Proof assistants based on dependent type theory provide expressive languages for both
programming and proving within the same system. However, all of the major …

We introduce MTT, a dependent type theory which supports multiple modalities. MTT is
parametrized by a mode theory which specifies a collection of modes, modalities, and …

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 …

We begin by recalling the essentially global character of universes in various models of
homotopy type theory, which prevents a straightforward axiomatization of their properties …

Cubical type theory provides a constructive justification to certain aspects of homotopy type
theory such as Voevodsky's univalence axiom. This makes many extensionality principles …

We prove normalization for MTT, a general multimodal dependent type theory capable of
expressing modal type theories for guarded recursion, internalized parametricity, and …

We define and develop two-level type theory (2LTT), a version of Martin-Löf type theory
which combines two different type theories. We refer to them as the 'inner'and the 'outer'type …

The theory of program modules is of interest to language designers not only for its practical
importance to programming, but also because it lies at the nexus of three fundamental …

We propose an abstract notion of a type theory to unify the semantics of various type
theories including Martin–Löf type theory, two-level type theory, and cubical type theory. We …

Directed type theory is an analogue of homotopy type theory where types represent
categories, generalizing groupoids. A bisimplicial approach to directed type theory …