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 canonicity property: all closed terms of boolean type evaluate to either true or false.
@InProceedings{angiuli_et_al:LIPIcs.CSL.2018.6, author = {Angiuli, Carlo and Hou (Favonia), Kuen-Bang and Harper, Robert}, title = {{Cartesian Cubical Computational Type Theory: Constructive Reasoning with Paths and Equalities}}, booktitle = {27th EACSL Annual Conference on Computer Science Logic (CSL 2018)}, pages = {6:1--6:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-088-0}, ISSN = {1868-8969}, year = {2018}, volume = {119}, editor = {Ghica, Dan R. and Jung, Achim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2018.6}, URN = {urn:nbn:de:0030-drops-96734}, doi = {10.4230/LIPIcs.CSL.2018.6}, annote = {Keywords: Homotopy Type Theory, Two-Level Type Theory, Computational Type Theory, Cubical Sets} }
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