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arXiv · 2003.01491

A Cubical Language for Bishop Sets

Jonathan Sterling·Carlo Angiuli·Daniel Gratzer
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Abstract

We present XTT, a version of Cartesian cubical type theory specialized for Bishop sets \`a la Coquand, in which every type enjoys a definitional version of the uniqueness of identity proofs. Using cubical notions, XTT reconstructs many of the ideas underlying Observational Type Theory, a version of intensional type theory that supports function extensionality. We prove the canonicity property of XTT (that every closed boolean is definitionally equal to a constant) using Artin gluing.

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Jonathan Sterling, Carlo Angiuli, Daniel Gratzer. 2020-03-03. A Cubical Language for Bishop Sets. https://doi.org/10.46298/lmcs-18(1%3A43)2022

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