Non-idempotent types for classical calculi in natural deduction style

Abstract

In the first part of this paper, we define two resource aware typing systems for the λμ-calculus based on non-idempotent intersection and union types. The non-idempotent approach provides very simple combinatorial arguments-based on decreasing measures of type derivations-to characterize head and strongly normalizing terms. Moreover, typability provides upper bounds for the lengths of the head reduction and the maximal reduction sequences to normal-form. In the second part of this paper, the λμ-calculus is refined to a small-step calculus called λμs, which is inspired by the substitution at a distance paradigm. The λμs-calculus turns out to be compatible with a natural extensionof the non-idempotent interpretations of λμ, i.e., λμs-reduction preserves and decreases typing derivations in an extended appropriate typing system. We thus derive a simple arithmetical characterization of strongly λμs-normalizing terms by means of typing.