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,

1 Introduction When searching for proofs of theorems which contain definitions, it is a significant problem to decide which instances of the definitions to instantiate. Often, one needs to instantiate some, but not all, of them, and if one does instantiate all of them, one can cause the search space to expand in a very undesirable way. This problem has been noted in [4] and [23], and treatments of it may be found in [6], [8] and [12]. We have found a partial solution to this problem; it involves making each instance of a definition accessible to the search procedure in both its instantiated and its uninstantiated form, and letting the search procedure decide which to use, with a bias in favor of the uninstantiated form. This is very effective in some cases.

In this paper we describe a theorem proving system called grover. grover is novel in that it may be guided in its search for a proof by information contained in a diagram. There are two parts to the system: the underlying theorem prover, called &, and the graphical subsystem which examines the diagram and makes calls to the underlying prover on the basis of the information found there. We have used grover to prove the Diamond Lemma, a non-trivial theorem from the theory of well-founded relations. Key words. Automated reasoning, graphical theorem proving, proof strategies. This material is based upon work supported by the National Science Foundation under award number ISI-8701133. 1 INTRODUCTION 2 1 Introduction Open almost any mathematics text book and you will find, along with the familiar symbolism of mathematics and motivational text, many diagrams which are included to help the reader visualize the particular point being made. One might be tempted to conclude that mathema...

This paper analyses a technique (called Gazing) for unfolding de nitions on the basis of a global plan built in an abstract space. Gazing's logical properties are studied inside a formal framework which relies on a more general theory of abstraction. Some experimental results con rming the theoretical ones are also presented.

Abstract. Set theory is the common language of mathematics. Therefore, set theory plays an important rôle in many important applications of automated deduction. In this paper, we present an improved tableau calculus for the decidable fragment of set theory called multi-level syllogistic with singleton (MLSS). Furthermore, we describe an extension of our calculus for the bigger fragment consisting of MLSS enriched with free (uninterpreted) function symbols (MLSSF). 1

this paper we give a decision procedure and a decidable tableau calculus for the extension of Multilevel Syllogistic