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Despite the vast body of research literature proposing algorithms with formal guarantees, the amount of verifiable code in today's systems remains minimal. This discrepancy stems from the inherent difficulty of verifying code, particularly…
Interactive theorem proving requires a lot of human guidance. Proving a property involves (1) figuring out why it holds, then (2) coaxing the theorem prover into believing it. Both steps can take a long time. We explain how to use GL, a…
Nowadays, formal theorem provers have made monumental progress on high-school and competition-level mathematics, but few of them generalize to more advanced mathematics. In this paper, we present REAL-Prover, a new open-source stepwise…
Mathematical reasoning remains a significant challenge for Large Language Models (LLMs) due to hallucinations. When combined with formal proof assistants like Lean, these hallucinations can be eliminated through rigorous verification,…
Formal theorem-proving benchmarks enable mechanically verifiable evaluation of mathematical reasoning in large language models. However, existing benchmarks mainly focus on Olympiad-style problems and algebraic domains, leaving…
A simple mathematical extension of quantum theory is presented. As well as opening the possibility of alternative methods of calculation, the additional formalism implies a new physical interpretation of the standard theory by providing a…
To study the dynamical behaviour of the engineering and physical systems, we often need to capture their continuous behaviour, which is modeled using differential equations, and perform the frequency-domain analysis of these systems.…
A foundational result in constructive quantum field theory is the construction of the free bosonic quantum field theory in four-dimensional Euclidean spacetime and the proof that it satisfies the Glimm-Jaffe axioms, a variant of the…
Approaching limitations of digital computing technologies have spurred research in neuromorphic and other unconventional approaches to computing. Here we argue that if we want to systematically engineer computing systems that are based on…
We present ZFLean, a Lean 4 library for doing core mathematics inside a model of ZFC with the ergonomics expected of typed Mathlib developments. Building on Mathlib's ZFC model, we contribute a relational calculus for sets with rewriting…
We investigate how large language models can be used as research tools in scientific computing while preserving mathematical rigor. We propose a human-in-the-loop workflow for interactive theorem proving and discovery with LLMs. Human…
We describe the formalization of the Ionescu-Tulcea theorem, showing the existence of a probability measure on the space of trajectories of a Markov chain, in the proof assistant Lean using the integrated library Mathlib. We first present a…
During 2024 and 2025 the discussion about the theorem-proving capabilities of large language models started reporting interesting success stories, mostly to do with difficult exercises (such as problems from the International Mathematical…
We introduce ChemPro, a progressive benchmark with 4100 natural language question-answer pairs in Chemistry, across 4 coherent sections of difficulty designed to assess the proficiency of Large Language Models (LLMs) in a broad spectrum of…
We illustrate the main features of a recently proposed method based on ensemble density functional theory to divide rigorously a complex molecular system into its parts [M.H. Cohen and A. Wasserman, J. Phys. Chem. A 111, 2229 (2007)]. The…
To obtain the highest confidence on the correction of numerical simulation programs implementing the finite element method, one has to formalize the mathematical notions and results that allow to establish the soundness of the method. The…
We formalize a proof of the irrationality of $\zeta(3)$ in Lean 4, using Beukers' method. To support this, we extend the Lean mathematical library (Mathlib) by formalizing shifted Legendre polynomials and important results in analytic…
Interactive theorem provers (ITPs) are powerful tools for the formal verification of mathematical proofs down to the axiom level. However, their lack of a natural language interface remains a significant limitation. Recent advancements in…
Isabelle is a generic theorem prover, designed for interactive reasoning in a variety of formal theories. At present it provides useful proof procedures for Constructive Type Theory, various first-order logics, Zermelo-Fraenkel set theory,…
Using AI to write formal proofs for mathematical problems is a challenging task that has seen some advancements in recent years. Automated systems such as Lean can verify the correctness of proofs written in formal language, yet writing the…