Canonicity for Cubical Type Theory
arXiv:1607.04156
Abstract
Cubical type theory is an extension of Martin-Löf type theory recently proposed by Cohen, Coquand, Mörtberg and the author which allows for direct manipulation of $n$-dimensional cubes and where Voevodsky's Univalence Axiom is provable. In this paper we prove canonicity for cubical type theory: any natural number in a context build from only name variables is judgmentally equal to a numeral. To achieve this we formulate a typed and deterministic operational semantics and employ a computability argument adapted to a presheaf-like setting.
34 pages. v2: Added section on propositional truncation; fixed typos. To appear in the Journal of Automated Reasoning