### 11.4 Functions and Variables for Predicates

Function: charfun (p)

  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 ·

Function: compare (x, y)

  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 ·

Function: equal (a, b)

  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

  Categories:Operators ·

Function: notequal (a, b)

  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 ·

Function: unknown (expr)

  Returns `true` if and only if expr contains an operator or function not recognized by the Maxima simplifier.

  Categories:Predicate functions ·Simplification functions ·

Function: zeroequiv (expr, v)

  Tests whether the expression expr in the variable v is equivalent to zero, returning `true`, `false`, or `dontknow`.

  `zeroequiv` has these restrictions:

  1. Do not use functions that Maxima does not know how to differentiate and evaluate.
  2. If the expression has poles on the real line, there may be errors in the result (but this is unlikely to occur).
  3. If the expression contains functions which are not solutions to first order differential equations (e.g. Bessel functions) there may be incorrect results.
  4. The algorithm uses evaluation at randomly chosen points for carefully selected subexpressions. This is always a somewhat hazardous business, although the algorithm tries to minimize the potential for error.

  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 ·