Related papers: Tutte's theorem as an educational formalization pr…
We present a formalization, in the theorem prover Lean, of the classification of solvable Lie algebras of dimension at most three over arbitrary fields. Lie algebras are algebraic objects which encode infinitesimal symmetries, and as such…
Verifying mathematical proofs is difficult, but can be automated with the assistance of a computer. Autoformalization is the task of automatically translating natural language mathematics into a formal language that can be verified by a…
We formalize Hall's Marriage Theorem in the Lean theorem prover for inclusion in mathlib, which is a community-driven effort to build a unified mathematics library for Lean. One goal of the mathlib project is to contain all of the topics of…
We provide a matrix-based formula for the Tutte symmetric function of a graph. In particular, for any graph $G$ with a designated head and tail vertex, we describe an infinite matrix $M_G$ from which the Tutte symmetric function can be…
There is a long tradition of fruitful interaction between logic and social choice theory. In recent years, much of this interaction has focused on computer-aided methods such as SAT solving and interactive theorem proving. In this paper, we…
Mathematics formalisation is the task of writing mathematics (i.e., definitions, theorem statements, proofs) in natural language, as found in books and papers, into a formal language that can then be checked for correctness by a program. It…
Matchings were among the earliest motivations for graph theory. They subsequently remained a central goal, inspiring the development of new tools that went well beyond problems directly concerning matchings. These tools proved widely…
Autoformalization has emerged as a term referring to the automation of formalization - specifically, the formalization of mathematics using interactive theorem provers (proof assistants). Its rapid development has been driven by progress in…
While large language models (LLMs) have shown progress in mathematical reasoning, they still face challenges in formalizing theorems that arise from instantiating abstract structures in concrete settings. With the goal of auto-formalizing…
A biform theory is a combination of an axiomatic theory and an algorithmic theory that supports the integration of reasoning and computation. These are ideal for formalizing algorithms that manipulate mathematical expressions. A theory…
We report on our experience formalizing differential geometry with mathlib, the Lean mathematical library. Our account is geared towards geometers with no knowledge of type theory, but eager to learn more about the formalization of…
Formalizing mathematical proofs using computerized verification languages like Lean 4 has the potential to significantly impact the field of mathematics, it offers prominent capabilities for advancing mathematical reasoning. However,…
The continuous functional calculus is perhaps the most fundamental construction in the theory of operator algebras, especially $C^{*}$-algebras. Here we document our formalization of the continuous functional calculus in Lean, which…
A realisation of a graph in the plane as a bar-joint framework is rigid if there are finitely many other realisations, up to isometries, with the same edge lengths. Each of these finitely-many realisations can be seen as a solution to a…
Lie algebras are an important class of algebras which arise throughout mathematics and physics. We report on the formalisation of Lie algebras in Lean's Mathlib library. Although basic knowledge of Lie theory will benefit the reader, none…
In this survey of graph polynomials, we emphasize the Tutte polynomial and a selection of closely related graph polynomials. We explore some of the Tutte polynomial's many properties and applications and we use the Tutte polynomial to…
A theory graph is a network of axiomatic theories connected with meaning-preserving mappings called theory morphisms. Theory graphs are well suited for organizing large bodies of mathematical knowledge. Traditional and formal proofs do not…
This paper is the blueprint underlying the Lean formalization of the proof of Carleson's classical result asserting almost everywhere convergence of Fourier series of continuous functions. We break up the proof into two steps, a reduction…
In this article we describe the formalisation of the Bruhat-Tits tree - an important tool in modern number theory - in the Lean Theorem Prover. Motivated by the goal of connecting to ongoing research, we apply our formalisation to verify a…
The chase is a sound, complete, but possibly non-terminating algorithm for reasoning with existential rules (aka. tuple-generating dependencies), a highly expressive knowledge representation language. Although the procedure appears simple,…