Swapping: a natural bridge between named and indexed explicit substitution calculi

This article is devoted to the presentation of lambda_rex, an explicit substitution calculus with de Bruijn indexes and a simple notation. By being isomorphic to lambda_ex – a recent formalism with variable names –, lambda_rex accomplishes simulation of beta-reduction (Sim), preservation of beta-strong normalization (PSN) and meta-confluence (MC), among other desirable properties. Our calculus is based on a novel presentation of lambda_dB, using a swap notion that was originally devised by de Bruijn. Besides lambda_rex, two other indexed calculi isomorphic to lambda_x and lambda_xgc are presented, demonstrating the potential of our technique when applied to the design of indexed versions of known named calculi.