A Faithful and Quantitative Notion of Distant Reduction for Generalized Applications (Long Version)
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We introduce a call-by-name lambda-calculus λ J with generalized applications which integrates a notion of distant reduction that allows to unblock β-redexes without resorting to the permutative conversions of generalized applications. We show strong normalization of simply typed terms, and we then fully characterize strong normalization by means of a quantitative typing system. This characterization uses a non-trivial inductive definition of strong normalization –that we relate to others in the literature–, which is based on a weak-head normalizing strategy. Our calculus relates to explicit substitution calculi by means of a translation between the two formalisms which is faithful, in the sense that it preserves strong normalization. We show that our calculus λ J and the well-know calculus Λ J determine equivalent notions of strong normalization. As a consequence, Λ J inherits a faithful translation into explicit substitutions, and its strong normalization can be characterized by the quantitative typing system designed for λ J, despite the fact that quantitative subject reduction fails for permutative conversions.
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