#184 abababbaba=bababa

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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