  Full Intuitionistic Linear Logic (FILL) is multiplicative intuitionistic
linear logic extended with par. Its proof theory has been notoriously difficult
to get right, and existing sequent calculi all involve inference rules with
complex annotations to guarantee soundness and cut-elimination. We give a
simple and annotation-free display calculus for FILL which satisfies Belnap's
generic cut-elimination theorem. To do so, our display calculus actually
handles an extension of FILL, called Bi-Intuitionistic Linear Logic (BiILL),
with an `exclusion' connective defined via an adjunction with par. We refine
our display calculus for BiILL into a cut-free nested sequent calculus with
deep inference in which the explicit structural rules of the display calculus
become admissible. A separation property guarantees that proofs of FILL
formulae in the deep inference calculus contain no trace of exclusion. Each
such rule is sound for the semantics of FILL, thus our deep inference calculus
and display calculus are conservative over FILL. The deep inference calculus
also enjoys the subformula property and terminating backward proof search,
which gives the NP-completeness of BiILL and FILL.
