Primal Grammars Driven Automated Induction

Automated induction is a powerful method for the validation of critical systems. However, the inductive proof process faces major challenges: it is undecidable and diverges even with small examples. Previous methods have proposed ad-hoc heuristics to speculate on additional lemmas that hopefully stop the divergence. Although these methods have succeeded in proving interesting theorems, they have significant limitations: in particular, they often fail to find appropriate lemmas, and the lemmas they provide may not be valid. We present a new method that allows us to perform inductive proofs in conditional theories. This method automatically detects divergence in proof traces and derives primal grammars as well as new lemmas that schematize the divergent sequence. This new construction allows us to break the divergence and complete the proof. Our method is presented as a set of inference rules whose soundness and refutational completeness have been formally proved. Unlike previous methods, our method is fully automated and has no risk of over-generalization. Moreover, our technique for capturing and schematizing divergence represents the most general decidable schematization, with respect to description power, among all known schematizations. Our method has been implemented in C++ and successfully proved over fifty complex examples that fail with well-known theorem provers (e.g., ACL2, Isabelle, PVS, SPIKE) and related methods for handling divergence in proofs by induction. Our method represents a significant contribution to the field of automated reasoning as it can be integrated with existing automated and interactive inductive proof systems to enhance their performance. Moreover, it has the potential to substantially reduce the time needed for the verification of critical systems.