  Multiplicative-Additive System Virtual (MAV) is a logic that extends
Multiplicative-Additive Linear Logic with a self-dual non-commutative operator
expressing the concept of "before" or "sequencing". MAV is also an extenson of
the the logic Basic System Virtual (BV) with additives. Formulas in BV have an
appealing reading as processes with parallel and sequential composition. MAV
adds internal and external choice operators. BV and MAV are also closely
related to Concurrent sKleene Algebras.
  Proof systems for MAV and BV are Deep Inference systems, which allow
inference rules to be applied anywhere inside a structure. As with any proof
system, a key question is whether proofs in MAV can be reduced to a normal
form, removing detours and the introduction of structures not present in the
original goal. In Sequent Calcluli systems, this property is referred to as Cut
Elimination. Deep Inference systems have an analogous Cut rule and other rules
that are not present in normalised proofs. Cut Elimination for Deep Inference
systems has the same metatheoretic benefits as for Sequent Calculi systems,
including consistency and decidability.
  Proofs of Cut Elimination for BV, MAV, and other Deep Inference systems
present in the literature have relied on intrincate syntactic reasoning and
complex termination measures.
  We present a concise semantic proof that all MAV proofs can be reduced to a
normal form avoiding the Cut rule and other "non analytic" rules. We also
develop soundness and completeness proofs of MAV (and BV) with respect to a
class of models. We have mechanised all our proofs in the Agda proof assistant,
which provides both assurance of their correctness as well as yielding an
executable normalisation procedure.
