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A Normalizing Computation Rule for Propositional Extensionality in Higher-Order Minimal Logic

The univalence axiom expresses the principle of extensionality for dependent type theory. However, if we simply add the univalence axiom to type theory, then we lose the property of canonicity - that every closed term computes to a canonical form. A computation becomes "stuck" when it reaches the point that it needs to evaluate a proof term that is an application of the...

Realizability at Work: Separating Two Constructive Notions of Finiteness

We elaborate in detail a realizability model for Martin-L�f dependent type theory with the purpose to analyze a subtle distinction between two constructive notions of finiteness of a set A. The two notions are: (1) A is Noetherian: the empty list can be constructed from lists over A containing duplicates by a certain inductive shortening process; (2) A is streamless: every...

The Univalence Axiom in Cubical Sets

In this note we show that Voevodsky’s univalence axiom holds in the model of type theory based on cubical sets as described in Bezem et al. (in: Matthes and Schubert (eds.) 19th international conference on types for proofs and programs (TYPES 2013), Leibniz international proceedings in informatics (LIPIcs), Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Dagstuhl, Germany, vol...

Non-Constructivity in Kan Simplicial Sets

Philosophical Society, 146:45–55, 2009. 9 10 11 12 13 14 Marc Bezem and Thierry Coquand . A Kripke model for simplicial sets . TCS , 2015 . doi:10 .1016/j.tcs. 2015 . 01 .035. Greg Friedman . An elementary

A Model of Type Theory in Cubical Sets

Takeuti-Gandy interpretation . To appear in Mathematical Structures in Computer Science , 2013 . Marc Bezem and Thierry Coquand . A Kripke model for simplicial sets . Preprint , 2013 . Erret Bishop ... -Holland, Amsterdam, 1975 . In Marc Bezem and Jan Friso Groote, editors, Proceedings of the conference Typed Lambda Calculi and Applications, number 664 in Lecture Notes in Computer Science , 1993 . Andrew M