### 11.2 Functions and Variables for Properties

Property: alphabetic

  `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 ·

Property: bindtest

  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

Property: constant

  `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.

  See `constantp` and `declare`.

  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 ·

Function: constantp (expr)

  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)

  Categories:Predicate functions ·Constants ·

Function: declare (a_1, p_1, a_2, p_2, …)

  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:

  ``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` is the same as `symmetric`.

  ``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`.

  ``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`.

  ``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 ·

Property: decreasing
Property: increasing

  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

  Categories:Declarations and inferences ·

Property: even
Property: odd

  `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

  Categories:Declarations and inferences ·

Property: feature

  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.

Function: featurep (a, f)

  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

  Categories:Predicate functions ·Declarations and inferences ·

Declaration: features

  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

  Categories:Declarations and inferences ·

Function: get (a, i)

  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.

  See also `put` and `qput`.

      (%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 ·

Property: integer
Property: noninteger

  `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 ·

Property: integervalued

  `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 ·

Property: nonarray

  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 ·

Property: nonscalar

  Makes atoms behave as does a list or matrix with respect to the dot operator.

  See also `declare`.

  Categories:Declarations and inferences ·Vectors ·Matrices ·

Function: nonscalarp (expr)

  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` and `declare`.

  Categories:Predicate functions ·Vectors ·Matrices ·

Property: posfun

  `declare (f, posfun)` declares `f` to be a positive function. `is (f(x) > 0)` yields `true`.

  See also `declare`.

  Categories:Declarations and inferences ·Operators ·

Function: printprops
`printprops` (a, i)
`printprops` ([a_1, …, a_n], i)
`printprops` (all, i)

  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`.

  Categories:Declarations and inferences ·Display functions ·

Function: properties (a)

  Returns a list of the names of all the properties associated with the atom a.

  Categories:Declarations and inferences ·

System variable: props

  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.

  Categories:Declarations and inferences ·Global variables ·

Function: propvars (prop)

  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.

  Categories:Declarations and inferences ·

Function: put (atom, value, indicator)

  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.

  See also `qput` and `get`.

  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 ·

Function: qput (atom, value, indicator)

  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

  Categories:Declarations and inferences ·

Property: rational
Property: irrational

  `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 ·

Property: real
Property: imaginary
Property: complex

  `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 ·

Function: rem (atom, indicator)

  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 ·

Function: remove
`remove` (a_1, p_1, …, a_n, p_n)
`remove` ([a_1, …, a_m], [p_1, …, p_n], …)
`remove` ("a", operator)
`remove` (a, transfun)
`remove` (all, p)

  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.

  Categories:Declarations and inferences ·

Property: scalar

  `declare(a, scalar)` tells Maxima to consider a a scalar variable.

  See also `declare`.

  Categories:Declarations and inferences ·

Function: scalarp (expr)

  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`.

  Categories:Predicate functions ·Vectors ·Matrices ·