Lemma Discovery and Middle-Out Reasoning for Automated Inductive Proofs

Lemma speculation has long been considered a promising technique to automate the discovery of missing lemmas for inductive proofs. This technique involves speculating a schematic lemma that becomes incrementally instantiated by unification as the proof continues. This synthesis process is known as middle-out reasoning. We have extended lemma speculation, and more generally middle-out reasoning, to dynamic rippling for higher-order domains, implemented it in the IsaPlanner system and improved the technique to ensure termination. This provides a practical basis for exploring the applications of middle-out reasoning. We demonstrate such an application by performing a critical and comparative evaluation of lemma speculation. This shows that when lemma speculation is applied it often finds the needed lemmas to complete the proof, but it is not applicable as often as initially expected. In comparison, we show that simpler proof methods combined with theory formation methods offer an effective alternative.