Proof by mathematical induction gives rise to various kinds of eureka steps, e.g. missing lemmata, generalization, etc. Most inductive theorem provers rely upon user intervention in supplying the required eureka steps.

In software engineering there is a growing demand for formal methods for the specification and validation of software systems. The formal development of a system might give rise to many proof obligations. We must prove the completeness of the specification and the validity of some inductive properties. In this framework, many provers have been developed. However they require much user interaction even for simple proof tasks. In this paper, we present new procedures to test sufficient completeness and to prove or disprove inductive properties automatically in parameterized conditional specifications. The method has been implemented in the prover SPIKE. Computer experiments illustrate the improvements in length and structure of proofs, due to parameterization. Moreover, SPIKE offers facilities to check and complete specifications.

Coinduction is a method of growing importance in reasoning about functional languages, due to the increasing prominence of lazy data structures. Through the use of bisimulations and proofs that observational equivalence is a congruence in various domains it can be used to proof the congruence of two processes. Several proof tools have been developed to aid coinductive proofs but all require user interaction. Crucially they require the user to supply an appropriate relation which the system can then prove to be a bisimulation. A method is proposed which uses the idea of proof plans to make a heuristic guess at a suitable relation. If the proof fails for that relation the reasons for failure are analysed using a proof critic and a new relation is proposed to allow the proof to go through.

Abstract. We define and discuss various conceivable notions of inductive validity for first-order equational clauses. This is done within the framework of constructor-based positive/negative conditional equational specifications which permits to treat negation and incomplete function definitions in an adequate and natural fashion. Moreover, we show that under some reasonable assumptions all these notions are monotonic w. r. t. consistent extension, in contrast to the case of inductive validity for initial semantics (of unconditional or positive conditional equations). In particular from a practical point of view, this monotonicity property is crucial since it allows for an incremental construction process of complex specifications where consistent extensions of specifications cannot destroy the validity of (already proved) inductive properties. Finally we show how various notions of inductive validity in the literature fit in or are related to our classification. 1 Introduction, Motivation

In this paper we are interested in an algebraic specification language that (1) allows for sufficient expessiveness, (2) admits a well-defined semantics, and (3) allows for formal proofs. To that end we study clausal specifications over built-in algebras. To keep things simple, we consider built-in algebras only that are given as the initial model of a Horn clause specification. On top of this Horn clause specification new operators are (partially) defined by positive/negative conditional equations. In the first part of the paper we define three types of semantics for such a hierarchical specification: model-theoretic, operational, and rewrite-based semantics. We show that all these semantics coincide, provided some restrictions are met. We associate a distinguished algebra A spec to a hierachical specification spec. This algebra is initial in the class of all models of spec. In the second part of the paper we study how to prove a theorem (a clause) valid in the distinguished algebra ...

We compare rippling and exhaustive rewriting using recursive path ordering, on a range of inductive proofs. We present statistics on success rates, branching rates and CPU times. We use these statistics to argue that rippling succeeds more often. However, these statistics also show that rippling and reduction are roughly the same in terms of average branching rate and that rippling often takes longer in terms of CPU time.

The efficient combining and augmenting of decision procedures are often very important for a successful use of theorem provers. There are several schemes for combining and augmenting decision procedures; some of them support handling uninterpreted functions, use of available lemmas, and the like. In this paper we introduce a general setting for describing different schemes for both combining and augmenting decision procedures. This setting is based on the macro inference rules used in different approaches. Some of these rules are abstraction, entailment, congruence closure and lemma invoking. The general setting gives a simple description and the key ideas of one scheme and makes different schemes comparable. Also, it makes easier combining ideas from different schemes. In this paper we describe several schemes via introduced macro inference rules and report on our prototype implementation.

. We describe the implementation of an automated theorem prover for algebraic specifications, in an algebraic specification setting using the ASF+SDF Meta-environment. The current implementation is based on the implicit induction approach implemented in SPIKE [1, 2]. We consider the implementation as a case study on the tool generation within ASF+SDF which also provides an experimental basis for the research on automated induction. We consider the issues of the simplification strategies of implicit induction, the user interaction/heuristics in implicit induction, the computational effectiveness of ASF+SDF specifications, and the plausible improvements of the ASF+SDF specification language. 1 On leave from the Institute of Cybernetics, Kiev, Ukraine. Supported by the Fulbright fellowship and the National Science Foundation under Grants CCR-9202838 and CCR-9357851. 2 Supported by the Netherlands Organization for Scientific Research (NWO) under the Generic Tools for Program Analysis a...

s of the papers from the ASF+SDF'95 Workshop CWI, Amsterdam, May 11 & 12, 1995 [1] J. Heering and P. Klint. The prehistory of ASF+SDF (1980-1984). pages 1--4. [2] P. Klint. The evolution of implementation techniques in the ASF+SDF metaenvironment. pages 5--26. Abstract: The ASF+SDF Meta-environment is an interactive development environment for formal language definitions. It is both a meta-environment supporting fully interactive editing of modular language definitions written in the formalism ASF+SDF and a generator for dedicated environments for defined languages. The actual development of this system started in 1985 as part of the GIPE (Generation of Interactive Programming Environments) projects [HKKL86]. Now, ten years later, it is worthwhile to assess what has been achieved and, more importantly, which problems are still to be addressed. A historical and at times methodological perspective is necessary in such an assessment. However, rather than evaluating all aspects of the sys...

Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.5.4 Special Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.6 Semantics of Programming Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.6.1 Semantics of Ada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.6.2 Action Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.7 Specification Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.7.1 Early Algebraic Specification Languages . . . . . . . . . . . . . . . . . . . . . . . . 53 4.7.2 Recent Algebraic Specification Languages . . . . . . . . . . . . . . . . . . . . . . . 55 4.7.3 The Common Framework Initiative. . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5 Methodology 57 5.1 Development Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.1.1 Applica...