The $::$ operator serves several purposes in Maple, all related to types.

When used in a procedure parameter declaration, the left-hand side of the operator specifies the parameter name, and the right-hand side specifies the expected type of the argument.

The return type of a procedure can be declared by following the procedure's parameter declaration sequence by $::T$. In this context, $::$ serves as an assertion that the procedure returns a value of the specified type, T. If assertions are enabled, returning a value not matching the type will raise an exception.

When used as the left-hand side of an assignment statement or as the first argument to the assign function, $x::T$ asserts that the value being assigned to x is of type T.

A local variable can be declared with a type by using the $::$ operator. Within the procedure in which the variable is declared, any assignment made to it are treated as if the left-hand side of the assignment were written using the $::$ declaration.

In the condition of an if or while statement, $::$ serves as a type testing operator. In this context, $x::T$ is equivalent to $\mathrm{type}\left(x\,T\right)$.

The $::$ operator also acts as a type test when used as an argument to any of the Boolean operators, and, or, not, xor, implies, or the $\mathrm{evalb}$ function.

In any other context, $::$ evaluates its arguments, but does not itself evaluate any further. Thus, it can be used as a data structure or part of a larger data structure. Although it is not required, it is customary to use $::$ as a data structure only when the right-hand side represents a type or a type-like concept. For example, see RealRange.

An expression using the $::$ operator is of type $\mathrm{::}$.

The :: operator is thread-safe as of Maple 15.

For more information on thread safety, see index/threadsafe.

f := proc( x::integer ) x^2 end proc:

$f\left(2\right)$

$4$

$f\left(3.4\right)$

Error, invalid input: f expects its 1st argument, x, to be of type integer, but received 3.4

f := proc( x::integer ) :: odd; x^2 end proc:

$\mathrm{kernelopts}\left(\mathrm{assertlevel}=2\right)\:$

$f\left(3\right)$

$9$

$f\left(4\right)$

Error, (in f) assertion failed: f expects its return value to be of type of odd, but computed 16

$x::\'\mathrm{integer}\'≔3.4$

Error, assertion failed in assignment, expected integer, got 3.4

f := proc( x ) local y::integer;
    y := x
end proc:

$f\left(4.5\right)$

Error, (in f) assertion failed in assignment, expected integer, got 4.5

f := proc( x )
    if x::integer then
    print("an integer")
    else
    print("not an integer")
    end if
end proc:

$f\left(2\right)$

$''an\; integer''$

$f\left(3.4\right)$

$''not\; an\; integer''$

$x≔2$

$x≔2$

$x::\'\mathrm{integer}\'$

$2::?$

$x::\&apos;\mathrm{integer}\&apos;\mathbf{and}x<3$

$\mathrm{true}$

$\mathrm{type}\left(x::\mathrm{symbol}\&comma;\mathrm{`::`}\right)$

$\mathrm{true}$

$\mathrm{evalb}\left(\left(x::\mathrm{string}\right)::\mathrm{`::`}\right)$

$\mathrm{true}$