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Original presentation

\( \left\langle a, b \;\middle|\; a^{4} b a b^{2} a^{2} = b a b a^{3} \right\rangle \)

Modified presentation

\( \left\langle a, b, c, d \;\middle|\; a^{4} c b c a = c^{2} a^{2},\; c = b a,\; d = b a b^{2} \right\rangle \)

Complete rewriting system

Using recursive path order (RPOCmp) with \(a < d < b < c\):

\( \begin{aligned} c &\rightarrow b a \\ b a b^{2} &\rightarrow d \\ b a b a^{3} &\rightarrow a^{4} d a^{2} \\ b a b d &\rightarrow d a b^{2} \\ d a b a^{3} &\rightarrow a^{4} d a^{3} d a^{2} \end{aligned} \)