Maxima Operator
=The equation operator.
An expression a = b, by itself, represents
an unevaluated equation, which might or might not hold.
Unevaluated equations may appear as arguments to solve and algsys
or some other functions.
The function is evaluates = to a Boolean value.
is(a = b) evaluates a = b to true when a and b
are identical. That is, a and b are atoms which are identical,
or they are not atoms and their operators are identical and their arguments are identical.
Otherwise, is(a = b) evaluates to false;
it never evaluates to unknown.
When is(a = b) is true, a and b are said to be syntactically equal,
in contrast to equivalent expressions, for which is(equal(a, b)) is true.
Expressions can be equivalent and not syntactically equal.
The negation of = is represented by #.
As with =, an expression a # b, by itself, is not evaluated.
is(a # b) evaluates a # b to
true or false.
In addition to is,
some other operators evaluate = and # to true or false,
namely if, and, or, and not.
Note that because of the rules for evaluation of predicate expressions
(in particular because not expr causes evaluation of expr),
not a = b is equivalent to is(a # b),
instead of a # b.
rhs and lhs return the right-hand and left-hand sides,
respectively, of an equation or inequation.
Examples:
An expression a = b, by itself, represents
an unevaluated equation, which might or might not hold.
(%i1) eq_1 : a * x - 5 * y = 17; (%o1) a x - 5 y = 17 (%i2) eq_2 : b * x + 3 * y = 29; (%o2) 3 y + b x = 29 (%i3) solve ([eq_1, eq_2], [x, y]); 196 29 a - 17 b (%o3) [[x = ---------, y = -----------]] 5 b + 3 a 5 b + 3 a (%i4) subst (%, [eq_1, eq_2]); 196 a 5 (29 a - 17 b) (%o4) [--------- - --------------- = 17, 5 b + 3 a 5 b + 3 a 196 b 3 (29 a - 17 b) --------- + --------------- = 29] 5 b + 3 a 5 b + 3 a (%i5) ratsimp (%); (%o5) [17 = 17, 29 = 29]
is(a = b) evaluates a = b to true when a and b
are syntactically equal (that is, identical).
Expressions can be equivalent and not syntactically equal.
(%i1) a : (x + 1) * (x - 1); (%o1) (x - 1) (x + 1) (%i2) b : x^2 - 1; 2 (%o2) x - 1 (%i3) [is (a = b), is (a # b)]; (%o3) [false, true] (%i4) [is (equal (a, b)), is (notequal (a, b))]; (%o4) [true, false]
Some operators evaluate = and # to true or false.
(%i1) if expand ((x + y)^2) = x^2 + 2 * x * y + y^2 then FOO else BAR; (%o1) FOO (%i2) eq_3 : 2 * x = 3 * x; (%o2) 2 x = 3 x (%i3) eq_4 : exp (2) = %e^2; 2 2 (%o3) %e = %e (%i4) [eq_3 and eq_4, eq_3 or eq_4, not eq_3]; (%o4) [false, true, true]
Because not expr causes evaluation of expr,
not a = b is equivalent to is(a # b).
(%i1) [2 * x # 3 * x, not (2 * x = 3 * x)]; (%o1) [2 x # 3 x, true] (%i2) is (2 * x # 3 * x); (%o2) true