相关论文: Schemes in Lean

We prove a new theorem on additive Levy processes and show that this theorem implies several proved theorems and a hard conjectured theorem.

An approach is shown that proves various theorems of plane geometry in an algorithmic manner. The approach affords transparent proofs of a generalization of the Theorem of Morley and other well known results by casting them in terms of…

The three sections of this paper illustrate the importance of rationalizing ISM theory and modeling with the observational information.

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…

Complex systems typically have many different parts and facets, with different characteristics. In a multi-paradigm approach to modeling, formalisms with different natures are used in combination to describe complementary parts and aspects…

In Machine-Assisted Theorem Proving, a theorem proving agent searches for a sequence of expressions and tactics that can prove a conjecture in a proof assistant. In this work, we introduce several novel concepts and capabilities to address…

We give a direct and elementary proof of the theorem on formal functions by studying the behaviour of the Godement resolution of a sheaf of modules under completion.

In the paper different kinds of proof of a given statement are discussed. Detailed descriptions of direct and indirect methods of proof are given. Logical models illustrate the essence of specific types of indirect proofs. Direct proofs of…

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…

We describe here our perception of complex systems, of how we feel the different layers of description are important part of a correct complex system simulation. We describe a rough models categorization between rules based and law based,…

We show that large language models (LLMs) exhibit an $\textit{internal chain-of-thought}$: they sequentially decompose and execute composite tasks layer-by-layer. Two claims ground our study: (i) distinct subtasks are learned at different…

The Abella interactive theorem prover has proven to be an effective vehicle for reasoning about relational specifications. However, the system has a limitation that arises from the fact that it is based on a simply typed logic:…

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…

Neural theorem proving has advanced rapidly in the past year, reaching IMO gold-medalist capabilities and producing formal proofs that span thousands of lines. Although such proofs are mechanically verified by formal systems like Lean,…

In model-driven development, an ordered model transformation is a nested set of transformations between source and target classes, in which each transformation is governed by its own pre and post- conditions, but structurally dependent on…

A line of work in planning uses LLM not to generate a plan, but to generate a formal representation in some planning language, which can be input into a symbolic solver to deterministically find a plan. While showing improved trust and…

This paper presents a methodology for constructing iterative schemes of any order of convergence for solving nonlinear systems of equations. It also provides formulas for the order of convergence of any iterative schemes constructed using…

Transform methods, like Laplace and Fourier, are frequently used for analyzing the dynamical behaviour of engineering and physical systems, based on their transfer function, and frequency response or the solutions of their corresponding…

This document reports on the use of an algebraic, visual, formal approach to the specification of patterns for the formalization of the GoF design patterns. The approach is based on graphs, morphisms and operations from category theory and…

The design and analysis of systems that combine computational behaviour with physical processes' continuous dynamics - such as movement, velocity, and voltage - is a famous, challenging task. Several theoretical results from programming…