare(a, rational)
or declare(a, irrational)
tells
Maxima to recognize a as a rational or irrational real variable.
See also declare
.
Categories: Declarations and inferences
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
.
Categories: Declarations and inferences
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.
Categories: Declarations and inferences
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
,
nofix
.
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
,
remrule
.
remove
quotes its arguments.
Categories: Declarations and inferences
declare(a, scalar)
tells Maxima to consider a a scalar
variable.
See also declare
.
Categories: Declarations and inferences
Returns true
if expr is a number, constant, or variable declared
scalar
declare
,
constants, and such variables, but not containing matrices or lists.
See also the predicate function nonscalarp
.
Categories: Predicate functions Vectors Matrices
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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
of contexts which are active by way of the activate
function.
Categories: Declarations and inferences
Default value: []
activecontexts
is a list of the contexts which are active
by way of the activate
they are subcontexts of the current context.
Categories: Declarations and inferences
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.
Categories: Declarations and inferences
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
.
Categories: Declarations and inferences
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
redunant
.
assume
evaluates its arguments.
See also is
,
facts
,
forget
,
context
,
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
Categories: Declarations and inferences
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.
Categories: Declarations and inferences
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.
Categories: Declarations and inferences
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
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
Categories: Declarations and inferences
Default value: initial
context
names the collection of facts maintained by assume
forget
.
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
Categories: Declarations and inferences
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
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
,
forget
.
Categories: Declarations and inferences
Deactivates the specified contexts context_1, …, context_n.
Categories: Declarations and inferences
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.
Categories: Declarations and inferences
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.
Categories: Declarations and inferences
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
.
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
,
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
Categories: Predicate functions Declarations and inferences
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
.
Categories: Declarations and inferences
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
.
Examples:
(%i1) maybe (x > 0); (%o1) unknown (%i2) assume (x > 1); (%o2) [x > 1] (%i3) maybe (x > 0); (%o3) true
Categories: Predicate functions Declarations and inferences
Creates a new, empty context, called name, which
has global
as its only subcontext. The newly-created context
becomes the currently active context.
newcontext
evaluates its argument.
newcontext
returns name.
Categories: Declarations and inferences
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).
Categories: Declarations and inferences
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.
Categories: Declarations and inferences
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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]
Categories: Mathematical functions
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.
Categories: Declarations and inferences
Represents equivalence, that is, equal value.
By itself, equal
does not evaluate or simplify.
The function is
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
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
.
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
,
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
Categories: Operators
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
Categories: Operators
Returns true
if and only if expr contains an operator or function
not recognized by the Maxima simplifier.
Categories: Predicate functions Simplification functions
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
.
Categories: Predicate functions
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