On the Integrity of a Repository of Formalized Mathematics

Rudnicki, Piotr; Trybulec, Andrzej

doi:10.1007/3-540-36469-2_13

Cited by 26 publications

(22 citation statements)

References 2 publications

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“…For example, in the Mizar library 4 (and even more in other collections like the Coq contribs) 5 , articles may be contributed as isolated developments, and only much later (or never) re-factored into a form that makes them work well with related developments. Such refactoring is often a nontrivial process [67]. In a directed development like Flyspeck, such integrity is a concern from the very beginning, and this concern should result in the theorems working better together to justify new conjectures that combine the areas covered by the development.…”

Section: Flyspeck As An Interesting Corpus For Semantic Ai Methodsmentioning

confidence: 99%

Learning-Assisted Automated Reasoning with Flyspeck

2014

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The considerable mathematical knowledge encoded by the Flyspeck project is combined with external automated theorem provers (ATPs) and machine-learning premise selection methods trained on the Flyspeck proofs, producing an AI system capable of proving a wide range of mathematical conjectures automatically. The performance of this architecture is evaluated in a bootstrapping scenario emulating the development of Flyspeck from axioms to the last theorem, each time using only the previous theorems and proofs. It is shown that 39 % of the 14185 theorems could be proved in a push-button mode (without any high-level advice and user interaction) in 30 seconds of real time on a fourteen-CPU workstation. The necessary work involves: (i) an implementation of sound translations of the HOL Light logic to ATP formalisms: untyped first-order, polymorphic typed first-order, and typed higher-order, (ii) export of the dependency information from HOL Light and ATP proofs for the machine learners, and (iii) choice of suitable representations and methods for learning from previous proofs, and their integration as advisors with HOL Light. This work is described and discussed here, and an initial analysis of the body of proofs that were found fully automatically is provided.

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Section: Flyspeck As An Interesting Corpus For Semantic Ai Methodsmentioning

confidence: 99%

Learning-Assisted Automated Reasoning with Flyspeck

2014

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“…What we were trying to do was to increase of the integrity of (the geometrical part of) the MML as pointed out in the paper of Piotr Rudnicki and Andrzej Trybulec [39] and this could be a kind of partial realization of their postulates. This was done mainly via the mechanism of revisions of the repository -stepwise refinement of items already included in MML, done not necessarily by authors themselves [19].…”

Section: Introductionmentioning

confidence: 99%

Tarski's geometry modelled in Mizar computerized proof assistant

Grabowski¹

2016

Annals of Computer Science and Information Systems

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Abstract-In the paper, we discuss the formal approach to Tarski geometry axioms modelled with the help of the Mizar computerized proof assistant system. Although our basic development was inspired by Julien Narboux's Coq pseudo-code and is dated back to 2014, there are significant steps in the formalization of geometry done in the last decade of the previous century. Taking this into account, we will propose the reuse of existing results within this new framework (including Hilbert's axiomatic approach), with the ultimate future goal to encode the textbook Metamathematische Methoden in der Geometrie by Schwabhäuser, Szmielew and Tarski. We try however to go much further from the use of simple predicates in the direction of the use of structures with their inheritance, attributes as a tool of more human-friendly namespaces for axioms, and registrations of clusters to obtain more automation (with the possible use of external equational theorem provers like Otter/Prover9).

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“…The articles [14], [38], [28], [32], [39], [11], [40], [13], [33], [12], [5], [4], [2], [6], [10], [37], [36], [25], [3], [15], [19], [35], [24], [30], [18], [34], [16], [9], [22], [21], [41], [17], [20], [7], [31], [29], [8], [23], and [27] provide the notation and terminology for this paper.…”

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confidence: 99%

The Perfect Number Theorem and Wilson's Theorem

Riccardi¹

2009

Formalized Mathematics

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Summary. This article formalizes proofs of some elementary theorems of number theory (see [1,26]): Wilson's theorem (that n is prime iff n > 1 and (n − 1)! ∼ = −1 (mod n)), that all primes (1 mod 4) equal the sum of two squares, and two basic theorems of Euclid and Euler about perfect numbers. The article also formally defines Euler's sum of divisors function φ, proves that φ is multiplicative and that PreliminariesWe adopt the following convention: k, n, m, l, p denote natural numbers and n 0 , m 0 denote non zero natural numbers.We now state several propositions: (1) 2 n+1 < 2 n+2 − 1.

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On the Integrity of a Repository of Formalized Mathematics

Cited by 26 publications

References 2 publications

Learning-Assisted Automated Reasoning with Flyspeck

Learning-Assisted Automated Reasoning with Flyspeck

Tarski's geometry modelled in Mizar computerized proof assistant

The Perfect Number Theorem and Wilson's Theorem

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