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Proof nets and explicit substitutions

Published online by Cambridge University Press:  20 May 2003

PPS (CNRS UMR 7126), Université Paris 7, 175, rue du Chevaleret, Paris, France Email:
LRI (CNRS UMR 8623), Bât 490, Université de Paris-Sud, 91405 Orsay Cedex, France Email:
PPS (CNRS UMR 7126), Université Paris 7, 175, rue du Chevaleret, Paris, France Email:


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).

Research Article
2003 Cambridge University Press

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Using various translations of the $\l$-calculus into proof nets, new abstract machines have been proposed that exploit the Geometry of Interaction and Dynamic Algebras (Girard 1989; Abramsky and Jagadeesan 1992; Danos 1990), leading to work on optimal reduction (Gonthier et al. 1992; Lamping 1990).