This paper presents some fundamental aspects of the design and the implementation of an automated prover for Zermelo-Fraenkel set theory within the well-known Theorema system. The method applies the “Prove-Compute-Solve”-paradigm as its major strategy for generating proofs in a natural style for statements involving constructs from set theory.

Abstract. We describe an environment that allows the users of the Theorema system to flexibly control aspects of computer-supported proof development. The environment supports the display and manipulation of proof trees and proof situations, logs the user activities (commands communicated with the system during the proving session), and presents (also unfinished) proofs in a human-oriented style. In particular, the user can navigate through the proof object, expand/remove proof branches, provide witness terms, develop several proofs concurrently, proceed step by step or automatically and so on. The environment enhances the effectiveness and flexibility of the reasoners of the Theorema system. 1

Approaching the problem of imperative program verification from a practical point of view has certain implications concerning [4]: the style of specifications, the programming language which is used, the help provided to the user for finding appropriate loop invariants, the theoretical frame used for formal verification, the language used for expressing generated verification theorems as well as the database of necessary mathematical knowledge, and finally the proving power, style and language. The Theorema system (www. theorema.org) [1] has certain capabilities which make it appropriate for such a practical approach: the logic language of the system is higher-order predicate logic expressed in natural style; the procedural language is simple and intuitive, yet sufficiently expressive and fully integrated in the logical frame of the system; the language and the style of the proofs are natural, similar to those used by humans; and finally the proving power of Theorema is enhanced by using specific provers for special domains, which are integrated with sophisticated mathematical algorithms. Our approach for imperative program verification in Theorema is based on the so-called Hoare–Logic, which verification process is characterized by (for a tutorial introduction see also [7, 8]): • an imperative program • and a logical specification

Abstract. We present a verification environment for imperative programs (using Hoare logic) and for functional programs (using fixpoint theory) in the frame of the Theorema system (www.theorema.org). In particular, we discuss some methods for finding the invariants of loops and of specifications of auxiliary tail recursive functions. These methods use algorithms from (polynomial) algebra and combinatorics, namely Groebner bases, variable elimination and symbolic summation (the Gosper algorithm, the technique of generating functions). The techniques are demonstrated on several examples which have been treated automatically by our implementation.

Observing is the process of obtaining new knowledge, expressed in language, by bringing the senses in contact with reality. Reasoning, in contrast, is the process of obtaining new knowledge from given knowledge, by applying certain general transformation rules that depend only on the form of

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We present a method that generates automatically algebraic invariant properties of a loop. The implementation and verification process is done in a prototype verification condition generator for imperative programs. This verification tool is integrated into the overall framework of the Theorema system, which is based on a version of higher order predicate logic and includes verification procedures for functional and rewrite algorithms but also for procedural programs. The main contribution of this paper is the algorithm that generates invariants for loops with conditionals. In the proposed algorithm program analysis is performed in order to transform the code into a form for which algebraic and combinatorial techniques (symbolic summation, variable elimination, polynomial algebra) can be applied to obtain an invariant property. The application of the method is demonstrated in few examples.

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à Specification of the problem Find an instantiation of variables (a collection of witness terms), which satisfies domain-specific constraints. E.g., in order to prove the formula

www.math.u-szeged.hu In mathematical models, we often search for objects whose components satisfy certain constraints. Among the constraint satisfaction problems the simplest ones are the systems containing only equations and inequalities as constraints among the unknowns. We can be interested for satisfiability, i.e., for the existence of solutions, which might be dependent on certain parameters or we can search for the constructive description of the entire solution set. In this article we consider general algebraic systems over the reals. It turns out that systems containing only algebraic equations and inequalities can be treated completely algorithmically and the constructive characterization of the solution sets are possible. The latter description is equivalent to an elimination of quantifiers from a logical formula which represents the system. The effective construction of the solution set is only tractable even for problems with modest input size with the best hardware and software environments. After the introduction of the problem (the elimination of quantifiers) and the algorithm for its solution (cylindrical algebraic decomposition), we illustrate some field of applications. 1.