Section: New Results

Foundational Proof Certificates

We have continued to explore a number of new aspects of framework we call Foundational Proof Certificates (FPCs). Besides having defined and implemented prototype checkers for FPCs in classical and intuitionistic logic [37] we have also extended the proof theory underlying numerous modal logics so that FPCs can be applied to modal logics [35] . We have also extended the notion of FPC to work also in the model checking setting [31] . In both the modal logic and model checking domains, the key to getting FPCs to work is to have descriptions of focused proof systems available for those logics.

Given that FPCs are declarative and semantically simple structures, it has been possible to find numerous applications of them outside the problem of simply checking them. It was shown, for example, that FPCs can be used to help define the semantics of the output from traditional theorem provers [23] . We have also used FPCs as proof outlines in order to define high-level tactics to direct proof search [24] .