  This paper presents the first in a series of results that allow us to develop
a theory providing finer control over the complexity of normalisation, and in
particular of cut elimination. By considering atoms as self-dual
non-commutative connectives, we are able to classify a vast class of inference
rules in a uniform and very simple way. This allows us to define simple
conditions that are easily verifiable and that ensure normalisation and cut
elimination by way of a general theorem. In this paper we define and consider
splittable systems, which essentially comprise a large class of linear logics,
including MLL and BV, and we prove for them a splitting theorem, guaranteeing
cut elimination and other admissibility results as corollaries. In papers to
follow, we will extend this result to non-linear logics. The final outcome will
be a comprehensive theory giving a uniform treatment for most existing logics
and providing a blueprint for the design of future proof systems.
