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The symmetric $\lambda \mu$-calculus is the $\lambda \mu$-calculus introducedby Parigot in which the reduction rule $\m'$, which is the symmetric of $\mu$,is added. We give arithmetical proofs of some strong normalization results forthis calculus. We show (this is a new result) that the $\mu\mu'$-reduction isstrongly normalizing for the un-typed calculus. We also show the strongnormalization of the $\beta\mu\mu'$-reduction for the typed calculus: this wasalready known but the previous proofs use candidates of reducibility where theinterpretation of a type was defined as the fix point of some increasingoperator and thus, were highly non arithmetical.

Arithmetical proofs of strong normalization results for the symmetric $\lambda \mu$-calculus