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Construction of Real Algebraic Numbers in Coq
[chapter]

2012
*
Lecture Notes in Computer Science
*

This paper shows a

doi:10.1007/978-3-642-32347-8_6
fatcat:vdpsbyj4ejhafadmv34tgn7cee
*construction**in**Coq**of*the set*of**real**algebraic**numbers*, together with a formal proof that this set has a structure*of*discrete Archimedean*real*closed field. ... This work also intends to be a basis for the*construction**of*complex*algebraic**numbers*and to be a reference implementation for the certification*of*numerous algorithms relying on*algebraic**numbers**in*... Acknowledgement I wish to thank Georges Gonthier for the numerous ideas which constitute the basis*of*this development and Russell O'Connor for discussions which helped me find the good way to state and ...##
###
A Constructive Algebraic Hierarchy in Coq

2002
*
Journal of symbolic computation
*

We have developed this framework as part

doi:10.1006/jsco.2002.0552
fatcat:l5tzxkehqbbmvk64l5jiyfxkmy
*of*the FTA project*in*Nijmegen,*in*which a*constructive*proof*of*the fundamental theorem*of**algebra*has been formalized*in**Coq*. ... We describe a framework*of**algebraic*structures*in*the proof assistant*Coq*. ...*Algebraic*Structures and Coercive Subtyping We have defined a*number**of*types*in**Coq*representing*algebraic*structures*of*which the carriers are*constructive*setoids. ...##
###
C-CoRN, the Constructive Coq Repository at Nijmegen
[chapter]

2004
*
Lecture Notes in Computer Science
*

We present C-CoRN, the

doi:10.1007/978-3-540-27818-4_7
fatcat:p6t6ka5ftjbphag25dv4mzmsay
*Constructive**Coq*Repository at Nijmegen. It consists*of*a mathematical library*of**constructive**algebra*and analysis formalized*in*the theorem prover*Coq*. ... C-CoRN wants to provide such a library, but it can also be seen as a case study*in*developing such a library*of*formalized mathematics and deriving its requirements. ... We would also like to thank the anonymous referees, whose valuable suggestions much helped improve the quality*of*this paper. This ...##
###
Constructive Reals in Coq: Axioms and Categoricity
[chapter]

2002
*
Lecture Notes in Computer Science
*

We describe a

doi:10.1007/3-540-45842-5_6
fatcat:erq3apmbtbcovigm5xvkp47ina
*construction**of*the*real**numbers*carried out*in*the*Coq*proof assistant. ... The basis is a set*of*axioms for the*constructive**real**numbers*as used*in*the FTA (Fundamental Theorem*of**Algebra*) project, carried out at Nijmegen University. ... Acknowledgements We want to thank Venanzio Capretta, Freek Wiedijk and Jan Zwanenburg for the many fruitful discussions and their useful*Coq*suggestions. ...##
###
A univalent formalization of the p-adic numbers

2015
*
Mathematical Structures in Computer Science
*

This formalization, which has been verified

doi:10.1017/s0960129514000541
fatcat:g7zfmcxsp5gs3hjg42hz3w2eki
*in*the*Coq*proof assistant, provides an approach to thep-adic*numbers**in**constructive**algebra*and analysis. ... The goal*of*this paper is to report on a formalization*of*thep-adic*numbers**in*the setting*of*the second author's univalent foundations program. ... Introduction*In*this paper, we present a formalization*of*the*construction**of*the p-adic*numbers**in*the*Coq*proof assistant. ...##
###
A constructive formalisation of Semi-algebraic sets and functions

2018
*
Proceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs - CPP 2018
*

We formally define

doi:10.1145/3176245.3167099
fatcat:bpdry2qnsbfyrgwzdxfkbofw5i
*in**Coq*the base operations on semi-*algebraic*sets and functions using embedded firstorder formulae over the language*of**real*closed fields, and we prove the correctness*of*their geometrical ...*In*doing so, we exploit a previous formalisation*of*quantifier elimination on such embedded formulae to guarantee the decidability*of*several first-order properties and keep our development*constructive*... Acknowledgments Pretty*Coq*code listing was done thanks to Assia Mahboubi's file [18] . ...##
###
A constructive formalisation of Semi-algebraic sets and functions

2018
*
Proceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs - CPP 2018
*

We formally define

doi:10.1145/3167099
dblp:conf/cpp/Djalal18
fatcat:cf35odpsirfjbhuxs2i4les7ne
*in**Coq*the base operations on semi-*algebraic*sets and functions using embedded firstorder formulae over the language*of**real*closed fields, and we prove the correctness*of*their geometrical ...*In*doing so, we exploit a previous formalisation*of*quantifier elimination on such embedded formulae to guarantee the decidability*of*several first-order properties and keep our development*constructive*... Acknowledgments Pretty*Coq*code listing was done thanks to Assia Mahboubi's file [18] . ...##
###
Computer Certified Efficient Exact Reals in Coq
[chapter]

