# The Delta-calculus: syntax and types

1 KAIROS - Logical Time for Formal Embedded System Design
Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués, CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : We present the $∆$-calculus, an explicitly typed $λ$-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-Salle', the Coppo-Dezani-Venneri and the Barendregt-Coppo-Dezani ones, producing a family of $∆$-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 a la Curry and the corresponding intersection typed systems a la Church by means of an essence function translating an explicitly typed $∆$-term into a pure $λ$-term one. We finally translate a $∆$-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 $∆$- calculus can be parametrized to take into account other intersection type theories as the ones in the Barendregt et al. book.
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https://hal.archives-ouvertes.fr/hal-01963662
Contributor : Luigi Liquori <>
Submitted on : Friday, December 21, 2018 - 2:42:20 PM
Last modification on : Tuesday, May 26, 2020 - 6:50:58 PM
Document(s) archivé(s) le : Friday, March 22, 2019 - 5:04:13 PM

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delta-calculus.pdf
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### Identifiers

• HAL Id : hal-01963662, version 1
• ARXIV : 1803.09660

### Citation

Luigi Liquori, Claude Stolze. The Delta-calculus: syntax and types. FSCD 2019 - 4th International Conference on Formal Structures for Computation and Deduction, 2019-06-24; 2019-06-24, Jun 2019, Dortmund, Germany. ⟨hal-01963662⟩

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