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A logic for specification and verification is derived from the axioms of Zermelo-Fraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higher-order syntax supports the definition of new binding operators. Unknowns in subgoals can be instantiated incrementally. The paper describes the derivation of rules for descriptions, relations and functions, and discusses interactive proofs of Cantor’s Theorem, the Composition of Homomorphisms challenge [9], and Ramsey’s Theorem [5]. A generic proof assistant can stand up against provers dedicated to particular logics. Key words. Isabelle, set theory, generic theorem proving, Ramsey’s Theorem,

A theory of recursive definitions has been mechanized in Isabelle's Zermelo-Fraenkel (ZF) set theory. The objective is to support the formalization of particular recursive definitions for use in verification, semantics proofs and other computational reasoning.

this article concerns autosegmental representations, and not the rules which are presumed to manipulate them. Due to the expository goals of this paper we have not attempted to carry out a detailed analysis of a large body of phonological data, however we acknowledge that this is an important task and it is one that we intend to undertake in future work. Deriving the No-Crossing Constraint Sagey defines three relations on temporal units: simultaneity, precedence and overlap. Certain facts about the first two relations (and presumably the third also) are taken to be `included in our knowledge of the world' (p.110). We begin with a brief review of these facts. Temporal overlap is a two-place relation which is reflexive, symmetric and nontransitive. If we employ the notation x ffi y for the statement `x overlaps y' then these facts about overlap can be stated as follows: (1) a. For any x, x ffi x overlap is reflexive

The analysis of set-based programs is sometimes facilitated by the computational model of a set machine; i.e., a uniform cost sequential RAM augmented with an assortment of primitives on finite sets, under the assumption that associative operations, e.g., set membership, take unit time. In this paper we give broad sufficient conditions in which to simulate a set machine on a RAM (without set primitives) in real time. Two variants of a RAM are considered. One allows for pointer and cursor access. The other permits only pointer access. Our translation method introduces a new programming methodology for data structure design and provides a new framework for investigating automatic data structure selection for set-based programs. November 10, 1994 ############### 1 Part of this work was done while the author was a summer faculty at IBM T.J. Watson Research Center. This work is also partly based on research supported by the Office of Naval Research under Contract No. N00014-87-K-0461 and b...

This manual describes how to use the theorem prover Isabelle. For beginners, it explains how to perform simple single-step proofs in the built-in logics. These include first-order logic, a classical sequent calculus, zf set theory, Constructive Type Theory, and higher-order logic. Each of these logics is described. The manual then explains how to develop advanced tactics and tacticals and how to derive rules. Finally, it describes how to define new logics within Isabelle. Acknowledgements. Isabelle uses Dave Matthews's Standard ml compiler, Poly/ml. Philippe de Groote wrote the first version of the logic lk. Funding and equipment were provided by SERC/Alvey grant GR/E0355.7 and ESPRIT BRA grant 3245. Thanks also to Philippe Noel, Brian Monahan, Martin Coen, and Annette Schumann. Contents 1 Basic Features of Isabelle 5 1.1 Overview of Isabelle : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 1.1.1 The representation of logics : : : : : : : : : : : : : : : : : : : 6 1.1.2 The...

Abstract. Fairly deep results of Zermelo-Frænkel (ZF) set theory have been mechanized using the proof assistant Isabelle. The results concern cardinal arithmetic and the Axiom of Choice (AC). A key result about cardinal multiplication is κ ⊗ κ = κ, where κ is any infinite cardinal. Proving this result required developing theories of orders, order-isomorphisms, order types, ordinal arithmetic, cardinals, etc.; this covers most of Kunen, Set Theory, Chapter I. Furthermore, we have proved the equivalence of 7 formulations of the Well-ordering Theorem and 20 formulations of AC; this covers the first two chapters of Rubin and Rubin, Equivalents of the Axiom of Choice, and involves highly technical material. The definitions used in the proofs are

iii This document and the work it represents was impossible without the support of my wife Ginger. Often one needs non-technical advice to make clear what one is contemplating. Also one always needs a financial supporter. My thesis advisor Erik Antonsson helped focus many of my thoughts. In addition to providing me with technical assistance, he as well provided instruction on the process of conducting academic research, the communication of ideas both orally and written, and the approach to a developing field. I also owe much to my colleagues in the Engineering and Applied Science Division at Caltech. Their comments and advice maintained my comprehension and rigor. Andrew Lewis in particular provided me with invaluable support. Many of the technical proofs were impossible without him. This material and the work it represented were made possible, in part, by a fellowship from the AT&T-Bell Laboratories Ph.D. scholar program, sponsored by the AT&T foundation. Also, the National Science Foundation provided funding under a Presidential Young