apply continuity analysis

This paper addresses the challenge of precisely characterizing data-state evolution and interference in parallel processes. Methodologically, it introduces the first framework systematically embedding rely/guarantee (R/G) logic into an imperative process algebra—specifically, an extension of ACP—by integrating predicate logic with operational program semantics. It designs R/G inference rules supporting partial, weak, and strong total correctness verification, and establishes a formally complete judgment system. Key contributions include: (i) the first deep integration of R/G logic with imperative process algebra; (ii) significantly enhanced expressiveness and precision in modeling interference among shared-variable parallel programs; and (iii) theoretical and empirical evidence demonstrating that, compared to Hoare logic, the framework yields more natural, accurate, compositional, and practical reasoning about concurrent interference.

This work addresses the longstanding gap in formal mathematics concerning the continuous functional calculus for C*-algebras. Using Lean 4 and the Mathlib library, we present the first fully verified formalization of this theory in any proof assistant: we rigorously define the continuous functional calculus on arbitrary C*-algebras, construct a framework for continuous maps from compact subsets of ℂ to bounded operators, and formally verify its fundamental properties—including the spectral mapping theorem, algebra homomorphism property, and continuity. Our design balances mathematical naturalness with seamless integration into Mathlib’s existing infrastructure; all results have been merged into the main Mathlib repository. This formalization establishes a foundational cornerstone for the mechanized development of C*-algebra theory and spectral theory, while providing highly reusable interfaces that significantly facilitate subsequent formalizations—such as spectral decomposition and the classification of normal operators.

本文提出了一种名为叙事的抽象框架，用于处理时间变化的数据，并通过三个案例展示了如何解决信息丢失、数据分解及多智能体系统建模等问题。

This work addresses the challenge of safety verification for complex dynamical systems—such as parametrized ordinary differential equations and partially observable Markov processes—by proposing a compositional verification framework grounded in category theory. The approach models systems as lenses and integrates assume-guarantee reasoning with advanced categorical structures, including symmetric monoidal double categories, fibrations, and 2-functors, to enable, for the first time, the compositional construction of local input-to-state stability (L)ISS Lyapunov functions. By employing contact conditions to unify diverse system models, the framework supports modular safety verification for generalized Moore machines, significantly enhancing both the expressiveness and applicability of compositional verification while maintaining strong scalability.

Compositional systems lack a structured, rewriteable mathematical foundation. Method: This work proposes a category-theoretic rewriting framework wherein structured cospans serve as the fundamental syntactic units; it introduces, for the first time, the coupling of structured cospans with double-pushout (DPO) rewriting, yielding a unified theory supporting both traced and trace-free semantics. The framework enables inductive, structure-preserving decomposition of closed systems and establishes a sound correspondence between syntax (structured cospans) and semantics (DPO rewriting). Contribution/Results: It provides the first categorical integration of structured cospans with DPO rewriting; defines two distinct rewriting paradigms—traceable and trace-free; and delivers the first mathematically rigorous, compositional, and rewriteable foundation for systems science, enabling cross-disciplinary modeling and formal analysis of complex systems.

This work addresses the lack of machine-verifiable foundations in control theory for cyber-physical systems by developing an open-source formal library within the Lean interactive theorem prover. The library formalizes Lyapunov stability theory and the small-gain theorem, supporting continuous, discrete, and hybrid dynamical systems. A key contribution is a unified formulation of Lyapunov’s theorem applicable to both points and sets, alongside a relational definition of input–output systems that avoids well-posedness assumptions, enabling a fully formalized proof of the small-gain theorem. Leveraging mathematical tools such as neighborhood filters, the project establishes a scalable verification framework for control theory, laying the groundwork for trustworthy, machine-checked validation of cyber-physical systems.

This work addresses the challenges of transferability and computational feasibility in discrete abstraction for symbolic model checking of cyber-physical systems by proposing a conservatism-first, four-step modular workflow to construct finite-state abstractions of closed-loop dynamical systems. The approach integrates state partitioning, conservative transition construction, spurious behavior elimination, and specification semantics lifting, enabling composable and replaceable subroutine design. Transition relations are built using axis-aligned bounding boxes, polyhedra, and sampling with PAC coverage certificates, combined with certified erasure and counterexample-guided refinement. Reliable lifting of LTL specifications is achieved through may–must semantics. Evaluation across three case studies demonstrates that the workflow effectively balances abstraction accuracy and verification efficiency while clearly revealing the impact of different design choices on the outcomes.

This work presents the first complete formalization in Lean 4, based on mathlib, of Dana Scott’s 1972 theory of continuous lattices and its application to modeling the untyped λ-calculus. The project rigorously reproduces the 43 core results from the first four sections of Scott’s original paper, covering essential constructions such as T₀-space embeddings, the Scott topology, the way-below relation, function spaces, and inverse limits, while also incorporating Milner’s corrections to the original proofs. By introducing foundational infrastructure—including bases of Scott-open sets, step functions, towers of function spaces, and the i_∞/j_∞ mapping pair—the formalization establishes the self-embedding theorem D_∞ ≅ [D_∞ → D_∞] under classical logic (with the axiom of choice), propositional extensionality, and quotient type soundness assumptions. All results are verified machine-checked without any use of “sorry”.

This project addresses the challenges of temporal causality and interpretability in reactive systems by proposing a general temporal interpretability framework. Methodologically, it unifies the concepts of sufficient reasons and contrastive explanations, extends neural network interpretation techniques to temporal logic specifications, constructs symbolic representations along the temporal dimension, and integrates formal verification with symbolic execution for solving. The primary theoretical contribution lies in establishing the computational complexity boundaries for various forms of temporal explanations. Practically, the effectiveness of the proposed approach is validated through a prototype system. Overall, this work provides a novel paradigm for the trustworthy analysis of complex reactive systems.

This work addresses the absence of a formally verified foundational library in mathematical finance, which has hindered rigorous and reusable theoretical development. Building upon Mathlib and the BrownianMotion package in Lean 4, the authors construct a comprehensive formal library spanning eleven core areas, including continuous-time stochastic calculus, derivative pricing, and risk and portfolio theory. The library comprises over 200 theorems proved without gaps in assumptions. Notably, it presents the first formal construction of the L² Itô integral and derives the risk-neutral measure within a proof assistant. Additionally, a fidelity auditing mechanism is introduced to explicitly track the axioms and assumptions underlying each theorem. This effort establishes the most extensive machine-verified infrastructure for mathematical finance to date, enabling unified certification and reliable reuse of classical results.