The relational model is injective for Multiplicative Exponential Linear Logic (without weakenings)

We show that for a suitable fragment of linear logic the syntactical equivalence relation on proofs induced by cut-elimination coincide with the semantic equivalence relation on proofs induced by the multiset based relational model: one says that the interpretation in the model (or the semantics) is injective. More precisely, we prove that two cut-free proofs of the multiplicative and exponential fragment of linear logic with the same interpretation in the multiset based relational model are the same ``up to the connections between the doors of exponential boxes''. This result is proven in an untyped framework: two cut-free untyped Proof-Structures (PS) have the same Linear Proof-Structure (LPS). An immediate consequence is that two cut-free PS without weakenings with the same relational interpretation are the same; hence relational semantics is injective.