MALL proof nets identify proofs modulo rule commutation

The proof nets for MALL (Multiplicative Additive Linear Logic [2], without units) introduced in [4, 5] solved numerous issues with monomial proof nets [3], for example: • There is a simple (deterministic) translation function from cut-free proofs to proof nets. • Cut elimination is simply defined and strongly normalising. • Proof nets form a semi (i.e., unit-free) star-autonomous category with (co)products. A proof net is a set of linkings on a sequent. Each linking is a set of links between complementary formula leaves (literal occurrences). Figure 1 illustrates the translation of a proof into a proof net. In this paper we prove that the translation precisely captures proofs modulo rule commutation: two proofs translate to the same proof net if and only if one can be obtained from the other by a succession of rule commutations. A rule commutation is a transposition of adjacent rules that preserves subproofs immediately above, with possible duplication/identification, for example