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

@inproceedings{Grgoire2008ProofCF,
  title={Proof Certificates for Algebra and Their Application to Automatic Geometry Theorem Proving},
  author={Benjamin Gr{\'e}goire and Lo{\"i}c Pottier and Laurent Th{\'e}ry},
  booktitle={Automated Deduction in Geometry},
  year={2008}
}
Integrating decision procedures in proof assistants in a safe way is a major challenge. In this paper, we describe how, starting from Hilbert's Nullstellensatz theorem, we combine a modified version of Buchberger's algorithm and some reflexive techniques to get an effective procedure that 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… 
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TLDR
The formalization of the arithmetization of Euclidean geometry in the Coq proof assistant is described, derived from Tarski's system of geometry a formal proof of the nine-point circle theorem using the Grobner basis method.
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