#267 ababbabbba=babbbaba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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