Natural Deduction for Intuitionistic Non-Commutative Linear Logic

We present a system of natural deduction and associated term calculus for intuitionistic non-commutative linear logic (INCLL) as a conservative extension of intuitionistic linear logic. We prove subject reduction and the existence of canonical forms in the implicational fragment. We also illustrate by means of an example how the proof term calculus can be employed in the setting of logical frameworks to capture useful representation invariants.