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Classical Cut-elimination in the π-calculus

We define a calculus LK a variant of the calculus X that enjoys the Curry-Howard correspondence for Gentzen’s calculus LK; the variant consists of allowing arbitrary progress of cut over cut . We study the π-calculus enriched with pairing, for which we define a notion of implicative type assignment, translate the terms of LK into this variant of π, and show that reduction and assignable types are preserved. This implies that all proofs in LK have a representation in π, and cut-elimination is effectively simulated by π’s synchronisation, congruence, and equivalence between processes. We present two interpretations for which we show soundness results, as well as type preservation. Using the second, we show that we preserve Gentzen’s Hauptsatz result, and prove completeness. We then enrich the logic with the connector ¬, and show that this also can be represented in π, whilst preserving the results.