Automating inversion of inductive predicates in Coq

Abstract

An inductive definition of a set is often informally presented by giving some rules that explain how to build the elements of the set. The closure property states that any object is in the set if and only if it has been generated according to the formation rules. This is enough to justify case analysis reasoning: we can read the formation rules backwards to derive the necessary conditions for a given instance to hold. The problem of inversion consists in finding out these conditions.

In this paper we address the problem of deriving inversion lemmas in logical frameworks based on Type Theory that have been extended with inductive definitions at the primitive level. These frameworks associate to each inductive definition a case analysis principle corresponding to the closure property. In this formal context, inversion lemmas can be seen as derived case analysis principles. Though they are intuitively simple they are curiously hard to formalize.

We relate first inversion to completion in logic programming. Then we discuss two general algorithms to generate inversion lemmas. They are presented in the proof assistant system Coq [8] but they can be adapted to any other proof assistant having a similar notion of inductive definition.