The Delta-calculus: Syntax and Types

Authors Luigi Liquori, Claude Stolze

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Luigi Liquori
  • Université Côte d'Azur, Inria, Sophia Antipolis, France
Claude Stolze
  • Université Côte d'Azur, Inria, Sophia Antipolis, France


We are grateful to Benjamin Pierce, Joe Wells, Furio Honsell, and the anonymous reviewers for the useful comments and remarks.

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Luigi Liquori and Claude Stolze. The Delta-calculus: Syntax and Types. In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 28:1-28:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We present the Delta-calculus, an explicitly typed lambda-calculus with strong pairs, projections and explicit type coercions. The calculus can be parametrized with different intersection type theories T, e.g. the Coppo-Dezani, the Coppo-Dezani-Sallé, the Coppo-Dezani-Venneri and the Barendregt-Coppo-Dezani ones, producing a family of Delta-calculi with related intersection typed systems. We prove the main properties like Church-Rosser, unicity of type, subject reduction, strong normalization, decidability of type checking and type reconstruction. We state the relationship between the intersection type assignment systems à la Curry and the corresponding intersection typed systems à la Church by means of an essence function translating an explicitly typed Delta-term into a pure lambda-term one. We finally translate a Delta-term with type coercions into an equivalent one without them; the translation is proved to be coherent because its essence is the identity. The generic Delta-calculus can be parametrized to take into account other intersection type theories as the ones in the Barendregt et al. book.

Subject Classification

ACM Subject Classification
  • Theory of computation → Lambda calculus
  • Theory of computation → Type theory
  • intersection types
  • lambda calculus à la Church and à la Curry
  • proof-functional logics


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