Surreal Numbers in Coq

Mamane, Lionel

doi:10.1007/11617990_11

Cited by 6 publications

(5 citation statements)

References 10 publications

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“…Mamane [19] develops the theory of surreal numbers in the proof assistant Coq, using an encoding to reduce the inductive-inductive definition to an ordinary inductive one. …”

Section: Examples Of Inductive-inductive Definitionsmentioning

confidence: 99%

A Finite Axiomatisation of Inductive-Inductive Definitions

Forsberg

Setzer²

2012

Logic, Construction, Computation

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Induction-induction is a principle for mutually defining data types A : Set and B : A → Set. Both A and B are defined inductively, and the constructors for A can refer to B and vice versa. In addition, the constructor for B can refer to the constructor for A. Induction-induction occurs in a natural way when formalising dependent type theory in type theory. We give some examples of inductive-inductive definitions, such as the set of surreal numbers. We then give a new finite axiomatisation of the principle of induction-induction, and prove its consistency by constructing a model.

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“…Mamane [19] develops the theory of surreal numbers in the proof assistant Coq, using an encoding to reduce the inductive-inductive definition to an ordinary inductive one. …”

Section: Examples Of Inductive-inductive Definitionsmentioning

confidence: 99%

A Finite Axiomatisation of Inductive-Inductive Definitions

Forsberg

Setzer²

2012

Logic, Construction, Computation

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“…The licensing situation in the world of free works 11 within the two categories (executable software code on the one hand, and documents for human consumption on the other hand) has clear "winners". On the code side, statistics [16,9,24] based on web scraping of contents of large free/open source software repositories (such as SourceForge or freshmeat) show that slightly more than a half of FLOSS 12 code is under a variant of the GNU General Public License: Data from the FLOSSmole project [4] as of March 2011 shows that out of a total of 43 470 FLOSS projects tracked by freshmeat, 24 366 are licensed under a version of the GNU GPL 13 . Of the projects hosted on SourceForge, 110 412 use a variant of the GPL, out of 174 227 that use a license approved by the Open Source Initiative.…”

Section: Code and Text Licensesmentioning

confidence: 99%

“…For example, the third author chose Coq over Mizar for formalisation of surreal (Conway) numbers[13] over such an issue, although Mizar's set theory base could have provided a more natural framework for surreal numbers than Coq's type theory.…”

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confidence: 99%

Licensing the Mizar Mathematical Library

Alama

Kohlhase

Mamane

et al. 2011

Lecture Notes in Computer Science

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The Mizar Mathematical Library (MML) is a large corpus of formalised mathematical knowledge. It has been constructed over the course of many years by a large number of authors and maintainers. Yet the legal status of these efforts of the Mizar community has never been clarified. In 2010, after many years of loose deliberations, the community decided to investigate the issue of licensing the content of the MML, thereby clarifying and crystallizing the status of the texts, the text's authors, and the library's long-term maintainers. The community has settled on a copyright and license policy that suits the peculiar features of Mizar and its community. In this paper we discuss the copyright and license solutions. We offer our experience in the hopes that the communities of other libraries of formalised mathematical knowledge might take up the legal and scientific problems that we addressed for Mizar.

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“…This transfinite nature can be seen in the way the arithmetic operation on the surreal numbers are evaluated. Therefore a straightforward formalisation of algorithms for computation on the surreal numbers needs a more powerful framework than most present programming languages and should best be done in a framework where higher order types are present, as it is done in Coq [18]. However, the algorithms that we present for the Stern-Brocot arithmetic can be implemented in ordinary programming languages.…”

Section: Remarkmentioning

confidence: 99%

Exact arithmetic on the Stern–Brocot tree

Niqui

2007

Journal of Discrete Algorithms

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In this paper we present the Stern-Brocot tree as a basis for performing exact arithmetic on rational numbers. There exists an elegant binary representation for positive rational numbers based on this tree [Graham et al., Concrete Mathematics, 1994]. We will study this representation by investigating various algorithms to perform exact rational arithmetic using an adaptation of the homographic and the quadratic algorithms that were first proposed by Gosper for computing with continued fractions. We will show generalisations of homographic and quadratic algorithms to multilinear forms in n variables. Finally, we show an application of the algorithms for evaluating polynomials. Historical backgroundRecently exact arithmetic has been considered as a suitable approach to the problem of dealing with round-off errors and building more reliable and versatile programming tools for computation with rational and real numbers. According to this approach, real numbers are represented as an infinite stream over a finite or infinite alphabet and the computation over them is done in a lazy manner (also called: on-line, call-by-need, corecursive, etc.): in order to compute a function on a real number, we start absorbing the first element of the stream representing the real number. At each step we output an element of the output stream or we may need more information about the input, in which case we absorb the next element of the input stream.There have been many theoretical and practical instances of applying this idea. A rather general theoretical approach is taken by Konečný [12], where the limitation theorems for IFS-representations of real numbers is given. IFS-representations are representations of real numbers using an infinite composition of contracting functions on a compact interval. These include the representations by means of Möbius maps that is developed by Edalat and Potts [6,23]. Edalat and Potts's work generalises earlier works by Gosper [7,8], Vuillemin [27] and Menissier-Morain [19]. Gosper, in his famous unpublished work [7], showed how to add, multiply, subtract and divide two continued fractions. He introduced the idea of using homographic and quadratic algorithms. His algorithms are presented for regular N-fractions of rational numbers and were the first instance of lazy exact arithmetic on rational numbers. *

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Surreal Numbers in Coq

Cited by 6 publications

References 10 publications

A Finite Axiomatisation of Inductive-Inductive Definitions

A Finite Axiomatisation of Inductive-Inductive Definitions

Licensing the Mizar Mathematical Library

Exact arithmetic on the Stern–Brocot tree

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