#332 abababaaaa=baaba

Original presentation

\( \left\langle a, b \;\middle|\; a b a b a b a^{4} = b a^{2} b a \right\rangle \)

Modified presentation

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

Complete rewriting system

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

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