Labelled l -calculi with Explicit Copy and Erase

We present two rewriting systems that define labelled explicit substitution A-calculi. Our work is motivated by the close correspondence between Levyʹs labelled A-calculus and paths in proof-nets, which played an important role in the understanding of the Geometry of Interaction. The structure of the labels in Levyʹs labelled A-calculus relates to the multiplicative information of paths; the novelty of our work is that we design labelled explicit substitution calculi that also keep track of exponential information present in call-by-value and call-by-name translations of the A-calculus into linear logic proof-nets.