A Cubical Language for Bishop Sets
Abstract
We present XTT, a version of Cartesian cubical type theory specialized for Bishop sets à 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.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2020
- DOI:
- 10.48550/arXiv.2003.01491
- arXiv:
- arXiv:2003.01491
- Bibcode:
- 2020arXiv200301491S
- Keywords:
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- Computer Science - Logic in Computer Science;
- Mathematics - Logic