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[1611.02108] Cubical Type Theory: a constructive interpretation of the univalenc...

 4 years ago
source link: https://arxiv.org/abs/1611.02108
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[Submitted on 7 Nov 2016]

Cubical Type Theory: a constructive interpretation of the univalence axiom

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This paper presents a type theory in which it is possible to directly manipulate n-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in a cubical set model. This enables new ways to reason about identity types, for instance, function extensionality is directly provable in the system. Further, Voevodsky's univalence axiom is provable in this system. We also explain an extension with some higher inductive types like the circle and propositional truncation. Finally we provide semantics for this cubical type theory in a constructive meta-theory.

Comments: To be published in the post-proceedings of the 21st International Conference on Types for Proofs and Programs, TYPES 2015 Subjects: Logic in Computer Science (cs.LO); Logic (math.LO) ACM classes: F.3.2; F.4.1 Cite as: arXiv:1611.02108 [cs.LO]   (or arXiv:1611.02108v1 [cs.LO] for this version)

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