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#040 baabbaabba=aa

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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