The Associativity of App

ACL2!>(let ((a '(1 2))
            (b '(3 4))
            (c '(5 6)))
        (equal (app (app a b) c)
               (app a (app b c))))

Observe that, for the particular a, b, and c above, (app (app a b) c) returns the same thing as (app a (app b c)). Perhaps app is associative. Of course, to be associative means that the above property must hold for all values of a, b, and c, not just the ones tested above.

Wouldn't it be cool if you could type

ACL2!>(equal (app (app a b) c)
             (app a (app b c)))

and have ACL2 compute the value T? Well, you can't! If you try it, you'll get an error message! The message says we can't evaluate that form because it contains free variables, i.e., variables not given values. Click here to see the message.

We cannot evaluate a form on an infinite number of cases. But we can prove that a form is a theorem and hence know that it will always evaluate to true.