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A Generic and Executable Formalization of Signature-Based Gr\"obner Basis Algorithms
We present a generic and executable formalization of signature-based
algorithms (such as Faug\`ere's ) for computing Gr\"obner bases, as well
as their mathematical background, in the Isabelle/HOL proof assistant. Said
algorithms are currently the best known algorithms for computing Gr\"obner
bases in terms of computational efficiency. The formal development attempts to
be as generic as possible, generalizing most known variants of signature-based
algorithms, but at the same time the implemented functions are effectively
executable on concrete input for efficiently computing mechanically verified
Gr\"obner bases. Besides correctness the formalization also proves that under
certain conditions the algorithms a-priori detect and avoid all useless
reductions to zero, and return minimal signature Gr\"obner bases.
To the best of our knowledge, the formalization presented here is the only
formalization of signature-based Gr\"obner basis algorithms in existence so
far