Proof Nets and Explicit Substitutions

We refine the simulation technique introduced in (Di Cosmo and Kesner 1997) to show strong normalization of λ-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 we show that cut elimination modulo this equivalence relation is terminating. We then show strong normalization of the typed version of the λws-calculus with de Bruijn indices (a calculus with full composition defined in (David and Guillaume 1999)) using a translation from typed λws to proof nets. Finally, we propose a version of typed λws with named variables which helps to better understand the complex mechanism of the explicit weakening notation introduced in the λws-calculus with de Bruijn indices (David and Guillaume 1999).