#484 ababbabaaa=baaba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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