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Connecting Gröbner Bases Programs with Coq to do Proofs in Algebra, Geometry and Arithmetics [article]

Loïc Pottier
2010 arXiv   pre-print
We describe how we connected three programs that compute Groebner bases to Coq, to do automated proofs on algebraic, geometrical and arithmetical expressions.  ...  The connection between Coq and the programs that compute Groebner bases is done using the "external" tactic of Coq that allows to call arbitrary programs accepting xml inputs and outputs.  ...  Acknowledgements: we thank anonymous referees for their suggestions on the redaction of this paper and bibliographical completions.  ... 
arXiv:1007.3615v1 fatcat:qwrbij6yxba7xj7mn33lvhqi5u

Connecting Gröbner Bases Programs with Coq to do Proofs in Algebra, Geometry and Arithmetics

Loic Pottier
2008 Logic Programming and Automated Reasoning  
We describe how we connected three programs that compute Gröbner bases [1] to Coq [11], to do automated proofs on algebraic, geometrical and arithmetical expressions.  ...  The connection between Coq and the programs that compute Gröbner bases is done using the "external" tactic of Coq that allows to call arbitrary programs accepting xml inputs and outputs.  ...  Acknowledgements: we thank anonymous referees for their suggestions on the redaction of this paper and bibliographical completions.  ... 
dblp:conf/lpar/Pottier08 fatcat:ofbhqipo4ve6jmwg5vc3ojstxu

Proof Certificates for Algebra and Their Application to Automatic Geometry Theorem Proving [chapter]

Benjamin Grégoire, Loïc Pottier, Laurent Théry
2011 Lecture Notes in Computer Science  
automatically produces formal proofs of theorems in geometry.  ...  The method is implemented in the Coq system but, since our specialised version of Buchberger's algorithm outputs explicit proof certificates, it could be easily adapted to other proof assistants.  ...  Yves Bertot helped us with his expertise of the GeoGebra system [19] . We thanks the referees for their constructive comments on the first version of this paper.  ... 
doi:10.1007/978-3-642-21046-4_3 fatcat:ssvv2zgqena3joe3tdtr7sus6y

Formalization of the arithmetization of Euclidean plane geometry and applications

Pierre Boutry, Gabriel Braun, Julien Narboux
2019 Journal of symbolic computation  
This paper describes the formalization of the arithmetization of Euclidean plane geometry in the Coq proof assistant.  ...  The arithmetization of geometry paves the way for the use of algebraic automated deduction methods in synthetic geometry.  ...  In doing so, they demonstrated the advantage of using an algebraic approach and the need for a connection with a synthetic approach.  ... 
doi:10.1016/j.jsc.2018.04.007 fatcat:7lxvzb4akrbnblfsd4nrgpwjrm

The "Seven Dwarfs" of Symbolic Computation [chapter]

Erich L. Kaltofen
2011 Texts & Monographs in Symbolic Computation  
Exact linear algebra, integer lattices SymDwf 2. Exact polynomial and differential algebra, Gröbner bases SymDwf 3. Inverse symbolic problems, e.g., interpolation and parameterization SymDwf 4.  ...  Tarski's algebraic theory of real geometry SymDwf 5. Hybrid symbolic-numeric computation SymDwf 6. Computation of closed form solutions SymDwf 7.  ...  ., methods with exact arithmetic, or logic programming, say rewriting via rules, altogether. However, they inspire to make a corresponding list, and here we will do so for symbolic computation.  ... 
doi:10.1007/978-3-7091-0794-2_5 fatcat:y6fjj6ghrzek5ps4vig7uv5vpy

Theorema: Towards computer-aided mathematical theory exploration

Bruno Buchberger, Adrian Crǎciun, Tudor Jebelean, Laura Kovács, Temur Kutsia, Koji Nakagawa, Florina Piroi, Nikolaj Popov, Judit Robu, Markus Rosenkranz, Wolfgang Windsteiger
2006 Journal of Applied Logic  
This survey paper illustrates the style of Theoremasupported mathematical theory exploration by a case study (the automated synthesis of an algorithm for the construction of Gröbner Bases) and gives an  ...  overview on some reasoners and organizational tools for theory exploration developed in the Theorema project.  ...  and Research in the frame of the project "e-Austria Timişoara".  ... 
doi:10.1016/j.jal.2005.10.006 fatcat:b2mfsckqqjhinp536yq4zqemxy

Visually Dynamic Presentation of Proofs in Plane Geometry

Zheng Ye, Shang-Ching Chou, Xiao-Shan Gao
2009 Journal of automated reasoning  
It solves not only the problem of identifying geometry elements in the proof text with those in the diagram, but also makes the proof more vividly visualized and intuitive.  ...  The manual input method mainly uses mouse clicks to create the dynamic geometry diagram and the proof text.  ...  The authors also wish to thank the referees for their helpful criticism and suggestions. The NSF grant provides Zheng Ye with an unique opportunity to work at WSU. X.S. Gao is supported by NSFC.  ... 
doi:10.1007/s10817-009-9162-5 fatcat:ey3y4rjngvfcfk7ba5sdrg64ry