2011
*
Lecture Notes in Computer Science
*

We provide an implementation

doi:10.1007/978-3-642-22673-1_7
fatcat:lysohwv64zgutc4xbsx452e7fm
*of*the exact*real**numbers**in*the*Coq*proof assistant. ... This appears to be the first time that type classes are used*in*heavy computation. We obtain over a 100 times speed up*of*the basic operations and indications for improving the*Coq*system. ... We thank Pierre Letouzey and Matthieu Sozeau for closing some*of*our bug reports. ...##
###
Type classes for efficient exact real arithmetic in Coq

2013
*
Logical Methods in Computer Science
*

Previously, we [Krebbers/Spitters 2011] provided a fast implementation

doi:10.2168/lmcs-9(1:1)2013
fatcat:cnkuw5wwqrgktpab7rrp2im5rq
*of*the exact*real**numbers**in*the*Coq*proof assistant. ... Our implementation improved on an earlier implementation by O'Connor by using type classes to describe an abstract specification*of*the underlying dense set from which the*real**numbers*are built. ... We are grateful to the anonymous referees who helped to improve the presentation*of*the paper. ...##
###
Dependently typed programming

2013
*
Progress in Informatics
*

Working with

doi:10.2201/niipi.2013.10.8
fatcat:7puu2vjqqfdbbnfz6p2jdwxe6q
*real**numbers**in*dependent type theory is difficult since one usually has to work with computational*real**numbers*, which behave differently from*real**numbers*occurring*in*set theory. ... One way to avoid this difficulty is to formulate the*reals*by postulating axioms stating the existence*of**real**numbers*, operations on them and axioms. These*real**numbers*remain abstract. ...##
###
Formal and Efficient Primality Proofs by Use of Computer Algebra Oracles

2001
*
Journal of symbolic computation
*

Finally, we discuss the implementation

doi:10.1006/jsco.2001.0457
fatcat:qtwaznb6tbh6tmoayhaahcgb4m
*of*this approach and tackle the proof*of*primality for some*of*the largest*numbers*expressible*in**Coq*. ... Then we present an algorithm*in*which computer*algebra*software is employed as oracle to the proof assistant to generate the necessary witnesses for applying the criterion. ... A grateful thanks to the anonymous referees for pointing out missing references*in*the initial version*of*the bibliography. ...##
###
Computing with Classical Real Numbers
[article]

2008
*
arXiv
*
pre-print

*constructively*valid

*real*

*numbers*. ... There are two incompatible

*Coq*libraries that have a theory

*of*the

*real*

*numbers*; the

*Coq*standard library gives an axiomatic treatment

*of*classical

*real*

*numbers*, while the CoRN library from Nijmegen defines ... The

*construction*

*of*the isomorphism

*In*this section we briefly present the

*algebraic*hierarchy present

*in*CoRN (it is described

*in*detail

*in*[5] and [3] ). ...

##
###
A Large-Scale Experiment in Executing Extracted Programs

2006
*
Electronical Notes in Theoretical Computer Science
*

Previous work has focused on the difficulties

doi:10.1016/j.entcs.2005.11.024
fatcat:7moezpct3vedvoue5cznrpc6w4
*in*obtaining a program from a formalization*of*the Fundamental Theorem*of**Algebra*inside the*Coq*proof assistant. ...*In*theory, this program allows one to compute approximations*of*roots*of*polynomials. However, as we show*in*this work, there is currently a big gap between theory and practice. ... The Fundamental Theorem*of**Algebra*(FTA) states that every non-con-stant polynomial over the complex*numbers*has a root. The proof*of*this statement*in*C-CoRN is a*constructive*proof due to H. ...##
###
Developing the Algebraic Hierarchy with Type Classes in Coq
[chapter]

2010
*
Lecture Notes in Computer Science
*

We present a new formalization

doi:10.1007/978-3-642-14052-5_35
fatcat:vyvsweu5yrhithutyuwgylz23y
*of*the*algebraic*hierarchy*in**Coq*, exploiting its new type class mechanism to make practical a solution formerly thought infeasible. ... Our approach addresses both traditional challenges as well as new ones resulting from our ambition to build upon this development a library*of**constructive*analysis*in*which abstraction penalties inhibiting ...*In*particular, we want to be able to effortlessly swap implementations*of**number*representations. ...##
###
Finite Groups Representation Theory with Coq
[chapter]

2009
*
Lecture Notes in Computer Science
*

*In*this paper we present a formalization

*of*finite groups representation theory

*in*the

*Coq*system that includes a formalization

*of*Maschke's theorem on reducible finite group

*algebra*. ... Representation theory is a branch

*of*

*algebra*that allows the study

*of*groups through linear applications, i.e. matrices. Thus problems

*in*abstract groups can be reduced to problems on matrices. ... The second is the C-CoRN hierarchy [19] , mainly devoted to a

*constructive*formalisation

*of*

*real*

*numbers*and including a proof

*of*the fundamental theorem

*of*

*algebra*. ...

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