Strong Sums in Focused Logic

A useful connective that has not previously been made to work in focused logic is the strong sum, a form of dependent sum that is eliminated by projection rather than pattern matching. This makes strong sums powerful, but it also creates a problem adapting them to focusing: The type of the right projection from a strong sum refers to the term being projected from, but due to the structure of focused logic, that term is not available. In this work we confirm that strong sums can be viewed as a negative connective in focused logic. The key is to resolve strong sums' dependencies eagerly, before projection can see them, using a notion of selfification adapted from module type systems. We validate the logic by proving cut admissibility and identity expansion. All the proofs are formalized in Coq.