Related papers: Structuring Definitions in Mathematical Libraries
Education in the practical applications of logic and proving such as the formal specification and verification of computer programs is substantially hampered by the fact that most time and effort that is invested in proving is actually…
We introduce MLFMF, a collection of data sets for benchmarking recommendation systems used to support formalization of mathematics with proof assistants. These systems help humans identify which previous entries (theorems, constructions,…
Mechanical reasoning is a key area of research that lies at the crossroads of mathematical logic and artificial intelligence. The main aim to develop mechanical reasoning systems (also known as theorem provers) was to enable mathematicians…
The rise of multi-paradigm languages challenges traditional classification methods, leading to practical software engineering issues like interoperability defects. This systematic literature review (SLR) maps the formal foundations of…
With the recent advances in the field of artificial intelligence, an increasing number of decision-making tasks are delegated to software systems. A key requirement for the success and adoption of such systems is that users must trust…
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…
A proof is one of the most important concepts of mathematics. However, there is a striking difference between how a proof is defined in theory and how it is used in practice. This puts the unique status of mathematics as exact science into…
The Lean mathematical library Mathlib is one of the fastest-growing libraries of formalised mathematics. We describe various strategies to manage this growth, while allowing for change and avoiding maintainer overload. This includes dealing…
We report on a formalization of the change of variables formula in integrals, in the mathlib library for Lean. Our version of this theorem is extremely general, and builds on developments in linear algebra, analysis, measure theory and…
Data science workflows often integrate functionalities from a diverse set of libraries and frameworks. Tasks such as debugging require data lineage that crosses library boundaries. The problem is that the way that "lineage" is represented…
Libraries of formal proofs are an important part of our mathematical heritage, but their usability and sustainability is poor. Indeed, each library is specific to a proof system, sometimes even to some version of this system. Thus, a…
Stepwise refinement of algebraic specifications is a well known formal methodology for program development. However, traditional notions of refinement based on signature morphisms are often too rigid to capture a number of relevant…
We report on the higher-order differential calculus library developed inside the Lean mathematical library mathlib. To support a broad range of applications, we depart in several ways from standard textbook definitions: we allow arbitrary…
This paper describes a formal proof library, developed using the Coq proof assistant, designed to assist users in writing correct diagrammatic proofs, for 1-categories. This library proposes a deep-embedded, domain-specific formal language,…
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,…
Thanks to their linguistic capabilities, LLMs offer an opportunity to bridge the gap between informal mathematics and formal languages through autoformalization. However, it is still unclear how well LLMs generalize to sophisticated and…
The purpose of a program analysis is to compute an abstract meaning for a program which approximates its dynamic behaviour. A compositional program analysis accomplishes this task with a divide-and-conquer strategy: the meaning of a program…
This thesis concerns the development of a framework that facilitates the design and analysis of formal systems. Specifically, this framework provides a specification language which supports the concise and direct description of formal…
A plausible definition of "reasoning" could be "algebraically manipulating previously acquired knowledge in order to answer a new question". This definition covers first-order logical inference or probabilistic inference. It also includes…
Although there is a somewhat standard formalization of computability on countable sets given by Turing machines, the same cannot be said about uncountable sets. Among the approaches to define computability in these sets, order-theoretic…