Guarded Cubical Type Theory
Artikel i vetenskaplig tidskrift, 2019

This paper improves the treatment of equality in guarded dependent type theory ((Formula presented.)), by combining it with cubical type theory ((Formula presented.)). (Formula presented.) is an extensional type theory with guarded recursive types, which are useful for building models of program logics, and for programming and reasoning with coinductive types. We wish to implement (Formula presented.) with decidable type checking, while still supporting non-trivial equality proofs that reason about the extensions of guarded recursive constructions. (Formula presented.) is a variation of Martin–Löf type theory in which the identity type is replaced by abstract paths between terms. (Formula presented.) provides a computational interpretation of functional extensionality, enjoys canonicity for the natural numbers type, and is conjectured to support decidable type-checking. Our new type theory, guarded cubical type theory ((Formula presented.)), provides a computational interpretation of extensionality for guarded recursive types. This further expands the foundations of (Formula presented.) as a basis for formalisation in mathematics and computer science. We present examples to demonstrate the expressivity of our type theory, all of which have been checked using a prototype type-checker implementation. We show that (Formula presented.) can be given semantics in presheaves on (Formula presented.), where (Formula presented.) is the cube category, and (Formula presented.) is any small category with an initial object. We then show that the category of presheaves on (Formula presented.) provides semantics for (Formula presented.).

Cubical type theory

Homotopy type theory

Guarded recursion


Lars Birkedal

Aarhus Universitet

Aleš Bizjak

Aarhus Universitet

Ranald Clouston

Aarhus Universitet

Hans Bugge Grathwohl

Aarhus Universitet

Bas Spitters

Aarhus Universitet

Andrea Vezzosi

Chalmers, Data- och informationsteknik, Datavetenskap

Journal of Automated Reasoning

0168-7433 (ISSN) 1573-0670 (eISSN)

Vol. 63 2 211-253


Algebra och logik


Matematisk analys



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