#409 aaababbaaa=baaaba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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