Non-idempotent types for classical calculi in natural deduction style

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 nonidempotent 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 resource aware interpretation called λμr, which is inspired by the substitution at a distance paradigm. The small-step λμr-calculus turns out to be compatible with a natural extension of the nonidempotent interpretations of λμ, i.e. λμr-reduction preserves and decreases typing derivations in an extended appropriate typing system. We thus derive a simple arithmetical characterization of strongly λμr-normalizing terms by means of typing.