An Extensible Ad Hoc Interface between Lean and Mathematica

Lewis, Robert Y.

doi:10.4204/eptcs.262.4

Cited by 8 publications

(8 citation statements)

References 18 publications

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“…More recently, this technique is used by Chyzak et al to formalize the proof of irrationality of ζ(3) [14], and by Harrison to verify proofs of hypergeometric sums found using the WZ method [22]. Similar approaches are implemented in Isabelle [8], PVS [3] and Lean [28]. Compared to this work, we present more complex proof automation for reconstructing proofs, as well as a user interface for allowing users to perform multi-step computations in a more familiar setting.…”

Section: Related Workmentioning

confidence: 99%

“…A common approach, proposed by Harrison and Théry [20,23], is to invoke a computer algebra system for computations that are difficult to perform, but whose results can be verified more easily. This greatly extends the capability of proof assistants for tasks such as factorization [23], linear arithmetic [28], etc. However, to use such a system, the user still needs expertise in the use of proof assistants, and the range of applicability is limited by the simple proof automation that is available for checking results.…”

Section: Introductionmentioning

confidence: 97%

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Verified Interactive Computation of Definite Integrals

Zhan

2021

Automated Deduction – CADE 28

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Symbolic computation is involved in many areas of mathematics, as well as in analysis of physical systems in science and engineering. Computer algebra systems present an easy-to-use interface for performing these calculations, but do not provide strong guarantees of correctness. In contrast, interactive theorem proving provides much stronger guarantees of correctness, but requires more time and expertise. In this paper, we propose a general framework for combining these two methods, and demonstrate it using computation of definite integrals. It allows the user to carry out step-by-step computations in a familiar user interface, while also verifying the computation by translating it to proofs in higher-order logic. The system consists of an intermediate language for recording computations, proof automation for simplification and inequality checking, and heuristic integration methods. A prototype is implemented in Python based on HolPy, and tested on a large collection of examples at the undergraduate level.

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Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Verified Interactive Computation of Definite Integrals

Zhan

2021

Automated Deduction – CADE 28

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“…The only manual input is the value to instantiate k. This value was computed using computer algebra software, and using a link to such software (e.g. [29]), even this step could potentially be automated.…”

Section: Polynomial Identitiesmentioning

confidence: 99%

A formal proof of Hensel's lemma over the p-adic integers

Lewis

2019

Proceedings of the 8th ACM SIGPLAN International Conference on Certified Programs and Proofs

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The field of p-adic numbers Q p and the ring of p-adic integers Z p are essential constructions of modern number theory. Hensel's lemma, described by Gouvêa as the "most important algebraic property of the p-adic numbers, " shows the existence of roots of polynomials over Z p provided an initial seed point. The theorem can be proved for the p-adics with significantly weaker hypotheses than for general rings. We construct Q p and Z p in the Lean proof assistant, with various associated algebraic properties, and formally prove a strong form of Hensel's lemma. The proof lies at the intersection of algebraic and analytic reasoning and demonstrates how the Lean mathematical library handles such a heterogeneous topic.

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“…Theorem provers built on top of computer algebra systems: These include Analytica, Theorema, RedLog, and logical extensions to the Axiom system [35,33,36,37,38] . [34,44,45,46,47,48,49,50]. The example given above, bridging Isabelle and Maple, is an example of an approach from this category.…”

Section: Iv1 a Taxonomy Of Approachesmentioning

confidence: 99%

“…Such systems may also prove to be useful tools for AI safety researchers in proving the functional correctness of other components of an AI architecture. Natural choices of systems to use would be interfaces for the Wolfram Language, the most widely used computer algebra system, with one of the HOL family of theorem provers or Coq, both of which have substantial repositories of formalized proofs [71,72,73,74], or a more modern ITP such as Lean [75,50].…”

Section: Cmentioning

confidence: 99%

Robust Computer Algebra, Theorem Proving, and Oracle AI

Sarma

Hay

2017

SSRN Journal

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In the context of superintelligent AI systems, the term "oracle" has two meanings. One refers to modular systems queried for domain-specific tasks. Another usage, referring to a class of systems which may be useful for addressing the value alignment and AI control problems, is a superintelligent AI system that only answers questions. The aim of this manuscript is to survey contemporary research problems related to oracles which align with long-term research goals of AI safety. We examine existing question answering systems and argue that their high degree of architectural heterogeneity makes them poor candidates for rigorous analysis as oracles. On the other hand, we identify computer algebra systems (CASs) as being primitive examples of domain-specific oracles for mathematics and argue that efforts to integrate computer algebra systems with theorem provers, systems which have largely been developed independent of one another, provide a concrete set of problems related to the notion of provable safety that has emerged in the AI safety community. We review approaches to interfacing CASs with theorem provers, describe well-defined architectural deficiencies that have been identified with CASs, and suggest possible lines of research and practical software projects for scientists interested in AI safety.

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An Extensible Ad Hoc Interface between Lean and Mathematica

Cited by 8 publications

References 18 publications

Verified Interactive Computation of Definite Integrals

Verified Interactive Computation of Definite Integrals

A formal proof of Hensel's lemma over the p-adic integers

Robust Computer Algebra, Theorem Proving, and Oracle AI

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