Since the very beginning of the theory of linear logic it is known how to represent the λ-calculus as linear logic proof nets. The two systems however have different granularities, in particular proof nets have an explicit notion of sharing-the exponentials-and a micro-step operational semantics, while the λcalculus has no sharing and a small-step operational semantics. Here we show that the linear substitution calculus, a simple refinement of the λ-calculus with sharing, is isomorphic to proof nets at the operational level. Nonetheless, two different terms with sharing can still have the same proof nets representation-a further result is the characterisation of the equality induced by proof nets over terms with sharing. Finally, such a detailed analysis of the relationship between terms and proof nets, suggests a new, abstract notion of proof net, based on rewriting considerations and not necessarily of a graphical nature.
IntroductionGirard's seminal paper on linear logic [22] showed how to represent intuitionistic logicand so the λ-calculus-inside linear logic. During the nineties, Danos and Regnier provided a detailed study of such a representation via proof nets [15,40,14,16], which is nowadays a cornerstone of the field. Roughly, linear logic gives first-class status to sharing, accounted for by the exponential layer of the logic, and not directly visible in the λ-calculus. In turn, cut-elimination in linear logic provides a micro-step refinement of the small-step operational semantics of the λ-calculus, that is, β-reduction.The mismatch. Some of the insights provided by proof nets cannot be directly expressed in the λ-calculus, because of the mismatch of granularities. Typically, there is a mismatch of states: simulation of β on proofs passes through intermediate states / proofs that cannot be expressed as λ-terms. The mismatch does not allow, for instance, expressing fine strategies such as linear head evaluation [34,17] in the λ-calculus, nor to see in which sense proof nets quotient terms, as such a quotient concerns only the intermediate proofs. And when one starts to have a closer look, there are other mismatches, of which the lack of sharing in the λ-calculus is only the most macroscopic one.Some minor issues are due to a mismatch of styles: the fact that terms and proofs, despite their similarities, have different representations of variables and notions of redexes. Typically, two occurrences of a same variable in a term are smoothly identified by simply using the same name, while for proofs there is an explicit rule, contraction,