Abstract. We refine the simulation technique introduced in [10] to show strong normalization of λ-calculi with explicit substitutions via termination of cut elimination in proof nets [13]. We first propose a notion of equivalence relation for proof nets that extends the one in [9], and we show that cut elimination modulo this equivalence relation is terminating. We then show strong normalization of the typed version of the λ l -calculus with de Bruijn indices (a calculus with full composition defined in [8]) using a translation from typed λ l to proof nets. Finally, we propose a version of typed λ l with named variables which helps to better understand the complex mechanism of the explicit weakening notation introduced in the λ l -calculus with de Bruijn indices [8].

Abstract. The λµμ-calculus, defined by Curien and Herbelin [7], is a variant of the λµ-calculus that exhibits symmetries such as term/context and call-by-name/call-by-value. Since it is a symmetric, and hence a non-deterministic calculus, usual proof techniques of normalization needs some adjustments to be made to work in this setting. Here we prove the strong normalization (SN) of simply typed λµμ-calculus with explicit substitutions. For that purpose, we first prove SN of simply typed λµμ-calculus (by a variant of the reducibility technique from Barbanera and Berardi [2]), then we formalize a proof technique of SN via PSN (preservation of strong normalization), and we prove PSN by the perpetuality technique, as formalized by Bonelli [5].

We refine the simulation technique introduced in Di Cosmo and Kesner (1997) to show strong normalisation of $\l$-calculi with explicit substitutions via termination of cut elimination in proof nets (Girard 1987). We first propose a notion of equivalence relation for proof nets that extends the one in Di Cosmo and Guerrini (1999), and show that cut elimination modulo this equivalence relation is terminating. We then show strong normalisation of the typed version of the $\ll$-calculus with de Bruijn indices (a calculus with full composition defined in David and Guillaume (1999)) using a translation from typed $\ll$ to proof nets. Finally, we propose a version of typed $\ll$ with named variables, which helps to give a better understanding of the complex mechanism of the explicit weakening notation introduced in the $\ll$-calculus with de Bruijn indices (David and Guillaume 1999).