#169 abbaababaa=baaba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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