New Uses of Linear Arithmetic in Automated Theorem Proving by Induction
Journal of Automated Reasoning, Special issue on Mathematical Induction, 16 (1-2), March, 1996, 39-78.
Deepak Kapur and M. Subramaniam

Zhang, Kapur and Krishnamoorthy introduced a cover set method for designing induction schemes for automating proofs by induction from specifications expressed as equations and conditional equations. This method has been implemented in the theorem prover Rewrite Rule Laboratory ( RRL) and a proof management system Tecton built on top of RRL, and it has been used to prove many nontrivial theorems and reason about sequential as well as parallel programs. The cover set method is based on the assumption that a function symbol is defined using a finite set of terminating (conditional or unconditional) rewrite rules. The termination ordering employed in orienting the rules is used to perform proofs by well-founded induction. The left side of the rules are used to design different cases of an induction scheme, and recursive calls to the function made in the right side can be used to design appropriate instantiations for generating induction hypotheses. A weakness of this method is that it relies on syntactic unification for generating an induction scheme for a conjecture. This paper goes a step further by proposing semantic analysis for generating an induction scheme for a conjecture from a cover set. We discuss the use of a decision procedure for Presburger arithmetic (quantifier-free theory of numbers with the addition operation and relational predicates >, <, not =, >=, <=) for performing semantic analysis about numbers. The decision procedure is used to generate appropriate induction schemes for a conjecture using cover sets of function taking numbers as arguments. This extension of the cover set method automates proofs of many theorems which otherwise, require human guidance and hints. The effectiveness of the method is demonstrated using some examples which commonly arise in reasoning about specifications and programs. It is also shown how semantic analysis using a Presburger arithmetic decision procedure can be used for checking the completeness of a cover set of a function defined using operations such as + and - on numbers. Using this check, many function definitions used in a proof of the prime factorization theorem stating that every number can be factored uniquely into prime factors, which had to be checked manually, can now be checked automatically in RRL. The use of the decision procedure for guiding generalization for generating conjectures and merging induction schemes is also illustrated.