Abstract Development of formal proofs of correctness of programs can increase actual and
perceived reliability and facilitate better understanding of program specifications and their …

Machine-checked proofs of properties of programming languages have become acritical
need, both for increased confidence in large and complex designsand as a foundation for …

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 …

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

An old and celebrated analogy says that writing programs is like proving theorems. This
analogy has been productive in both directions, but in particular has demonstrated …

We show how to combine a general purpose type system for an existing language with
support for programming with binders and contexts by refining the type system of ML with a …

We argue that locally Cartesian closed categories form a suitable doctrine for defining
dependent type theories, including non-extensional ones. Using the theory of sketches, one …

There are so many different notions of “space”(topological spaces, manifolds, schemes,
stacks, and so on, as discussed in various other chapters of this book and its companion …

Dependent types help programmers write highly reliable code. However, this reliability
comes at a cost: it can be challenging to write new prototypes in (or migrate old code to) …

Over the past three decades, a variety of meta-reasoning systems which support reasoning
about higher-order abstract specifications have been designed and developed. In this …