Cubical type theory provides a constructive justification of homotopy type theory. A crucial ingredient of cubical type theory is a path lifting operation which is explained computationally by induction on the type involving several non-canonical choices. We present in this article two canonicity results, both proved by a sconing argument: a homotopy canonicity result, every natural number is path equal to a numeral, even if we take away the equations defining the lifting operation on the type structure, and a canonicity result, which uses these equations in a crucial way. Both proofs are done internally in a presheaf model.

@article{paperbot1726,
  title = {Canonicity and homotopy canonicity for cubical type theory},
  author = {Thierry Coquand and Simon Huber and Christian Sattler},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 18, Issue 1},
  year = {2022},
  doi = {10.46298/lmcs-18(1:28)2022}
}