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Since it was realized that the Curry-Howard isomorphism can be extended tothe case of classical logic as well, several calculi have appeared ascandidates for the encodings of proofs in classical logic. One of the mostextensively studied among them is the $\lambda\mu$-calculus of Parigot. In thispaper, based on the result of Xi presented for the $\lambda$-calculus Xi, wegive an upper bound for the lengths of the reduction sequences in the$\lambda\mu$-calculus extended with the $\rho$- and $\theta$-rules.Surprisingly, our results show that the new terms and the new rules do not addto the computational complexity of the calculus despite the fact that$\mu$-abstraction is able to consume an unbounded number of arguments by virtueof the $\mu$-rule.

An estimation for the lengths of reduction sequences of the $\lambda\mu\rho\theta$-calculus