Lawrence C. Paulson

The paper briefly describes a method of formalizing non-well- founded data structures in standard ZF set theory. Examples include lists of n elements, the accessible part of a relation and the set of primitive recursive functions. One example of a coinductive definition is bisimulations for lazy lists. Recursive datatypes are examined in detail, as well as one example of a codatatype: lazy lists. The appendices are simple user's manuals for this Isabelle/ZF package.

A Fixedpoint Approach to Implementing (co)inductive Definitions

This book is concerned with techniques for formal theorem-proving, with particular reference to Cambridge LCF (Logic for Computable Functions). Cambridge LCF is a computer program for reasoning about computation. It combines the methods of mathematical logic with domain theory, the basis of the denotational approach to specifying the meaning of program statements. Cambridge LCF is based on an earlier theorem-proving system, Edinburgh LCF, which introduced a design that gives the user flexibility to use and extend the system. A goal of this book is to explain the design, which has been adopted in several other systems. The book consists of two parts. Part I outlines the mathematical preliminaries, elementary logic and domain theory, and explains them at an intuitive level, giving reference to more advanced reading; Part II provides sufficient detail to serve as a reference manual for Cambridge LCF. It will also be a useful guide for implementors of other programs based on the LCF approach.

Interactive Proof with Cambridge LCF

Logic and Computation

Larry C. Paulson

ML for the Working Programmer

Abstract: "Informal arguments that cryptographic protocols are secure can be made rigorous using inductive definitions. The approach is based on ordinary predicate calculus and copes with infinite-state systems. Proofs are generated using Isabelle/HOL. The human effort required to analyze a protocol can be as little as a week or two, yielding a proof script that takes a few minutes to run. Protocols are inductively defined as sets of traces. A trace is a list of communication events, perhaps comprising many interleaved protocol runs. Protocol descriptions incorporate attacks and accidental losses. The model spy knows some private keys and can forge messages using components decrypted from previous traffic. Three protocols are analyzed below: Otway-Rees (which uses shared-key encryption), Needham-Schroeder (which uses public-key encryption), and a recursive protocol [9] (which is of variable length). One can prove that event ev always precedes event ev ́or that property P holds provided X remains secret. Properties can be proved from the viewpoint of the various principals: say, if A receives a final message from B then the session key it conveys is good."

The Inductive Approach to Verifying Cryptographic Protocols

Abstract: "The inductive approach to verifying security protocols, previously applied to shared-key encryption, is here applied to the public key version of the Needham-Schroeder protocol. As before, mechanized proofs are performed using Isabelle/HOL. Both the original, flawed version and Lowe's improved version are studied; the properties proved highlight the distinctions between the two versions. The results are compared with previous analyses of the same protocol. The analysis reported below required only 30 hours of the author's time. The proof scripts execute in under three mintues."

Needham-Schroeder with Public Keys

Mechanized Proofs of Security Protocols

As a generic theorem prover, Isabelle supports a variety of logics. Distinctive features include Isabelle's representation of logics within a meta-logic and the use of higher-order unification to combine inference rules. Isabelle can be applied to reasoning in pure mathematics or verification of computer systems. This volume constitutes the Isabelle documentation. It begins by outlining theoretical aspects and then demonstrates the use in practice. Virtually all Isabelle functions are described, with advice on correct usage and numerous examples. Isabelle's built-in logics are also described in detail. There is a comprehensive bebliography and index. The book addresses prospective users of Isabelle as well as researchers in logic and automated reasoning.

A Generic Theorem Prover

Isabelle

Tobias Nipkow

Markus Wenzel

This volume is a self-contained introduction to interactive proof in high- order logic (HOL), using the proof assistant Isabelle 2002. Compared with existing Isabelle documentation, it provides a direct route into higher-order logic, which most people prefer these days. It bypasses ?rst-order logic and minimizes discussion of meta-theory. It is written for potential users rather than for our colleagues in the research world. Another departure from previous documentation is that we describe Markus Wenzel’s proof script notation instead of ML tactic scripts. The l- ter make it easier to introduce new tactics on the ?y, but hardly anybody does that. Wenzel’s dedicated syntax is elegant, replacing for example eight simpli?cation tactics with a single method, namely simp, with associated - tions. The book has three parts. – The ?rst part, Elementary Techniques, shows how to model functional programs in higher-order logic. Early examples involve lists and the natural numbers. Most proofs are two steps long, consisting of induction on a chosen variable followed by the auto tactic. But even this elementary part covers such advanced topics as nested and mutual recursion. – The second part, Logic and Sets, presents a collection of lower-level tactics that you can use to apply rules selectively. It also describes I- belle/HOL’s treatment of sets, functions, and relations and explains how to de?ne sets inductively. One of the examples concerns the theory of model checking, and another is drawn from a classic textbook on formal languages.

A Proof Assistant for Higher-Order Logic

Isabelle/HOL

Matt Kaufmann

This book constitutes the refereed proceedings of the First International Conference on Interactive Theorem proving, ITP 2010, held in Edinburgh, UK, in July 2010. The 33 revised full papers presented were carefully reviewed and selected from 74 submissions. The papers are organized in topics such as counterexample generation, hybrid system verification, translations from one formalism to another, and cooperation between tools. Several verification case studies were presented, with applications to computational geometry, unification, real analysis, etc.

First International Conference, ITP 2010 Edinburgh, UK, July 11-14, 2010, Proceedings

Interactive Theorem Proving

Abstract: "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."

Lawrence C. Paulson

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