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The theory part of this paper is sketched as follows. Based on columnstochastic average matrix $T_n$ selected as a basic substitution matrix, themethod of advanced successive difference substitution is established. Then, aset of necessary and sufficient conditions for deciding positive semi-definiteform on $\R^n_+$ is derived from this method. And furthermore, it is provedthat the sequence of SDS sets of a positive definite form is positivelyterminating.  Worked out according to these results, the Maple program TSDS3 not onlyautomatically proves the polynomial inequalities, but also outputs counterexamples for the false. Sometimes TSDS3 does not halt, but it is very useful byexperimenting on so many examples.

Successive Difference Substitution Based on Column Stochastic Matrix and Mechanical Decision for Positive Semi-definite Forms