#237 ababbaba=bababa

Original presentation

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

Modified presentation

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

Complete rewriting system

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

\( \begin{aligned} b a &\rightarrow c \\ a d &\rightarrow c^{3} \\ c b c^{2} &\rightarrow d \\ b c^{3} &\rightarrow c d \\ d b c^{2} &\rightarrow c b c d \\ c^{2} d &\rightarrow d c \\ a c b c d &\rightarrow d c \\ c^{2} b c d &\rightarrow b d c \\ b d c^{2} &\rightarrow c d c d \\ b c d c &\rightarrow c d^{2} \\ c d b c d &\rightarrow b d^{2} \\ c b d c &\rightarrow d^{2} \\ c b^{2} d c &\rightarrow d b c d \\ d b^{2} d c &\rightarrow c b^{2} d^{2} \\ b c^{2} b d^{2} &\rightarrow c d^{2} b c d \\ b c d b d^{2} &\rightarrow c d^{3} b c d \\ b d^{2} b c d &\rightarrow c d b^{2} d^{2} \\ b d c b d^{2} &\rightarrow c d c d^{2} b c d \\ b d^{2} c &\rightarrow c d c d^{2} \\ c^{3} b^{2} d^{2} &\rightarrow d^{2} b c d \\ d c b c d &\rightarrow c b d^{2} \\ a c b^{2} d^{2} &\rightarrow c b d^{2} \\ b d^{3} &\rightarrow c d c^{2} b d^{2} \\ b c b d^{2} &\rightarrow c^{2} b^{2} d^{2} \\ b c d^{2} &\rightarrow c^{2} b d^{2} \\ b d^{2} b d^{2} &\rightarrow c d c d^{3} b c d \\ b d b d^{2} &\rightarrow c d c^{2} b^{2} d^{2} \\ d b d c &\rightarrow c^{3} b d^{2} \\ a c^{3} b d^{2} &\rightarrow d c d \\ a c b d^{2} &\rightarrow c d^{2} \\ c^{5} b d^{2} &\rightarrow d^{3} \\ b d c d &\rightarrow c^{4} b d^{2} \end{aligned} \)