  We investigate the proof theory of regular expressions with fixed points,
construed as a notation for (omega-)context-free grammars. Starting with a
hypersequential system for regular expressions due to Das and Pous, we define
its extension by least fixed points and prove soundness and completeness of its
non-wellfounded proofs for the standard language model. From here we apply
proof-theoretic techniques to recover an infinitary axiomatisation of the
resulting equational theory, complete for inclusions of context-free languages.
Finally, we extend our syntax by greatest fixed points, now computing
omega-context-free languages. We show the soundness and completeness of the
corresponding system using a mixture of proof-theoretic and game-theoretic
techniques.
