A sequent calculus for signed interval logic

We propose and discuss a complete sequent calculus formulation for Signed Interval Logic (SIL) with the chief purpose of improving proof support for SIL in practice. The main theoretical result is a simple characterization of the limit between decidability and undecidability of quantifier-free SIL. We present a mechanization of SIL in the generic proof assistant Isabelle and consider techniques for automated reasoning. Many of the results and ideas of this report are also applicable to traditional (non-signed) interval logic and, hence, to Duration Calculus.

Formalizing Basic Number Theory

This document describes a formalization of basic number theory including two theorems of Fermat and Wilson. Most of this have (in some context) been formalized before but we present a new generalized approach for handling some central parts, based on concepts which seem closer to the original mathematical intuition and likely to be useful in other (similar) developments. Our formalization has been mechanized in the Isabelle/HOL system. Contents 1 Introduction 2 2 Basic Number Theory 2 2.1 The Chinese Remainder Theorem . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Fermat's Little Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Wilson's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Formalization 8 3.1 Bijection Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Fermat's Little Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2.1 BoyerMoore's proof . . . . . . . . . . . . . . . . . . . . ..

An Inductive Approach to Formalizing Notions of Number Theory Proofs

Abstract. In certain proofs of theorems of, e.g., number theory and the algebra of finite fields, one-to-one correspondences and the “pairing off” of elements often play an important role. In textbook proofs these con-cepts are often not made precise but if one wants to develop a rigorous formalization they have to be. We have, using an inductive approach, developed constructs for handling these concepts. We illustrate their usefulness by considering formalizations of Euler-Fermat’s and Wilson’s Theorems. The formalizations have been mechanized in Isabelle/HOL, making a comparison with other approaches possible.

Signed Interval Logic

Signed Interval Logic (SIL) is an extension of Interval Temporal Logic (ITL) with the introduction of the notion of a direction of an interval. We develop syntax, semantics, and proof system of SIL, and show that this proof system is sound and complete. The proof system of SIL is not more complicated than that of ITL but SIL is (contrary to ITL) capable of specifying liveness properties. Other interval logics capable of this (such as Neighbourhood Logic) have more complicated proof systems. We discuss how to de ne future intervals in SIL for the specification of liveness properties. To characterize the expressive power of SIL we relate SIL to arrow logic and relational algebra

and Niels Hellraiser Christensen for being good friends and colleagues.

As any other Ph.D. student, I have met plenty of people during the course of my studies, that should be thanked in some way or the other. First and foremost, my heartfelt thanks to Neil D. Jones for being my strict supervisor and kind boss, during both my master’s thesis and my Ph.D. studies. Also to Klaus Grue, who suggested the topic to me in early 1999, when I was complaining about total functions in HOL, and for helping me with numerous technical questions regarding map theory. Tobias Nipkow at Technische Universität München having me as a guest in the first half of 2000. Thanks also to the rest of the Isabelle group