Focused Inductive Theorem Proving

Focused proof systems provide means for reducing and structuring the non-determinism involved in searching for sequent calculus proofs. We present a focused proof system for a first-order logic with inductive and co-inductive definitions in which the introduction rules are partitioned into an asynchronous phase and a synchronous phase. These focused proofs allow us to naturally see proof search as being organized around interleaving intervals of computation and more general deduction. For example, entire Prolog-like computations can be captured using a single synchronous phase and many model-checking queries can be captured using an asynchronous phase followed by a synchronous phase. Leveraging these ideas, we have developed an interactive proof assistant, called Tac, for this logic. We describe its high-level design and illustrate how it is capable of automatically proving many theorems using induction and coinduction. Since the automatic proof procedure is structured using focused proofs, its behavior is often rather easy to anticipate and modify. We illustrate the strength of Tac with several examples of proved theorems, some achieved entirely automatically and others achieved with user guidance.