A Faithful and Quantitative Notion of Distant Reduction for the Lambda-Calculus with Generalized Applications - Santo, José Espírito; Kesner, Delia; Peyrot, Loïc (Logical methods in computer science, 2024)
We introduce a call-by-name lambda-calculus $λ Jn$ with generalized applications which is equipped with distant reduction. This allows to unblock $β$-redexes without resorting to the standard permutative conversions of generalized applications used in the original $Λ J$-calculus with generalized applications of Joachimski and Matthes. We show strong normalization of simply-typed terms, and we then fully characterize strong normalization by means of a quantitative (i.e. non-idempotent intersection) typing system. This characterization uses a non-trivial inductive definition of strong normalization --related to others in the literature--, which is based on a weak-head normalizing strategy. We also show that our calculus $λ Jn$ relates to explicit substitution calculi by means of a faithful translation, in the sense that it preserves strong normalization. Moreover, our calculus $λ Jn$ and the original $Λ J$-calculus determine equivalent notions of strong normalization. As a consequence, $λ J$ inherits a faithful translation into explicit substitutions, and its strong normalization can also be characterized by the quantitative typing system designed for $λ Jn$, despite the fact that quantitative subject reduction fails for permutative conversions.