Deepak Kapur and M. Subramaniam

Speculating intermediate lemmas is one of the main reason of user interaction/guidance while mechanically attempting proofs by induction. An approach for generating intermediate lemmas is developed, and its effectiveness is demonstrated while proving properties of recursively defined functions. The approach is guided by the paradigm of attempting to generate a proof of the conclusion subgoal in an induction step by the application of an induction hypothesis(es). Generation of intermediate conjectures is motivated by attempts to find appropriate instantiations for non-induction variables in the main conjecture. In case, the main conjecture does not have any non-induction variables, such variables are introduced by attempting its generalization. A constraint based paradigm is proposed for guessing the missing side of an intermediate conjecture by identifying constraints on the term schemes introduced for the missing side. Definitions and properties of functions are judiciously used for generating instantiations and intermediate conjectures. Heuristics are identified for performing such analysis. The approach fails if appropriate instantiations of non-induction variables cannot be generated. Otherwise, proofs of intermediate conjectures are attempted and the proposed method is recursively applied. The method has proven to be surprisingly very effective in speculating intermediate lemmas for tail-recursive programs. The method is demonstrated using a number of examples on numbers and lists.