This paper presents a type theory in which it is possible to directly manipulate $ n $-
dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of …

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 …

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 …

In their usual form, representation independence metatheorems provide an external
guarantee that two implementations of an abstract interface are interchangeable when they …

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

Polymorphic type systems such as System F enjoy the parametricity property: polymorphic
functions cannot inspect their type argument and will therefore apply the same algorithm to …

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 …

Reasoning modulo equivalences is natural for everyone, including mathematicians.
Unfortunately, in proof assistants based on type theory, which are frequently used to …

We present a cubical type theory based on the Cartesian cube category (faces,
degeneracies, symmetries, diagonals, but no connections or reversal) with univalent …

Homotopy Type Theory promises a unification of the concepts of equality and equivalence in
Type Theory, through the introduction of the univalence principle. However, existing proof …