4,980 research outputs found

    Representing Isabelle in LF

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    LF has been designed and successfully used as a meta-logical framework to represent and reason about object logics. Here we design a representation of the Isabelle logical framework in LF using the recently introduced module system for LF. The major novelty of our approach is that we can naturally represent the advanced Isabelle features of type classes and locales. Our representation of type classes relies on a feature so far lacking in the LF module system: morphism variables and abstraction over them. While conservative over the present system in terms of expressivity, this feature is needed for a representation of type classes that preserves the modular structure. Therefore, we also design the necessary extension of the LF module system.Comment: In Proceedings LFMTP 2010, arXiv:1009.218

    Encoding a Dependent-Type Lambda-Calculus in a Logic Programming Language

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    Various forms of typed λ-calculi have been proposed as specification languages for representing wide varieties of object logics. The logical framework, LF, is an example of such a dependent-type λ-calculus. A small subset of intuitionistic logic with quantification over simply typed λ-calculus has also been proposed as a framework for specifying general logics. The logic of hereditary Harrop formulas with quantification at all non-predicate types, denoted here as hhω, is such a meta-logic that has been implemented in both the Isabelle theorem prover and the λProlog logic programming language. Both frameworks provide for specifications of logics in which details involved with free and bound variable occurrences, substitutions, eigenvariables, and the scope of assumptions within object logics are handled correctly and elegantly at the meta level. In this paper, we show how LF can be encoded into hhω in a direct and natural way by mapping the typing judgments in LF into propositions in the logic of hhω. This translation establishes a very strong connection between these two languages: the order of quantification in an LF signature is exactly the order of a set of hhω clauses, and the proofs in one system correspond directly to proofs in the other system. Relating these two languages makes it possible to provide implementations of proof checkers and theorem provers for logics specified in LF by using standard logic programming techniques which can be used to implement hhω

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

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    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

    Translating HOL to Dedukti

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    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

    Encoding a dependent-type -calculus in a logic programming language

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    Various forms of typed l-calculi have been proposed as specification languages for representing wide varieties of object logics. The logical framework, LF is an example of such a dependent-type l-calculus. A small subset of intuitionistic logic with quantification over simply typed l-calculus has also been proposed as a framework for specifying general logics. The logic of hereditary Harrop formulas with quantification at all non-predicate types, denoted here as hhw is such a meta-logic that has been implemented in both the Isabelle theorem prover and the lProlog logic programming language. In this paper, we show how LF can be encoded into hhw in a direct and natural way by mapping the typing judgments in LF into propositions in the logic of hhw. This translation establishes a strong connection between these two languages : the order of quantification in an LF signature is exactly the order of a set of hhw clauses and the proofs in one system correspond directly to proofs in the other system

    A Case Study on Logical Relations using Contextual Types

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    Proofs by logical relations play a key role to establish rich properties such as normalization or contextual equivalence. They are also challenging to mechanize. In this paper, we describe the completeness proof of algorithmic equality for simply typed lambda-terms by Crary where we reason about logically equivalent terms in the proof environment Beluga. There are three key aspects we rely upon: 1) we encode lambda-terms together with their operational semantics and algorithmic equality using higher-order abstract syntax 2) we directly encode the corresponding logical equivalence of well-typed lambda-terms using recursive types and higher-order functions 3) we exploit Beluga's support for contexts and the equational theory of simultaneous substitutions. This leads to a direct and compact mechanization, demonstrating Beluga's strength at formalizing logical relations proofs.Comment: In Proceedings LFMTP 2015, arXiv:1507.0759
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