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Isabelle: the next 700 theorem provers
The theorem prover Isabelle is described briefly and informally. Its historical development is traced from Edinburgh LCF to the present day. The main issues are unification, quantifiers, and the representation of inference rules. The Edinburgh Logical Framework is also described, for a comparison with Isabelle. An appendix presents several Isabelle logics, including set theory and Constructive Type Theory, with examples of theorems.
In this paper we discuss the similarities between program specialisation and inductive theorem proving, and then show how program specialisation can be used to perform inductive theorem proving. We then study this relationship in more detail for the particular problem of verifying infinite state systems in order to establish a clear link between program specialisation and inductive theorem proving. Indeed, Ecce is a program specialisation system which can be used to automatically generate abstractions for the model checking of infinite state systems. We show that to verify the abstractions generated by Ecce we may employ the proof assistant Isabelle. Thereby Ecce is used to generate the specification, hypotheses and proof script in Isabelle's theory format. Then, in many cases, Isabelle can automatically execute these proof scripts and thereby verify the soundness of Ecce's abstraction. In this work we focus on the specification and verification of Petri nets.
The Foundation of a Generic Theorem Prover
2000
Isabelle is an interactive theorem prover that supports a variety of logics. It represents rules as propositions (not as functions) and builds proofs by combining rules. These operations constitute a meta-logic (or 'logical framework') in which the object-logics are formalized. Isabelle is now based on higher-order logic -a precise and well-understood foundation.
Applications of Proof Theory to Isabelle
1996
Abstract Isabelle [3, 4] is a generic theorem prover. It suppports interactive proof in several formal systems, including first-order logic (intuitionistic and classical), higher-order logic, Martin-Löf type theory, and Zermelo-Fraenkel set theory. New logics can be introduced by specifying their syntax and rules of inference. Both natural deduction and sequent calculi are allowed.
Focus on Isabelle: From Specification to Verification
2008
This approach introduces a coupling of a specification framework with a verification system. Given a system, represented in a formal specification framework, one can verify its properties by translating the specification to a Higher-Order Logic and subsequently using the theorem prover Isabelle/HOL or the point of disagreement will be found. Moreover, using this approach one can validate the refinement relation between two given systems, as well as make automatic correctness proofs of syntactic interfaces for specified system components. The approach uses particularly the idea of refinement-based verification, where a verification of system properties can be treated as a validation of a system specification with respect to the specification of the properties.
The Rely-Guarantee Method in Isabelle/HOL
European Symposium on Programming, 2003
We present the formalization of the rely-guarantee method in the theorem prover Isabelle/HOL. This method consists of a Hoarelike system of rules to verify concurrent imperative programs with shared variables in a compositional way. Syntax, semantics and proof rules are de.ned in higher-order logic. Soundness of the proof rules w.r.t. the semantics is proven mechanically. Also parameterized programs, where the number of parallel components is a parameter, are included in the programming language and thus can be verified directly in the system. We prove that the system is complete for parameterized programs. Finally, we show by an example how the formalization can be used for verifying concrete programs.
A Consistent Foundation for Isabelle/HOL
Lecture Notes in Computer Science, 2015
The interactive theorem prover Isabelle/HOL is based on the well understood Higher-Order Logic (HOL), which is widely believed to be consistent (and provably consistent in set theory by a standard semantic argument). However, Isabelle/HOL brings its own personal touch to HOL: overloaded constant definitions, used to achieve Haskell-like type classes in the user space. These features are a delight for the users, but unfortunately are not easy to get right as an extension of HOL-they have a history of inconsistent behavior. It has been an open question under which criteria overloaded constant definitions and type definitions can be combined together while still guaranteeing consistency. This paper presents a solution to this problem: non-overlapping definitions and termination of the definition-dependency relation (tracked not only through constants but also through types) ensures relative consistency of Isabelle/HOL.
Towards program development, specification and verification with Isabelle
1995
The purpose of this paper is to report on our experiments to use Isabelle | a generic theorem prover | as a universal environment within which speci cation, development and veri cation of imperative programs can be performed. The use of a theorem prover for the programming tasks is most appropriate when the processes of program speci cation, development a n d v eri cation can be presented as logical activities. In our case this is achieved by adopting pLSD | a novel programming logic.
2008
Isabelle, which is available from http://isabelle. in. tum. de, is a generic framework for interactive theorem proving. The Isabelle/Pure meta-logic allows the formalization of the syntax and inference rules of a broad range of object-logics following the general idea of natural deduction [32, 33]. The logical core is implemented according to the well-known “LCF approach” of secure inferences as abstract datatype constructors in ML [16]; explicit proof terms are also available [8].
A Survey on Theorem Provers in Formal Methods
ArXiv, 2019
Mechanical reasoning is a key area of research that lies at the crossroads of mathematical logic and artificial intelligence. The main aim to develop mechanical reasoning systems (also known as theorem provers) was to enable mathematicians to prove theorems by computer programs. However, these tools evolved with time and now play vital role in the modeling and reasoning about complex and large-scale systems, especially safety-critical systems. Technically, mathematical formalisms and automated reasoning based-approaches are employed to perform inferences and to generate proofs in theorem provers. In literature, there is a shortage of comprehensive documents that can provide proper guidance about the preferences of theorem provers with respect to their designs, performances, logical frameworks, strengths, differences and their application areas. In this work, more than 40 theorem provers are studied in detail and compared to present a comprehensive analysis and evaluation of these to...