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While model checking has often been considered as a practical alternative to building formal proofs, we argue here that the theory of sequent calculus proofs can be used to provide an appealing foundation for model checking. Since the emphasis of model checking is on establishing the truth of a property in a model, we rely on the proof theoretic notion of additive inference rules, since such rules allow provability to directly describe truth conditions. Unfortunately, the additive treatment of quantifiers requires inference rules to have infinite sets of premises and the additive treatment of model descriptions provides no natural notion of state exploration. By employing a focused proof system, it is possible to construct large scale, synthetic rules that also qualify as additive but contain elements of multiplicative inference. These additive synthetic rules—essentially rules built from the description of a model—allow a direct treatment of state exploration. This proof theoretic framework provides a natural treatment of reachability and non-reachability problems, as well as tabled deduction, bisimulation, and winning strategies.

We apply the foundational proof certificate (FPC) framework to the problem of designing high-level outlines of proofs. The FPC framework provides a means to formally define and check a wide range of proof evidence. A focused proof system is central to this framework and such a proof system provides an interesting approach to proof reconstruction during the process of proof checking (relying on an underlying logic programming implementation). Here, we illustrate how the FPC framework can be used to design proof outlines and then to exploit proof checkers as a means for expanding outlines into fully detailed proofs. In order to validate this approach to proof outlines, we have built the ACheck system that allows us to take a sequence of theorems and apply the proof outline "do the obvious induction and close the proof using previously proved lemmas".