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