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We prove cut-elimination for a sequent-style proof system which is sound and complete for the equational theory of Kleene algebra, and where proofs are (potentially) non-wellfounded infinite trees. We extend these results to systems with meets and residuals, capturing `star-continuous' action lattices in a similar way. We recover the equational theory of all action lattices by restricting to regular proofs (with cut) - those proofs that are unfoldings of finite graphs.
Das et al. (Mon,) studied this question.