About the Admission of Recursive Definitions

You can't just add any formula as an axiom or definition and expect the logic to stay sound! The purported ``definition'' of APP must have several properties to be admitted to the logic as a new axiom

The key property a recursive definition must have is that the recursion terminate. This, along with some syntactic criteria, ensures us that there exists a function satisfying the definition.

Termination must be proved before the definition is admitted. This is done in general by finding a measure of the arguments of the function and a well-founded relation such that the arguments ``get smaller'' every time a recursive branch is taken.

For app the measure is the ``size'' of the first argument, x, as determined by the primitive function acl2-count . The well-founded relation used in this example is o-p , which is the standard ordering on the ordinals less than ``epsilon naught.'' These particular choices for app were made ``automatically'' by ACL2. But they are in fact determined by various ``default'' settings. The user of ACL2 can change the defaults or specify a ``hint'' to the defun command to specify the measure and relation.

You should now return to the Walking Tour.