A Faithful and Quantitative Notion of Distant Reduction for Generalized Applications (Long Version)

We introduce a call-by-name lambda-calculus $\lambda J$ with generalized applications which integrates a notion of distant reduction that allows to unblock $\beta$-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 $\lambda J$ and the well-know calculus $\Lambda J$ determine equivalent notions of strong normalization. As a consequence, $\Lambda J$ inherits a faithful translation into explicit substitutions, and its strong normalization can be characterized by the quantitative typing system designed for $\lambda J$, despite the fact that quantitative subject reduction fails for permutative conversions.