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Local fields, and fields complete with respect to a discrete valuation, are essential objects in commutative algebra, with applications to number theory and algebraic geometry. We formalize in Lean the basic theory of discretely valued fields. In ...

We present the formalization of Doob’s martingale convergence theorems in the mathlib library for the Lean theorem prover. These theorems give conditions under which (sub)martingales converge, almost everywhere or in L¹. In order to formalize those ...

The generality and pervasiveness of category theory in modern mathematics makes it a frequent and useful target of formalization. It is however quite challenging to formalize, for a variety of reasons. Agda currently (i.e. in 2020) does not have a ...

The Mizar Mathematical Library (MML) is a large corpus of formalised mathematical knowledge. It has been constructed over the course of many years by a large number of authors and maintainers. Yet the legal status of these efforts of the Mizar community ...

Automated theorem provers (ATPs) struggle to solve problems with large sets of possibly superfluous axiom. Several algorithms have been developed to reduce the number of axioms, optimally only selecting the necessary axioms. However, most of these ...

This paper discusses the semi-formal language of mathematics and presents the Naproche CNL, a controlled natural language for mathematical authoring. Proof Representation Structures, an adaptation of Discourse Representation Structures, are used to ...

This talk will consider rigorous mathematics and the nature of proof. It begins with an historical perspective and follows the development of formal mathematics. The talk will conclude with examples demonstrating that understanding the relationship ...

The paper describes the general philosophy and the main architectural and technological solutions adopted in the HELM Project for the management of large repositories of mathematical knowledge. The leitmotiv is the extensive use of XML technology, and ...

We present simple calculational proofs of Church-Rosser theorems for equational theories, quasiorderings and nonsymmetric transitive relations in Kleene algebra. We also calculate the abstract part of two standard proofs of Church-Rosser theorems in the ...

We survey a substantial body of knowledge about lambda calculus and Pure Type Systems, formally developed in a constructive type theory using the LEGO proof system. On lambda calculus, we work up to an abstract, simplified proof of standardization for ...

It is argued that formal verifications of programs, no matter how obtained, will not play the same key role in the development of computer science and software engineering as proofs do in mathematics. Furthermore the absence of continuity, the ...