#339 aaabaaabba=baaaba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

\( \begin{aligned} b c a &\rightarrow c^{2} b a \\ c b c &\rightarrow d \\ a^{3} b &\rightarrow c \\ b c^{2} &\rightarrow c d \\ a^{3} c^{2} b a &\rightarrow c^{2} a \\ d b c &\rightarrow c b d \\ b c d &\rightarrow c^{2} b d \\ c^{2} d &\rightarrow d c \\ c^{3} &\rightarrow a^{3} c d \\ a^{3} c d b a &\rightarrow d a \\ b d c &\rightarrow c d^{2} \\ a^{3} c d b d &\rightarrow d^{2} \\ c d c &\rightarrow a^{3} c d^{2} \\ a^{3} c^{2} b d &\rightarrow d c \\ a^{6} c d^{2} b a &\rightarrow c d a \\ c a^{3} c d &\rightarrow a^{6} c d^{2} \\ d c^{2} &\rightarrow a^{6} c d^{3} \\ c a^{6} c d^{2} &\rightarrow a^{6} c d^{2} c \\ b a^{3} c d &\rightarrow a^{3} c d^{2} \\ b d a &\rightarrow a^{3} c d^{2} b a \\ a^{3} c d^{2} c d &\rightarrow c d^{2} c \\ a^{6} c d^{2} b d &\rightarrow c d^{2} \\ b a^{6} c d^{2} &\rightarrow a^{3} c d^{2} c \\ a^{3} c d^{2} a^{3} c d^{3} &\rightarrow c d^{3} c \\ a^{3} c d^{2} a^{3} c d^{2} c &\rightarrow c d^{2} a^{3} c d \\ b d^{2} &\rightarrow a^{3} c d^{2} b d \\ a^{6} c d^{4} &\rightarrow d^{2} c \\ a^{6} c d^{2} a^{6} c d^{3} &\rightarrow d a^{6} c d^{2} \\ a^{6} c d^{3} c &\rightarrow d a^{3} c d \\ c d a^{3} c d^{2} &\rightarrow a^{3} c d^{3} c \\ c d a^{6} c d^{2} &\rightarrow a^{3} c d^{2} a^{3} c d \\ a^{6} c d^{2} c b a &\rightarrow d c a \\ a^{6} c d^{2} c b d &\rightarrow d c d \\ a^{6} c d^{2} a^{3} c d^{2} &\rightarrow c d a^{3} c d \\ a^{6} c d^{3} a^{6} c d^{2} &\rightarrow d a^{3} c d a^{3} c d \\ a^{6} c d^{3} a^{3} c d &\rightarrow d a^{6} c d^{2} c \\ a^{3} c d^{2} a^{3} c d^{2} a^{3} c d &\rightarrow c d^{2} a^{6} c d^{2} c \\ a^{3} c d^{2} a^{3} c d^{2} a^{6} c d^{2} &\rightarrow c d^{2} a^{3} c d a^{3} c d \\ c d a^{3} c d a^{3} c d &\rightarrow a^{3} c d^{2} a^{6} c d^{2} c \end{aligned} \)