2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/229647The symmetric $λμ$-calculus is the $λμ$-calculus introduced by Parigot in which the reduction rule $\m'$, which is the symmetric of $μ$, is added. We give arithmetical proofs of some strong normalization results for this calculus. We show (this is a new result) that the $μμ'$-reduction is strongly normalizing for the un-typed calculus. We also show the strong normalization of the $βμμ'$-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.Logictext