## Section: Application Domains

### Proof Certificates

Members of the Parsifal team have shown how to specify a large variety of proof systems—including natural deduction, the sequent calculus, and various tableau and free deduction systems—uniformly using either focused linear logic [52] , [50] or focused intuitionistic logic [43] as the meta-language. In the presence of induction and co-induction, arbitrary finite computations can be embedded into single synthetic steps [34] . Additional work [8] shows that this same framework can also capture resolution refutations as well as Pratt primality certificates.

An important application then of this work in designing synthetic
inference systems based on classical and intuitionistic logic is
that of designing a *broad spectrum proof certificate*. The
definition of proof certificates can be remarkably flexible within
the simple setting of focused proofs.

The most important implications of such a certificate format would be that most of the worlds theorem provers should be able to print out their proofs and communication them to other provers: these other provers could then check such certificates by expanding the synthetic connectives they contain down into a small and fixed set of “micro” inference rules.