Related papers: Introduction to Univalent Foundations of Mathemati…
Agda is a dependently-typed functional programming language, based on an extension of intuitionistic Martin-L\"of type theory. We implement first order natural deduction in Agda. We use Agda's type checker to verify the correctness of…
Agda is a dependently-typed programming language and a proof assistant, pivotal in proof formalization and programming language theory. This paper extends the Agda ecosystem into machine learning territory, and, vice versa, makes…
We offer an introduction for mathematicians to the univalent foundations of Vladimir Voevodsky, aiming to explain how he chose to encode mathematics in type theory and how the encoding reveals a potentially viable foundation for all of…
The Agda Universal Algebra Library (agda-algebras) is a library of types and programs (theorems and proofs) we developed to formalize the foundations of universal algebra in dependent type theory using the Agda programming language and…
In recent years, the interest in using proof assistants to formalise and reason about mathematics and programming languages has grown. Type-logical grammars, being closely related to type theories and systems used in functional programming,…
It is discussed a practical possibility of a provable programming of mathematics basing on intuitionism and the dependent types feature of a programming language.The principles of constructive mathematics and provable programming are…
The Agda Universal Algebra Library (UALib) is a library of types and programs (theorems and proofs) we developed to formalize the foundations of universal algebra in dependent type theory using the Agda programming language and proof…
Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of mathematics. On the one hand, Voevodsky's subtle and…
Recent discoveries have been made connecting abstract homotopy theory and the field of type theory from logic and theoretical computer science. This has given rise to a new field, which has been christened "homotopy type theory". In this…
The term UniMath refers both to a formal system for mathematics, as well as a computer-checked library of mathematics formalized in that system. The UniMath system is a core dependent type theory, augmented by the univalence axiom. The…
The Agda Universal Algebra Library (UALib) is a library of types and programs (theorems and proofs) we developed to formalize the foundations of universal algebra in dependent type theory using the Agda programming language and proof…
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 dependent type theory in a cubical set model. This enables new ways…
In this short note we give a glimpse of homotopy type theory, a new field of mathematics at the intersection of algebraic topology and mathematical logic, and we explain Vladimir Voevodsky's univalent interpretation of it. This…
This is an introductory textbook to univalent mathematics and homotopy type theory, a mathematical foundation that takes advantage of the structural nature of mathematical definitions and constructions. It is common in mathematical practice…
Proof assistant software has recently been used to verify proofs of major theorems, yet even the libraries of some of the most prominent proof assistants lack much of undergraduate mathematics. In particular, the Agda proof assistant has no…
Theorem provers are tools that help users to write machine readable proofs. Some of this tools are also interactive. The need of such softwares is increasing since they provide proofs that are more certified than the hand written ones. Agda…
The Univalence Principle is the statement that equivalent mathematical structures are indistinguishable. We prove a general version of this principle that applies to all set-based, categorical, and higher-categorical structures defined in a…
The received Hilbert-style axiomatic foundations of mathematics has been designed by Hilbert and his followers as a tool for meta-theoretical research. Foundations of mathematics of this type fail to satisfactory perform more basic and more…
Proof assistants and programming languages based on type theories usually come in two flavours: one is based on the standard natural deduction presentation of type theory and involves eliminators, while the other provides a syntax in…
Pure type systems arise as a generalisation of simply typed lambda calculus. The contemporary development of mathematics has renewed the interest in type theories, as they are not just the object of mere historical research, but have an…