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 …

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

Fascinating links between the semantics of concurrent programs and algebraic topology
have been discovered and developed since the 1990s, motivated by the hope that each field …

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 …

The goal of this thesis is to prove that $\pi_4 (S^ 3)\simeq\mathbb {Z}/2\mathbb {Z} $ in
homotopy type theory. In particular it is a constructive and purely homotopy-theoretic proof …

Higher inductive types (HITs) in Homotopy Type Theory allow the definition of datatypes
which have constructors for equalities over the defined type. HITs generalise quotient types …

We present guarded dependent type theory, gDTT gDTT, an extensional dependent type
theory with a 'later'modality and clock quantifiers for programming and proving with guarded …

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

This paper discusses the formalization of synthetic cohomology theory in a cubical extension
of Agda which natively supports univalence and higher inductive types. This enables …

Formal constructive type theory has proved to be an effective language for mechanized
proof. By avoiding non-constructive principles, such as the law of the excluded middle, type …