We present two rewriting systems that define labelled explicit substitution
lambda-calculi. Our work is motivated by the close correspondence between
Levy's labelled lambda-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 Levy's labelled lambda-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
lambda-calculus into linear logic proof-nets
