We present Quantitative Type Theory, a Type Theory that records usage information for each
variable in a judgement, based on a previous system by McBride. The usage information is …

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 combine homotopy type theory with axiomatic cohesion, expressing the latter internally
with a version of 'adjoint logic'in which the discretization and codiscretization modalities are …

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

Modalities are everywhere in programming and mathematics! Despite this, however, there
are still significant technical challenges in formulating a core dependent type theory with …

We present calf, ac ost-a ware l ogical f ramework for studying quantitative aspects of
functional programs. Taking inspiration from recent work that reconstructs traditional aspects …

Fitch-style modal deduction, in which modalities are eliminated by opening a subordinate
proof, and introduced by shutting one, were investigated in the 1990s as a basis for lambda …

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 …

In recent years, we have seen several new models of dependent type theory extended with
some form of modal necessity operator, including nominal type theory, guarded and clocked …

Dependent type theory allows us to write programs and to prove properties about those
programs in the same language. However, some properties do not require much proof, as …