Extending Nunchaku to Dependent Type Theory

Nunchaku is a new higher-order counterexample generator based on a sequence of transformations from polymorphic higher-order logic to first-order logic. Unlike its predecessor Nitpick for Isabelle, it is designed as a stand-alone tool, with frontends for various proof assistants. In this short paper, we present some ideas to extend Nunchaku with partial support for dependent types and type classes, to make frontends for Coq and other systems based on dependent type theory more useful.Comment: In Proceedings HaTT 2016, arXiv:1606.0542

HOL(y)Hammer: Online ATP Service for HOL Light

HOL(y)Hammer is an online AI/ATP service for formal (computer-understandable) mathematics encoded in the HOL Light system. The service allows its users to upload and automatically process an arbitrary formal development (project) based on HOL Light, and to attack arbitrary conjectures that use the concepts defined in some of the uploaded projects. For that, the service uses several automated reasoning systems combined with several premise selection methods trained on all the project proofs. The projects that are readily available on the server for such query answering include the recent versions of the Flyspeck, Multivariate Analysis and Complex Analysis libraries. The service runs on a 48-CPU server, currently employing in parallel for each task 7 AI/ATP combinations and 4 decision procedures that contribute to its overall performance. The system is also available for local installation by interested users, who can customize it for their own proof development. An Emacs interface allowing parallel asynchronous queries to the service is also provided. The overall structure of the service is outlined, problems that arise and their solutions are discussed, and an initial account of using the system is given

Comprehending Isabelle/HOL's consistency

The proof assistant Isabelle/HOL is based on an extension of Higher-Order Logic (HOL) with ad hoc overloading of constants. It turns out that the interaction between the standard HOL type definitions and the Isabelle-specific ad hoc overloading is problematic for the logical consistency. In previous work, we have argued that standard HOL semantics is no longer appropriate for capturing this interaction, and have proved consistency using a nonstandard semantics. The use of an exotic semantics makes that proof hard to digest by the community. In this paper, we prove consistency by proof-theoretic means—following the healthy intuition of definitions as abbreviations, realized in HOLC, a logic that augments HOL with comprehension types. We hope that our new proof settles the Isabelle/HOL consistency problem once and for all. In addition, HOLC offers a framework for justifying the consistency of new deduction schemas that address practical user needs

Comprehending Isabelle/HOL's consistency

The proof assistant Isabelle/HOL is based on an extension of Higher-Order Logic (HOL) with ad hoc overloading of constants. It turns out that the interaction between the standard HOL type definitions and the Isabelle-specific ad hoc overloading is problematic for the logical consistency. In previous work, we have argued that standard HOL semantics is no longer appropriate for capturing this interaction, and have proved consistency using a nonstandard semantics. The use of an exotic semantics makes that proof hard to digest by the community. In this paper, we prove consistency by proof-theoretic means—following the healthy intuition of definitions as abbreviations, realized in HOLC, a logic that augments HOL with comprehension types. We hope that our new proof settles the Isabelle/HOL consistency problem once and for all. In addition, HOLC offers a framework for justifying the consistency of new deduction schemas that address practical user needs

Safety and conservativity of definitions in HOL and Isabelle/HOL

Definitions are traditionally considered to be a safe mechanism for introducing concepts on top of a logic known to be consistent. In contrast to arbitrary axioms, definitions should in principle be treatable as a form of abbreviation, and thus compiled away from the theory without losing provability. In particular, definitions should form a conservative extension of the pure logic. These properties are crucial for modern interactive theorem provers, since they ensure the consistency of the logic, as well as a valid environment for total/certified functional programming. We prove these properties, namely, safety and conservativity, for Higher-Order Logic (HOL), a logic implemented in several mainstream theorem provers and relied upon by thousands of users. Some unique features of HOL, such as the requirement to give non-emptiness proofs when defining new types and the impossibility to unfold type definitions, make the proof of these properties, and also the very formulation of safety, nontrivial. Our study also factors in the essential variation of HOL definitions featured by Isabelle/HOL, a popular member of the HOL-based provers family. The current work improves on recent results which showed a weaker property, consistency of Isabelle/HOL’s definitions

Programmiersprachen und Rechenkonzepte

Die GI-Fachgruppe 2.1.4 "Programmiersprachen und Rechenkonzepte" veranstaltete vom 3. bis 5. Mai 2004 im Physikzentrum Bad Honnef ihren jährlichen Workshop. Dieser Bericht enthält eine Zusammenstellung der Beiträge. Das Treffen diente wie in jedem Jahr gegenseitigem Kennenlernen, der Vertiefung gegenseitiger Kontakte, der Vorstellung neuer Arbeiten und Ergebnisse und vor allem der intensiven Diskussion. Ein breites Spektrum von Beiträgen, von theoretischen Grundlagen über Programmentwicklung, Sprachdesign, Softwaretechnik und Objektorientierung bis hin zur überraschend langen Geschichte der Rechenautomaten seit der Antike bildete ein interessantes und abwechlungsreiches Programm. Unter anderem waren imperative, funktionale und funktional-logische Sprachen, Software/Hardware-Codesign, Semantik, Web-Programmierung und Softwaretechnik, generative Programmierung, Aspekte und formale Testunterstützung Thema. Interessante Beiträge zu diesen und weiteren Themen gaben Anlaß zu Erfahrungsaustausch und Fachgesprächen auch mit den Teilnehmern des zeitgleich im Physikzentrum Bad Honnef stattfindenden Workshops "Reengineering". Allen Teilnehmern möchte ich dafür danken, daß sie mit ihren Vorträgen und konstruktiven Diskussionsbeiträgen zum Gelingen des Workshops beigetragen haben. Dank für die Vielfalt und Qualität der Beiträge gebührt den Autoren. Ein Wort des Dankes gebührt ebenso den Mitarbeitern und der Leitung des Physikzentrums Bad Honnef für die gewohnte angenehme und anregende Atmosphäre und umfassende Betreuung

Extending Coq with Imperative Features and its Application to SAT Verification

This work was supported in part by the french ANR DECERT initiativeInternational audienceCoq has within its logic a programming language that can be used to replace many deduction steps into a single computation, this is the so-called reflection. In this paper, we present two extensions of the evaluation mechanism that preserve its correctness and make it possible to deal with cpu-intensive tasks such as proof checking of SAT traces