We prove normalization for (univalent, Cartesian) cubical type theory, closing the last major
open problem in the syntactic metatheory of cubical type theory. Our normalization result is …

Parametricity is a property of the syntax of type theory implying, eg, that there is only one
function having the type of the polymorphic identity function. Parametricity is usually proven …

Hofmann and Streicher famously showed how to lift Grothendieck universes into presheaf
topoi, and Streicher has extended their result to the case of sheaf topoi by sheafification. In …

Two approaches exist to incorporate parametricity into proof assistants based on dependent
type theory. On the one hand, parametricity translations conveniently compute parametricity …

We define a computational type theory combining the contentful equality structure of
cartesian cubical type theory with internal parametricity primitives. The combined theory …

We contribute XTT, a cubical reconstruction of Observational Type Theory which extends
Martin-L\" of's intensional type theory with a dependent equality type that enjoys function …

arXiv:2305.00893v2 [math.CT] 14 Jul 2023 Page 1 arXiv:2305.00893v2 [math.CT] 14 Jul
2023 CARTESIAN CUBICAL MODEL CATEGORIES STEVE AWODEY Abstract. The …

Formalized 1-category theory forms a core component of various libraries of mathematical
proofs. However, more sophisticated results in fields from algebraic topology to theoretical …

Dependent type theories are a family of logical systems that serve as expressive functional
programming languages and as the basis of many proof assistants. In the past decade, type …

We present XTT, a version of Cartesian cubical type theory specialized for Bishop setsa la
Coquand, in which every type enjoys a definitional version of the uniqueness of identity …