Related papers: Schemes in Lean
Traditional approaches for validating molecular simulations rely on making software open source and transparent, incorporating unit testing, and generally employing human oversight. We propose an approach that eliminates software errors…
In this paper we present a new "external checker" for the Lean theorem prover, written in Lean itself. This is the first complete typechecker for Lean 4 other than the reference implementation in C++ used by Lean itself, and our new checker…
LLMs have demonstrated strong mathematical reasoning abilities by leveraging reinforcement learning with long chain-of-thought, yet they continue to struggle with theorem proving due to the lack of clear supervision signals when solely…
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…
Three formal first-order finite dialectical schemes are investigated. It is shown that schemes 1 and 2 have significantly different finite models. Further, an infinite natural number model for schemes 1, 2, 3 is constructed, and it is shown…
Most of the engineering and physical systems are generally characterized by differential and difference equations based on their continuous-time and discrete-time dynamics, respectively. Moreover, these dynamical models are analyzed using…
We present in this position paper a methodology to validate legal governance regulatory models from an empirical approach, as illustrated by means of three diagrams: (i) a scheme drawing the rule and meta-rule of law; (ii) a metamodel for…
We describe an approach to learn, in a term-rewriting setting, function definitions from input/output equations. By confining ourselves to structurally recursive definitions we obtain a fairly fast learning algorithm that often yields…
Formally verifying properties of software code has been a highly desirable task, especially with the emergence of LLM-generated code. In the same vein, they provide an interesting avenue for the exploration of formal verification and…
We formalize a complete proof of the regular case of Fermat's Last Theorem in the Lean4 theorem prover. Our formalization includes a proof of Kummer's lemma, that is the main obstruction to Fermat's Last Theorem for regular primes. Rather…
The development and application of formal methods is a long standing research topic within the field of computer science. One particular challenge that remains is the uptake of formal methods into industrial practices. This paper introduces…
We present three projects concerned with applications of proof assistants in the area of programming language theory and mathematics. The first project is about a certified compilation technique for a domain-specific programming language…
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…
Recent advances in automated theorem proving use Large Language Models (LLMs) to translate informal mathematical statements into formal proofs. However, informal cues are often ambiguous or lack strict logical structure, making it hard for…
We consider the geometric optics problem of finding a system of two reflectors that transform a spherical wavefront into a beam of parallel rays with prescribed intensity distribution. Using techniques from optimal transportation theory, it…
We introduce MerLean, a fully automated agentic framework for autoformalization in quantum computation. MerLean extracts mathematical statements from \LaTeX{} source files, formalizes them into verified Lean~4 code built on Mathlib, and…
In this paper we propose a new perspective on the evolution and history of the idea of mathematical proof. Proofs will be studied at three levels: syntactical, semantical and pragmatical. Computer-assisted proofs will be give a special…
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…
This article is about the formalization of synthetic differential geometry with the Lean proof assistant and the mathematical library mathlib. The main result we prove and formalize is a Taylor theorem for functions of several variables,…
We offer a mathematical proof of consistency for Peano Arithmetic PA formalizable in PA. This result is compatible with Goedel's Second Incompleteness Theorem since our consistency proof does not rely on the representation of consistency as…