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.