Towards the Formalization of Fractional Calculus in Higher-Order Logic

Fractional calculus is a generalization of classical theories of integration and differentiation to arbitrary order (i.e., real or complex numbers). In the last two decades, this new mathematical modeling approach has been widely used to analyze a wide class of physical systems in various fields of science and engineering. In this paper, we describe an ongoing project which aims at formalizing the basic theories of fractional calculus in the HOL Light theorem prover. Mainly, we present the motivation and application of such formalization efforts, a roadmap to achieve our goals, current status of the project and future milestones.Comment: 9 page

A Verified Certificate Checker for Finite-Precision Error Bounds in Coq and HOL4

Being able to soundly estimate roundoff errors of finite-precision computations is important for many applications in embedded systems and scientific computing. Due to the discrepancy between continuous reals and discrete finite-precision values, automated static analysis tools are highly valuable to estimate roundoff errors. The results, however, are only as correct as the implementations of the static analysis tools. This paper presents a formally verified and modular tool which fully automatically checks the correctness of finite-precision roundoff error bounds encoded in a certificate. We present implementations of certificate generation and checking for both Coq and HOL4 and evaluate it on a number of examples from the literature. The experiments use both in-logic evaluation of Coq and HOL4, and execution of extracted code outside of the logics: we benchmark Coq extracted unverified OCaml code and a CakeML-generated verified binary

Proof of the basic theorem on concept lattices in Isabelle/HOL

This paper presents a machine-checked proof of the Basic Theorem on Concept Lattices, which appears in the book "Formal Concept Analysis" by Canter and Wille, in the Isabelle/HOL Proof Assistant. As a by-product, the underlying lattice theory by Kammueller has been extended

Essential Incompleteness of Arithmetic Verified by Coq

A constructive proof of the Goedel-Rosser incompleteness theorem has been completed using the Coq proof assistant. Some theory of classical first-order logic over an arbitrary language is formalized. A development of primitive recursive functions is given, and all primitive recursive functions are proved to be representable in a weak axiom system. Formulas and proofs are encoded as natural numbers, and functions operating on these codes are proved to be primitive recursive. The weak axiom system is proved to be essentially incomplete. In particular, Peano arithmetic is proved to be consistent in Coq's type theory and therefore is incomplete.Comment: This paper is part of the proceedings of the 18th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2005). For the associated Coq source files see the TeX sources, or see <http://r6.ca/Goedel20050512.tar.gz

A Modeling and Formal Approach for the Precise Specification of Security Patterns

International audienceNon-functional requirements such as Security and Dependability (S &D) become more important as well as more difficult to achieve. In fact, the integration of security features requires the availability of both application domain specific knowledge and security expertise at the same time. Hence, capturing and providing this expertise by the way of security patterns can support the integration of S&D features by design to foster reuse during the process of software system development.The solution envisaged here is based on combining metamodeling techniques and formal methods to represent security pattern at two levels of abstraction fostering reuse during the process of pattern development and during the process of pattern-based development. The contribution of this work is twofold: (1) An improvement of our previous pattern modeling language for representing security pattern in the form of a subsystem providing appropriate interfaces and targeting security properties, (2) Formal specification and validation of pattern properties, using the interactive Isabelle/HOL proof assistant. The resulting validation artifacts may mainly complete the definitions, and provide semantics for the interfaces and the properties in the context of S&D. As a result, validated patterns will be used as bricks to build applications through a Model-Driven engineering approach