Automatic Case Analysis in Proof by Induction

Abstract

We propose a new procedure for proof by induction in conditional theories where case analysis is simulated by term rewriting. This technique reduces considerably the number of variables of a conjecture to be considered for applying induction schemes (inductive positions). Our procedure is presented as a set of inference rules whose correctness has been formally proved. Moreover, when the axioms are ground convergent and the defined functions are completely defined over free constructors, it is possible to apply the system for refuting conjectures. The procedure is even refutationally complete for conditional equations with boolean preconditions (under the same hypotheses). The method is entirely implemented in the prover SPIKE. This system has proved interesting examples in a completely automatic way, that is, without interaction with the user and without ad-hoc heuristics. It has also proved the challenging Gilbreath card trick, with only 5 easy lemmas.