An element from reducibles is reduced by polynomials from reducers and the basis. If this reduction yields a non-zero polynomial p, then p is adjoined to reducers and all polynomials in reducers and in the basis, whose head term can be reduced by p, change to reducibles. This can lead to a cascade of reductions until reducibles is empty. When this state is reached, the ``reducers'' are adjoined to the basis and the pair set is updated.
A list containing the basis and the updated pair set is returned.