The rewriting steps transforming a candidate abstract proof structure into a proof net are of two kinds.  The logical contractions correspond to identities A ⊢ A, for complex formulas A; they reduce a configuration of a matching tensor and cotensor link to a point. The structural conversions perform an internal rewiring of a proof structure with hypotheses H1,…,Hn and conclusions C1,…,Cm to a structure with some permutation of the Hi as hypotheses and some permutation of the Ci as conclusions. Copying and deletion, in other words, are ruled out. As an example, the rewriting corresponding to one of the Grishin interaction principles discussed in §3.3, allowing us to infer C ⦸ A ⊢ D/B from A ⊗ B ⊢ C ⊕ D. Hypotheses A, B are on the left, conclusions C, D on the right.  The net representation brings out in a particularly clear way that these principles are structure-preserving: they leave the order of hypotheses and conclusions untouched.