We study the expressivity of Parigot’s λµ-calculus, and show that each statement Γ ⊢ LK ∆ that is provable in Gentzen’s LK has a proof in λµ. This result is obtained through defining an interpretation from nets from the X-calculus into both the λ-calculus and λµ; X enjoys the full Curry-Howard isomorphism for (the implicative fragment of) LK, and cut-elimination in LK is represented by reduction in X. This interpretation will be shown to preserve reduction in X via equality in the target calculi, and to preserve typeability using the standard double negation translation technique of types. Using the fact that, in λµ, we can inhabit ¬¬A→A for all types A, a completeness result as well as a consistency result are shown for λµ.
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