Two New Ways to Formally Prove Dandelin-Gallucci's Theorem

David Braun, Nicolas Magaud, Pascal Schreck
2021 Proceedings of the 2021 on International Symposium on Symbolic and Algebraic Computation  
Second, we use an original method based on matroid theory to generate a proof script which is then checked by the Coq proof assistant.  ...  We study and compare two new and very different approaches to prove this theorem. First, we propose a new proof based on the well-known Wu's method.  ...  The user (or a program written by the user) will generate a formal statement and a proof script in the formal language of Coq and the system will automatically type-check (i.e. proof-check) it to verify  ... 
doi:10.1145/3452143.3465550 fatcat:fnan4jlafbcn3fjwcxhnmorc7m

Electronic Communication of Mathematics and the Interaction of Computer Algebra Systems and Proof Assistants

Henk Barendregt, Arjeh M. Cohen
2001 Journal of symbolic computation  
Present day computer algebra systems (CASs) and proof assistants (PAs) are specialized programs that help humans with mathematical computations and deductions.  ...  The PAs have a limited power for computing and hence also for assistance with proofs.  ...  We thank Olga Caprotti and Martijn Oostdijk for providing us with large examples of formally proved prime numbers, the referees for their comments and Don Taylor and Weldon Lodwick for suggesting some  ... 
doi:10.1006/jsco.2001.0455 fatcat:ezzf6s6ehjakhbtguxnbwa3r7m

Proof and Computation in Geometry [chapter]

Michael Beeson
2013 Lecture Notes in Computer Science  
We might try to produce such proofs directly, or we might try to develop a "backtranslation" from algebra to geometry, following Descartes but with computer in hand.  ...  We consider the relationships between algebra, geometry, computation, and proof. Computers have been used to verify geometrical facts by reducing them to algebraic computations.  ...  Among the ways to compute, we mention Gröbner bases and the Collins CAD algorithm [7, 6] .  ... 
doi:10.1007/978-3-642-40672-0_1 fatcat:syvmbmvp7je7rpppujtew7jo7a

Proof-checking Euclid

Michael Beeson, Julien Narboux, Freek Wiedijk
2019 Annals of Mathematics and Artificial Intelligence  
HOL Light and Coq.  ...  We used axioms as close as possible to those of Euclid, in a language closely related to that used in Tarski's formal geometry.  ...  Within a few years three different groups began to use Gröbner bases to do the algebraic work, instead of the Wu-Ritt algorithm. See [15, 26, 27] .  ... 
doi:10.1007/s10472-018-9606-x fatcat:xpmopttt3vfvrnwyr42o7neku4

Proof-checking Euclid [article]

Michael Beeson and Julien Narboux and Freek Wiedijk
2018 arXiv   pre-print
HOL Light and Coq.  ...  We used axioms as close as possible to those of Euclid, in a language closely related to that used in Tarski's formal geometry.  ...  Within a few years three different groups began to use Gröbner bases to do the algebraic work, instead of the Wu-Ritt algorithm. See [26] , [27] , [15] .  ... 
arXiv:1710.00787v2 fatcat:aa7l3r4ncvhztddsi6pzkeqbva

Four Decades of Mizar

Adam Grabowski, Artur Korniłowicz, Adam Naumowicz
2015 Journal of automated reasoning  
This special issue is dedicated to works related to Mizar, the theorem proving project started by Andrzej Trybulec in the 1970s, and other automated proof checking systems used for formalizing mathematics  ...  Mizar 1 is the name of both a formal language in which the mathematics is written, and of an entire software system that checks the texts for logical correctness and manages the data base of Mizar articles  ...  Among used systems we can point out such tools as SAT-solvers and systems computing Gröbner bases [17] .  ... 
doi:10.1007/s10817-015-9345-1 fatcat:wclwjz6eqzbvbis7w5tdolu5n4

Formalizing complex plane geometry

Filip Marić, Danijela Petrović
2014 Annals of Mathematics and Artificial Intelligence  
Apart from applications in formalizing mathematics and in education, this work serves as a ground for formally investigating various non-Euclidean geometries and their intimate connections.  ...  We discuss different approaches to formalization and discuss the major advantages of the more algebraically oriented approach.  ...  Acknowledgements The authors are grateful to Pascal Schreck, Pierre Boutry, Julien Narboux, and anonymous reviewers of AMAI journal for many valuable suggestions and advice.  ... 
doi:10.1007/s10472-014-9436-4 fatcat:vfc4c5y6ufgovglq2r2jlvkzbm

Theory-Specific Automated Reasoning [chapter]

Andrea Formisano, Eugenio G. Omodeo
2010 Lecture Notes in Computer Science  
In discussing these and related issues, this paper refers to Set Theory as the unifying background, and to a specific proof-checker based on a slightly unorthodox formalization of it as an arena for experimentation  ...  In designing a large-scale computerized proof system, one is often confronted with issues of two kinds: issues regarding an underlying logical calculus, and issues that refer to theories, either specified  ...  Acknowledgements We are indebted to prof. Jacob T. Schwartz for his encouragement in pursuing research on proof-verification.  ... 
doi:10.1007/978-3-642-14309-0_3 fatcat:4yubeyj6nzevtn7orc24x5cpyi
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