Cartesian Cubical Computational Type Theory: Constructive Reasoning with Paths and Equalities

Authors Carlo Angiuli , Kuen-Bang Hou (Favonia) , Robert Harper



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Carlo Angiuli
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
Kuen-Bang Hou (Favonia)
  • School of Mathematics, Institute for Advanced Study, Princeton, NJ, USA
Robert Harper
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA

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Carlo Angiuli, Kuen-Bang Hou (Favonia), and Robert Harper. Cartesian Cubical Computational Type Theory: Constructive Reasoning with Paths and Equalities. In 27th EACSL Annual Conference on Computer Science Logic (CSL 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 119, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.CSL.2018.6

Abstract

We present a dependent type theory organized around a Cartesian notion of cubes (with faces, degeneracies, and diagonals), supporting both fibrant and non-fibrant types. The fibrant fragment validates Voevodsky's univalence axiom and includes a circle type, while the non-fibrant fragment includes exact (strict) equality types satisfying equality reflection. Our type theory is defined by a semantics in cubical partial equivalence relations, and is the first two-level type theory to satisfy the canonicity property: all closed terms of boolean type evaluate to either true or false.

Subject Classification

ACM Subject Classification
  • Theory of computation → Type theory
Keywords
  • Homotopy Type Theory
  • Two-Level Type Theory
  • Computational Type Theory
  • Cubical Sets

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