Matching concepts across HOL libraries

Many proof assistant libraries contain formalizations of the same mathematical concepts. The concepts are often introduced (defined) in different ways, but the properties that they have, and are in turn formalized, are the same. For the basic concepts, like natural numbers, matching them between libraries is often straightforward, because of mathematical naming conventions. However, for more advanced concepts, finding similar formalizations in different libraries is a non-trivial task even for an expert. In this paper we investigate automatic discovery of similar concepts across libraries of proof assistants. We propose an approach for normalizing properties of concepts in formal libraries and a number of similarity measures. We evaluate the approach on HOL based proof assistants HOL4, HOL Light and Isabelle/HOL, discovering 398 pairs of isomorphic constants and types

Translating HOL to Dedukti

Dedukti is a logical framework based on the lambda-Pi-calculus modulo rewriting, which extends the lambda-Pi-calculus with rewrite rules. In this paper, we show how to translate the proofs of a family of HOL proof assistants to Dedukti. The translation preserves binding, typing, and reduction. We implemented this translation in an automated tool and used it to successfully translate the OpenTheory standard library.Comment: In Proceedings PxTP 2015, arXiv:1507.0837

The Common HOL Platform

The Common HOL project aims to facilitate porting source code and proofs between members of the HOL family of theorem provers. At the heart of the project is the Common HOL Platform, which defines a standard HOL theory and API that aims to be compatible with all HOL systems. So far, HOL Light and hol90 have been adapted for conformance, and HOL Zero was originally developed to conform. In this paper we provide motivation for a platform, give an overview of the Common HOL Platform's theory and API components, and show how to adapt legacy systems. We also report on the platform's successful application in the hand-translation of a few thousand lines of source code from HOL Light to HOL Zero.Comment: In Proceedings PxTP 2015, arXiv:1507.0837

Sharing HOL4 and HOL Light proof knowledge

New proof assistant developments often involve concepts similar to already formalized ones. When proving their properties, a human can often take inspiration from the existing formalized proofs available in other provers or libraries. In this paper we propose and evaluate a number of methods, which strengthen proof automation by learning from proof libraries of different provers. Certain conjectures can be proved directly from the dependencies induced by similar proofs in the other library. Even if exact correspondences are not found, learning-reasoning systems can make use of the association between proved theorems and their characteristics to predict the relevant premises. Such external help can be further combined with internal advice. We evaluate the proposed knowledge-sharing methods by reproving the HOL Light and HOL4 standard libraries. The learning-reasoning system HOL(y)Hammer, whose single best strategy could automatically find proofs for 30% of the HOL Light problems, can prove 40% with the knowledge from HOL4

Towards MKM in the Large: Modular Representation and Scalable Software Architecture

MKM has been defined as the quest for technologies to manage mathematical knowledge. MKM "in the small" is well-studied, so the real problem is to scale up to large, highly interconnected corpora: "MKM in the large". We contend that advances in two areas are needed to reach this goal. We need representation languages that support incremental processing of all primitive MKM operations, and we need software architectures and implementations that implement these operations scalably on large knowledge bases. We present instances of both in this paper: the MMT framework for modular theory-graphs that integrates meta-logical foundations, which forms the base of the next OMDoc version; and TNTBase, a versioned storage system for XML-based document formats. TNTBase becomes an MMT database by instantiating it with special MKM operations for MMT.Comment: To appear in The 9th International Conference on Mathematical Knowledge Management: MKM 201

A Foundational View on Integration Problems

The integration of reasoning and computation services across system and language boundaries is a challenging problem of computer science. In this paper, we use integration for the scenario where we have two systems that we integrate by moving problems and solutions between them. While this scenario is often approached from an engineering perspective, we take a foundational view. Based on the generic declarative language MMT, we develop a theoretical framework for system integration using theories and partial theory morphisms. Because MMT permits representations of the meta-logical foundations themselves, this includes integration across logics. We discuss safe and unsafe integration schemes and devise a general form of safe integration

A Vernacular for Coherent Logic

We propose a simple, yet expressive proof representation from which proofs for different proof assistants can easily be generated. The representation uses only a few inference rules and is based on a frag- ment of first-order logic called coherent logic. Coherent logic has been recognized by a number of researchers as a suitable logic for many ev- eryday mathematical developments. The proposed proof representation is accompanied by a corresponding XML format and by a suite of XSL transformations for generating formal proofs for Isabelle/Isar and Coq, as well as proofs expressed in a natural language form (formatted in LATEX or in HTML). Also, our automated theorem prover for coherent logic exports proofs in the proposed XML format. All tools are publicly available, along with a set of sample theorems.Comment: CICM 2014 - Conferences on Intelligent Computer Mathematics (2014

Initial Experiments with TPTP-style Automated Theorem Provers on ACL2 Problems

This paper reports our initial experiments with using external ATP on some corpora built with the ACL2 system. This is intended to provide the first estimate about the usefulness of such external reasoning and AI systems for solving ACL2 problems.Comment: In Proceedings ACL2 2014, arXiv:1406.123