#481 babbabbbba=abbba

Original presentation

\( \left\langle a, b \;\middle|\; b a b^{2} a b^{4} a = a b^{3} a \right\rangle \)

Modified presentation

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

Complete rewriting system

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

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