A Heuristic Prover for Real Inequalities

Avigad, Jeremy; Lewis, Robert Y.; Roux, Cody

doi:10.1007/978-3-319-08970-6_5

Cited by 3 publications

(5 citation statements)

References 26 publications

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“…In a prior version of this paper [5], our test suite included only 51 problems that could be solved without invoking any of the modules described in Section 6 other than the congruence closure module. Polya solved these in about 2 seconds on an ordinary desktop using the polytope packages, and in about 5.5 seconds using Fourier-Motzkin.…”

Section: Successesmentioning

confidence: 99%

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A Heuristic Prover for Real Inequalities

2016

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We describe a general method for verifying inequalities between real-valued expressions, especially the kinds of straightforward inferences that arise in interactive theorem proving. In contrast to approaches that aim to be complete with respect to a particular language or class of formulas, our method establishes claims that require heterogeneous forms of reasoning, relying on a Nelson-Oppen-style architecture in which special-purpose modules collaborate and share information. The framework is thus modular and extensible. A prototype implementation shows that the method works well on a variety of examples, and complements techniques that are used by contemporary interactive provers.

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Section: Successesmentioning

confidence: 99%

“…This paper is a revised and expanded version of the conference paper [5]. The extensions described in this paper, chiefly the additional modules described in Section 6, are due to Avigad and Lewis.…”

Section: Introductionmentioning

confidence: 99%

A Heuristic Prover for Real Inequalities

2016

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“…It takes roughly 70 lines of Lean code to perform these computations, compared to roughly 10 in the informal presentation. Many of these computations fall under the scope of the tool Polya [2] developed by the author. In the future, such a tool could be used to significantly condense this portion of our proof.…”

Section: Defining the Newton Sequencementioning

confidence: 99%

“…The formalization described in this paper is incorporated into the Lean mathematical library, available on GitHub. 2 Since this library is regularly changing, we preserve a snapshot of its status at the time this paper was submitted. This snapshot, and a map between this paper and the formalization, can be found on the author's website.…”

Section: Introductionmentioning

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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“…A similar "blackboard" approach is used for heuristic theorem proving by Avigad et al [1], where the focus is on proving real inequalities. The portion of our system concerning inequalities is not as sophisticated as what is implemented there.…”

Section: Related Workmentioning

confidence: 99%

AUTO2, A Saturation-Based Heuristic Prover for Higher-Order Logic

Zhan

2016

Lecture Notes in Computer Science

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Abstract. We introduce a new theorem prover for classical higherorder logic named auto2. The prover is designed to make use of humanspecified heuristics when searching for proofs. The core algorithm is a best-first search through the space of propositions derivable from the initial assumptions, where new propositions are added by user-defined functions called proof steps. We implemented the prover in Isabelle/HOL, and applied it to several formalization projects in mathematics and computer science, demonstrating the high level of automation it can provide in a variety of possible proof tasks.

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A Heuristic Prover for Real Inequalities

Cited by 3 publications

References 26 publications

A Heuristic Prover for Real Inequalities

A Heuristic Prover for Real Inequalities

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

AUTO2, A Saturation-Based Heuristic Prover for Higher-Order Logic

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