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Connecting Gröbner bases programs with Coq to do proofs in algebra, geometry and arithmetics
"... 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 result is a set of Coq tactics and a certificate mechanism 1. The programs are: F4 [5], GB [4], and gbcoq [10]. F4 and GB are the ..."
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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 result is a set of Coq tactics and a certificate mechanism 1. The programs are: F4 [5], GB [4], and gbcoq [10]. F4 and GB are the fastest (up to our knowledge) available programs that compute Gröbner bases. Gbcoq is slow in general but is proved to be correct (in Coq), and we adapted it to our specific problem to be efficient. The automated proofs concern equalities and nonequalities on polynomials with coefficients and indeterminates in R or Z, and are done by reducing to Gröbner computation, via Hilbert’s Nullstellensatz. We adapted also the results of [7], to allow to prove some theorems about modular arithmetics. 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. We also produce certificates in order to make the proof scripts independant from the external programs. 1
Parametric linear arithmetic over ordered fields in Isabelle/HOL
"... We use higherorder logic to verify a quantifier elimination procedure for linear arithmetic over ordered fields, where the coefficients of variables are multivariate polynomials over another set of variables, we call parameters. The procedure generalizes Ferrante and Rackoff’s algorithm for the non ..."
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We use higherorder logic to verify a quantifier elimination procedure for linear arithmetic over ordered fields, where the coefficients of variables are multivariate polynomials over another set of variables, we call parameters. The procedure generalizes Ferrante and Rackoff’s algorithm for the nonparametric case. The formalization is based on axiomatic type classes and automatically carries over to e.g. the rational, real and nonstandard real numbers. It is executable, can be applied to HOL formulae by reflection and performs well on practical examples.
Formalization of Wu’s simple method in Coq
, 2011
"... We present in this paper the integration within the Coq proof assistant, of a method for automatic theorem proving in geometry. We use an approach based on the validation of a certificate. The certificate is generated by an implementation in Ocaml of a simple version of Wu’s method. ..."
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We present in this paper the integration within the Coq proof assistant, of a method for automatic theorem proving in geometry. We use an approach based on the validation of a certificate. The certificate is generated by an implementation in Ocaml of a simple version of Wu’s method.