Mathematical Induction in Otter-Lambda

Abstract

Otter-lambda is Otter modified by adding code to implement an algorithm for lambda unification. Otter is a resolution-based, clause-language first-order prover that accumulates deduced clauses and uses strategies to control the deduction and retention of clauses. This is the first time that such a first-order prover has been combined in one program with a unification algorithm capable of instantiating variables to lambda terms to assist in the deductions. The resulting prover has all the advantages of the proof-search algorithm of Otter (speed, variety of inference rules, excellent handling of equality) and also the power of lambda unification. We illustrate how these capabilities work well together by using Otter-lambda to find proofs by mathematical induction. Lambda unification instantiates the induction schema to find a useful instance of induction, and then Otter's first-order reasoning can be used to carry out the base case and induction step. If necessary, induction can be used for those, too. We present and discuss a variety of examples of inductive proofs found by Otter-lambda: some in pure Peano arithmetic, some in Peano arithmetic with defined predicates, some in theories combining algebra and the natural numbers, some involving algebraic simplification (used in the induction step) by simplification code from MathXpert, and some involving list induction instead of numerical induction. These examples demonstrate the feasibility and usefulness of adding lambda unification to a first-order prover.