  We study a system, called NEL, which is the mixed commutative/non-commutative
linear logic BV augmented with linear logic's exponentials. Equivalently, NEL
is MELL augmented with the non-commutative self-dual connective seq. In this
paper, we show a basic compositionality property of NEL, which we call
decomposition. This result leads to a cut-elimination theorem, which is proved
in the next paper of this series. To control the induction measure for the
theorem, we rely on a novel technique that extracts from NEL proofs the
structure of exponentials, into what we call !-?-Flow-Graphs.
