  In sequential functional languages, sized types enable termination checking
of programs with complex patterns of recursion in the presence of mixed
inductive-coinductive types. In this paper, we adapt sized types and their
metatheory to the concurrent setting. We extend the semi-axiomatic sequent
calculus, a subsuming paradigm for futures-based functional concurrency, and
its underlying operational semantics with recursion and arithmetic refinements.
The latter enables a new and highly general sized type scheme we call sized
type refinements. As a widely applicable technical device, we type recursive
programs with infinitely deep typing derivations that unfold all recursive
calls. Then, we observe that certain such derivations can be made infinitely
wide but finitely deep. The resulting trees serve as the induction target of
our termination result, which we develop via a novel logical relations
argument.
