• Corpus ID: 2188945

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

@article{Pottier2008ConnectingGB,
  title={Connecting Gr{\"o}bner Bases Programs with Coq to do Proofs in Algebra, Geometry and Arithmetics},
  author={Lo{\"i}c Pottier},
  journal={ArXiv},
  year={2008},
  volume={abs/1007.3615}
}
  • L. Pottier
  • Published 1 November 2008
  • Computer Science, Mathematics
  • ArXiv
We describe how we connected three programs that compute Groebner bases to Coq, to do automated proofs on algebraic, geometrical and arithmetical expressions. The result is a set of Coq tactics and a certificate mechanism (downloadable at this http URL). The programs are: F4, GB \, and gbcoq. F4 and GB are the fastest (up to our knowledge) available programs that compute Groebner bases. Gbcoq is slow in general but is proved to be correct (in Coq), and we adapted it to our specific problem to… 

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

This paper describes how a modified version of Buchberger's algorithm and some reflexive techniques are combined to get an effective procedure that automatically produces formal proofs of theorems in geometry.

Investigations in Computer-Aided Mathematics : Experimentation, Computation, and Certification. (Investigations en Mathématiques Assistées par Ordinateur : Expérimentation, Calcul et Certification)

This thesis presents three contributions to the topic of computer-assisted proofs: both in the case of proofs relying on computations and of formal proofs produced and verified using a piece of

Certified Verification of Algebraic Properties on Low-Level Mathematical Constructs in Cryptographic Programs

A certified technique is developed to verify low-level mathematical constructs in X25519, the default elliptic curve Diffie-Hellman key exchange protocol used in OpenSSH, and an algebraic specification of mathematical constructs is translated into angebraic problem.

Formalization of real analysis: a survey of proof assistants and libraries †

This survey presents how real numbers have been defined in these various provers and how the notions of real analysis described above have been formalized.

On the formalization of foundations of geometry

A new proof is exposed that Euclid’s parallel postulate is not derivable from the other axioms of first-order Euclidean geometry, and Pejas’ classification of parallel postulates is refined.

A Formal Proof of the Irrationality of $\zeta(3)$

The crux of this proof is to establish that some sequences satisfy a common recurrence, and formally prove this result by an a posteriori verification of calculations performed by computer algebra algorithms in a Maple session.

Proof Assistant Decision Procedures for Formalizing Origami

Issues that arise when proving properties of origami constructions using proof assistant decision procedures are described and ad-hoc decision procedures that can be used to optimize the proofs are shown.

Frex: dependently-typed algebraic simplification

An extensible, mathematically-structured algebraic simplification library design that guarantees simplification modules, even user-defined ones, are terminating, sound, and complete with respect to a well-specified class of equations is presented.

Formalization and Implementation of Algebraic Methods in Geometry

This work describes the ongoing project of formalization of algebraic methods for geometry theorem proving (Wu's method and the Groebner bases method), their implementation and integration in educational tools, and development of a new, open-source Java implementation of thegebraic methods.

A formal quantifier elimination for algebraically closed fields

We prove formally that the first order theory of algebraically closed fields enjoys quantifier elimination, and hence is decidable. This proof is organized in two modular parts. We first reify the

References

SHOWING 1-10 OF 15 REFERENCES

A new efficient algorithm for computing Gröbner bases (F4)

Automating Elementary Number-Theoretic Proofs Using Gröbner Bases

A uniform algorithm for proving automatically a fairly wide class of elementary facts connected with integer divisibility, based on a reduction to ideal membership assertions that are then solved using Grobner bases.

Verifying Nonlinear Real Formulas Via Sums of Squares

This work describes its experience with an implementation in HOL Light, noting some successes as well as difficulties, and describes a new approach to the univariate case that can handle some otherwise difficult examples.

Context Aware Calculation and Deduction

This work addresses some aspects of a system architecture for mathematical assistants that integrates calculations and deductions by common infrastructure within the Isabelle theorem proving environment, and is able to implement proof methods that operate on abstract theories and a range of particular theory interpretations.

Towards Self-verification of HOL Light

This work has performed a full verification of an imperfect but quite detailed model of the basic HOL Light core, without definitional mechanisms, and this verification is entirely conducted with respect to a set-theoretic semantics within HOL Light itself.

Commutative Algebra: with a View Toward Algebraic Geometry

Introduction.- Elementary Definitions.- I Basic Constructions.- II Dimension Theory.- III Homological Methods.- Appendices.- Hints and Solutions for Selected Exercises.- References.- Index of

Connecting Gröbner bases programs with Coq Pottier

  • Connecting Gröbner bases programs with Coq Pottier

Gb: une procédure de décision pour le système coq

  • Journées Francaises des Langages Applicatifs
  • 2004

Gb: une procédure de décision pour le système coq. Journées Francaises des Langages Applicatifs, Sainte-Marie-de-Ré

  • Gb: une procédure de décision pour le système coq. Journées Francaises des Langages Applicatifs, Sainte-Marie-de-Ré
  • 2004