Cartesian Cubical Computational Type Theory: Constructive Reasoning with Paths and Equalities

Carlo Angiuli, Hou (Favonia), Kuen-Bang, Robert Harper, Michael Wagner
2018 Annual Conference for Computer Science Logic  
We present a dependent type theory organized around a Cartesian notion of cubes (with faces, degeneracies, and diagonals), supporting both fibrant and non-fibrant types. The fibrant fragment validates Voevodsky's univalence axiom and includes a circle type, while the non-fibrant fragment includes exact (strict) equality types satisfying equality reflection. Our type theory is defined by a semantics in cubical partial equivalence relations, and is the first two-level type theory to satisfy the
more » ... nonicity property: all closed terms of boolean type evaluate to either true or false.
doi:10.4230/lipics.csl.2018.6 dblp:conf/csl/AngiuliF018 fatcat:56hfopefhfghdk6ty2zyy54hq4