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1 MARELLE - Mathematical, Reasoning and Software CRISAM - Inria Sophia Antipolis - Méditerranée 2 CSE - Department of Computer Science and Engineering Göteborg 3 School of Mathematics Princeton

Abstract : 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.

Keywords : Mathematical Logic Univalence Axiom Dependent Type Theory Cubical Sets

Autor: Cyril Cohen - Thierry Coquand - Simon Huber - Anders Mörtberg -



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