Separation with streams in the Λμ-calculus

The λμ-calculus is an extension of the λ-calculus introduced in 1992 by Parigot [17] in order to generalize the Curry-Howard isomorphism to classical logic. Two versions of the calculus are usually considered in the literature: Parigot’s original syntax and an alternative syntax introduced by de Groote. In 2001, David and Py [5] proved that the Separation Property (also referred to as Böhm theorem) fails for Parigot’s λμ-calculus. By analyzing David & Py’s result, we exhibit an extension of Parigot’s λμ-calculus, the Λμ-calculus, for which the Separation Property holds and which is built as an intermediate language between Parigot’s and de Groote’s λμ-calculi. We prove the theorem and describe how Λμcalculus can be considered as a calculus of terms and streams. We then illustrate Separation in showing how in Λμ-calculus it is possible to separate the counter-example