#344 babbbabba=abbba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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