From Display Calculi to Deep Nested Sequent Calculi: Formalised for Full Intuitionistic Linear Logic

Proof theory for a logic with categorical semantics can be developed by the following methodology: define a sound and complete display calculus for an extension of the logic with additional adjunctions; translate this calculus to a shallow inference nested sequent calculus; translate this calculus to a deep inference nested sequent calculus; then prove this final calculus is sound with respect to the original logic. This complex chain of translations between the different calculi require proofs that are technically intricate and involve a large number of cases, and hence are ideal candidates for formalisation. We present a formalisation of this methodology in the case of Full Intuitionistic Linear Logic (FILL), which is multiplicative intuitionistic linear logic extended with par.