Non-idempotent types for classical calculi in natural deduction style
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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 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 non-idempotent 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.
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