  Fair termination is the property of programs that may diverge "in principle"
but that terminate "in practice", i.e. under suitable fairness assumptions
concerning the resolution of non-deterministic choices. We study a conservative
extension of $\mu$MALL$^\infty$, the infinitary proof system of the
multiplicative additive fragment of linear logic with least and greatest fixed
points, such that cut elimination corresponds to fair termination. Proof terms
are processes of $\pi$LIN, a variant of the linear $\pi$-calculus with
(co)recursive types into which binary and (some) multiparty sessions can be
encoded. As a result we obtain a behavioral type system for $\pi$LIN (and
indirectly for session calculi through their encoding into $\pi$LIN) that
ensures fair termination: although well-typed processes may engage in
arbitrarily long interactions, they are fairly guaranteed to eventually perform
all pending actions.
