Isabelle: The next seven hundred theorem provers

Paulson, Lawrence C.

doi:10.1007/bfb0012891

Cited by 85 publications

(99 citation statements)

References 12 publications

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“…Isabelle/Isar is based on the existing logical framework of Isabelle/Pure [9], which provides a generic platform for higher-order natural deduction. The generic approach of Pure inference systems is extended by Isar towards actual proof texts.…”

Section: Generic Natural Deductionmentioning

confidence: 99%

“…The Pure logic [9] is an intuitionistic fragment of higher-order logic. In type-theoretic parlance, there are three levels of λ-calculus with corresponding arrows: ⇒ for syntactic function space (terms depending on terms), for universal quantification (proofs depending on terms), and =⇒ for implication (proofs depending on proofs).…”

Section: The Pure Frameworkmentioning

confidence: 99%

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Structured Induction Proofs in Isabelle/Isar

Wenzel

2006

Lecture Notes in Computer Science

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Abstract. Isabelle/Isar is a generic framework for human-readable formal proof documents, based on higher-order natural deduction. The Isar proof language provides general principles that may be instantiated to particular object-logics and applications. We discuss specific Isar language elements that support complex induction patterns of practical importance. Despite the additional bookkeeping required for induction with local facts and parameters, definitions, simultaneous goals and multiple rules, the resulting Isar proof texts turn out well-structured and readable. Our techniques can be applied to non-standard variants of induction as well, such as co-induction and nominal induction. This demonstrates that Isar provides a viable platform for building domain-specific tools that support fully-formal mathematical proof composition. Motivation The Isar philosophyIsabelle/Isar [15,16,7,17] is intended as a generic framework for developing formal mathematical documents with full proof checking. The Isabelle/Isar system is well integrated with existing theorem prover interface technology [1] and document preparation based on PDF-L A T E X. 1 The main objective is the design of a human-readable structured proof language, which is called the "primary proof format" in Isar terminology. Such a primary proof language is somewhere in the middle between the extremes of primitive proof objects and actual natural language. In this respect, Isar is a bit more formalistic than Mizar [12,10], using explicit logical connectives for certain reasoning schemes where Mizar would prefer English words; see [19,18] for further comparisons of these systems. We argue that any effort of building a library of formalized mathematics heavily depends on a reasonable notion of structured proofs -the Mizar Mathematical Library [6] provides some empirical evidence for this. So Isar challenges the traditional way of recording informal proofs in mathematical prose, as well as the common tendency to see fully formal proofs directly as objects of some logical calculus (e.g. λ-terms in a version of type theory). In fact, Isar is better understood as an interpreter of a simple block-structured language for describing data flow of local facts and goals, interspersed with occasional invocations of proof methods.1 In fact, the present paper has been prepared as an Isabelle/Isar theory document, which means that the proofs and proof outlines encountered here have been checked by the system.

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Section: Generic Natural Deductionmentioning

confidence: 99%

Section: The Pure Frameworkmentioning

confidence: 99%

Structured Induction Proofs in Isabelle/Isar

Wenzel

2006

Lecture Notes in Computer Science

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“…We make use of the Isabelle logical framework [13] to specify the foundations of Mizar [14]. We further define a number of mechanisms that help to translate the Mizar definitions and proofs [15].…”

Section: Introductionmentioning

confidence: 99%

Progress in the Independent Certiﬁcation of Mizar Mathematical Library in Isabelle

Kaliszyk

Pąk

2017

Proceedings of the 2017 Federated Conference on Computer Science and Information Systems

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Abstract-The Mizar Mathematical Library is one of the largest collections of machine understandable formal proofs encompassing many areas of today mathematics including results from algebra, analysis, topology, and lattice theory. The Mizar system has so far been the only tool able to completely process, certify, and make use of these developments. In this paper, we present the progress in the development of an independent certification mechanism of Mizar proofs based on the Isabelle logical framework. The approach allows rechecking the Mizar formal proofs based on a more succinct and more precisely specified formal infrastructure. Additionally, it necessitates a full formal specification of the mechanisms that ensure the correctness of the defined objects, in particular, the proofs that such mechanisms are correct. The development already covers an important part of the Mizar library foundations. We improve the mechanism for defining Mizar structures and show that it permits simpler validation of proof developments involving such objects. To demonstrate this, we perform a complete translation of the Mizar net of basic algebraic structures including their attributes and certify all the corresponding proofs in Isabelle.

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“…This approach has been pionneered in this context by Nipkow [15] and is available in Isabelle [18]. Its main feature is the use of higher-order pattern matching for firing rules.…”

Section: Introductionmentioning

confidence: 99%