15-816 Linear Logic
Using the represention of linear natural deduction and sequent calculus from the previous lecture, we now present and implement the proofs that the two systems are equivalent in the sense that a proposition is provable in natural deduction iff it is provable in sequent calculus.
The implementation takes the form of two relations, one for each direction. These relations (implemented as type families) can be computed under the logic programming interpretation to map natural deductions to sequent derivations and vice versa.
However, type-checking is not sufficient to guarantee that these relations represent proofs. For that, we need to check three properties of the LLF signature: