On Template-Based Inference of Rich Invariants in Leon

We present an approach for inferring rich invariants involving user-defined recursive functions over numerical and algebraic data types. In our approach, the developer provides the desired shape of the invariant using a set of templates. The templates are quantifier-free affine predicates with unknown coefficients. We also provide an enumeration based strategy for automatically inferring some of the templates. We present a scalable counter-example driven algorithm that finds the unknown coefficients in templates and thus computes expressive inductive invariants. Our algorithm incrementally solves a set of quantified constraints involving recursive functions and data structures. We discuss several optimizations that make the approach scale to complex programs and present an empirical evaluation. Our implementation proves correctness properties as well as symbolic bounds on running times of recursive programs. For example, the implementation establishes that the time taken to insert into a red-black tree is bounded by the logarithm of its size.