A Cubical Language for Bishop Sets

@article{Sterling2020ACL,
  title={A Cubical Language for Bishop Sets},
  author={Jonathan Sterling and Carlo Angiuli and Daniel Gratzer},
  journal={Log. Methods Comput. Sci.},
  year={2020},
  volume={18}
}
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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