We have to consider  For the converse, (for technical reasons and to ease proofs) we use both free group and free pregroup models of NL since a sequent is valid in NL only if its interpretation is valid in both models.
the free group translation for the types of G n is: ] we get : the free pregroup translation for the types of G n is: we get: Proof: Let τ n denote the type assignment by the rigid grammar G n .