  This work concerns the proof theory of (left) skew monoidal categories and
their variants (e.g. closed monoidal, symmetric monoidal), continuing the line
of work initiated in recent years by Uustalu et al. Skew monoidal categories
are a weak version of Mac Lane's monoidal categories, where the structural laws
are not required to be invertible, they are merely natural transformations with
a specific orientation. Sequent calculi which can be modelled in such
categories can be identified as deductive systems for restricted substructural
fragments of intuitionistic linear logic. These calculi enjoy cut elimination
and admit a focusing strategy, sharing resemblance with Andreoli's
normalization technique for linear logic. The focusing procedure is useful for
solving the coherence problem of the considered categories with skew structure.
  Here we investigate possible extensions of the sequent calculi of Uustalu et
al. with additive connectives. As a first step, we extend the sequent calculus
with additive conjunction and disjunction, corresponding to studying the proof
theory of skew monoidal categories with binary products and coproducts
satisfying a left-distributivity condition. We introduce a new focused sequent
calculus of derivations in normal form, which employs tag annotations to reduce
non-deterministic choices in bottom-up proof search. The focused sequent
calculus and the proof of its correctness have been formalized in the Agda
proof assistant. We also discuss extensions of the logic with additive units, a
form of skew exchange and linear implication.
