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Construction of Real Algebraic Numbers in Coq [chapter]

Cyril Cohen
2012 Lecture Notes in Computer Science  
This paper shows a 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  ... 
doi:10.1007/978-3-642-32347-8_6 fatcat:vdpsbyj4ejhafadmv34tgn7cee

A Constructive Algebraic Hierarchy in Coq

Herman Geuvers, Randy Pollack, Freek Wiedijk, Jan Zwanenburg
2002 Journal of symbolic computation  
We have developed this framework as part 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.  ... 
doi:10.1006/jsco.2002.0552 fatcat:l5tzxkehqbbmvk64l5jiyfxkmy

C-CoRN, the Constructive Coq Repository at Nijmegen [chapter]

Luís Cruz-Filipe, Herman Geuvers, Freek Wiedijk
2004 Lecture Notes in Computer Science  
We present C-CoRN, the 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  ... 
doi:10.1007/978-3-540-27818-4_7 fatcat:p6t6ka5ftjbphag25dv4mzmsay

Constructive Reals in Coq: Axioms and Categoricity [chapter]

Herman Geuvers, Milad Niqui
2002 Lecture Notes in Computer Science  
We describe a 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.  ... 
doi:10.1007/3-540-45842-5_6 fatcat:erq3apmbtbcovigm5xvkp47ina

A univalent formalization of the p-adic numbers

ÁLVARO PELAYO, VLADIMIR VOEVODSKY, MICHAEL A. WARREN
2015 Mathematical Structures in Computer Science  
This formalization, which has been verified 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.  ... 
doi:10.1017/s0960129514000541 fatcat:g7zfmcxsp5gs3hjg42hz3w2eki

A constructive formalisation of Semi-algebraic sets and functions

Boris Djalal
2018 Proceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs - CPP 2018  
We formally define 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] .  ... 
doi:10.1145/3176245.3167099 fatcat:bpdry2qnsbfyrgwzdxfkbofw5i

A constructive formalisation of Semi-algebraic sets and functions

Boris Djalal
2018 Proceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs - CPP 2018  
We formally define 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] .  ... 
doi:10.1145/3167099 dblp:conf/cpp/Djalal18 fatcat:cf35odpsirfjbhuxs2i4les7ne

Computer Certified Efficient Exact Reals in Coq [chapter]

Robbert Krebbers, Bas Spitters
2011 Lecture Notes in Computer Science  
We provide an implementation 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.  ... 
doi:10.1007/978-3-642-22673-1_7 fatcat:lysohwv64zgutc4xbsx452e7fm

Type classes for efficient exact real arithmetic in Coq

Robbert Krebbers, Bas Spitters, Martín Escardó
2013 Logical Methods in Computer Science  
Previously, we [Krebbers/Spitters 2011] provided a fast implementation 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.  ... 
doi:10.2168/lmcs-9(1:1)2013 fatcat:cnkuw5wwqrgktpab7rrp2im5rq

Dependently typed programming

Shin-CHENG MU, Conor MCBRIDE, Stephanie WEIRICH
2013 Progress in Informatics  
Working with 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.  ... 
doi:10.2201/niipi.2013.10.8 fatcat:7puu2vjqqfdbbnfz6p2jdwxe6q

Formal and Efficient Primality Proofs by Use of Computer Algebra Oracles

Olga Caprotti, Martijn Oostdijk
2001 Journal of symbolic computation  
Finally, we discuss the implementation 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.  ... 
doi:10.1006/jsco.2001.0457 fatcat:qtwaznb6tbh6tmoayhaahcgb4m

Computing with Classical Real Numbers [article]

Cezary Kaliszyk, Russell O'Connor
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] ).  ... 
arXiv:0809.1644v1 fatcat:wusktwnutbfelnjau5tfyxmvje

A Large-Scale Experiment in Executing Extracted Programs

Luís Cruz-Filipe, Pierre Letouzey
2006 Electronical Notes in Theoretical Computer Science  
Previous work has focused on the difficulties 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.  ... 
doi:10.1016/j.entcs.2005.11.024 fatcat:7moezpct3vedvoue5cznrpc6w4

Developing the Algebraic Hierarchy with Type Classes in Coq [chapter]

Bas Spitters, Eelis van der Weegen
2010 Lecture Notes in Computer Science  
We present a new formalization 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.  ... 
doi:10.1007/978-3-642-14052-5_35 fatcat:vyvsweu5yrhithutyuwgylz23y

Finite Groups Representation Theory with Coq [chapter]

Sidi Ould Biha
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.  ... 
doi:10.1007/978-3-642-02614-0_34 fatcat:krsg5ecsabdofedtlm4rg3qe7y
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