In the first example, we use the keyword linear:

To define commutative and associative operations, use the keywords orderless and flat. orderless means that the arguments of a function have no order, and flat means that f(a,f(b,c)) is equal to f(a,b,c).

One can use define for functional programming:

A nice way to define the greatest common divisor of integers:

The keyword multilinear can be used to define multilinear functions:

Now we use the definemore command to add to the existing definition of H the rule of commutativity (keyword orderless), and a simplification rule for some special arguments:

Define an integration procedure: integration is linear,   equals    when  a  does not depend on  .

We now define the integral of  :

And now the integral for powers of  :

An example with the keyword diff and the command diff :
We define the derivative of  to be :

Define properties of a function  which is linear, has a derivative of , and for which .

Check the derivative of  :

Given that  is linear:

Now Maple can compute the following integral using the fact that  is linear and has derivative :

Even nested functions with  can be integrated:

Since the derivative is given, we can compute limit and series:

Using the neutral operator &m, we define:

We can add more properties: