  We present a proof-theoretic analysis of the logic NL$\lambda$ (Barker \&
Shan 2014, Barker 2019). We notably introduce a novel calculus of proof nets
and prove it is sound and complete with respect to the sequent calculus for the
logic. We study decidability and complexity of the logic using this new
calculus, proving a new upper bound for complexity of the logic (showing it is
in NP) and a new lower bound for the class of formal language generated by the
formalism (mildly context-sensitive languages extended with a permutation
closure operation). Finally, thanks to this new calculus, we present a novel
comparison between NL$\lambda$ and the hybrid type-logical grammars of Kubota
\& Levine (2020). We show there is an unexpected convergence of the natural
language analyses proposed in the two formalism. In addition to studying the
proof-theoretic properties of NL$\lambda$, we greatly extends its linguistic
coverage.
