#124 bbaabbbbaa=aabbbba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

\( \begin{aligned} b^{4} &\rightarrow c \\ c a^{3} &\rightarrow d \\ b c &\rightarrow c b \\ a^{2} c a &\rightarrow c a^{2} d \\ b^{2} a^{2} d &\rightarrow c a^{2} d a \\ a^{2} c^{2} a^{2} d &\rightarrow c a^{2} d a c a \\ c b a^{3} &\rightarrow b d \\ c^{2} a^{2} d a d &\rightarrow d a c a \\ c a^{2} d a^{2} &\rightarrow a^{2} d \\ c a c a^{2} d &\rightarrow d c a \\ c b^{2} a^{3} &\rightarrow b^{2} d \\ c^{2} b a^{2} d a d &\rightarrow b d a c a \\ c b a^{2} d a^{2} &\rightarrow b a^{2} d \\ c b a c a^{2} d &\rightarrow b d c a \\ c a^{2} d a c a^{2} d &\rightarrow a^{2} d a c a \\ c a^{2} d a d a^{2} &\rightarrow a^{4} d \\ c a^{2} d c a^{2} d &\rightarrow a^{2} d c a \\ a^{6} d &\rightarrow c a^{2} d a d a d a^{2} \\ c b^{3} a^{3} &\rightarrow b^{3} d \\ b^{2} d a c^{2} a^{2} d &\rightarrow c^{2} a^{2} d^{2} a c a \\ b^{2} d a c a &\rightarrow c^{2} a^{2} d^{2} \\ c b^{2} a c a^{2} d &\rightarrow b^{2} d c a \\ c b a^{2} d a c a^{2} d &\rightarrow b a^{2} d a c a \\ c b a^{2} d a d a^{2} &\rightarrow b a^{4} d \\ c b a^{2} d c a^{2} d &\rightarrow b a^{2} d c a \\ c a^{2} d a d a c a^{2} d &\rightarrow a^{4} d a c a \\ c a^{2} d a d c a^{2} d &\rightarrow a^{4} d c a \\ b^{2} d a^{5} d &\rightarrow c^{2} a^{2} d^{2} a d a d a^{2} \\ b^{2} d a^{3} d &\rightarrow c^{2} a^{2} d^{2} a d a^{2} \\ b^{2} d a d &\rightarrow c^{2} a^{2} d^{2} a^{2} \\ c b^{3} a c a^{2} d &\rightarrow b^{3} d c a \\ b^{2} a^{4} d &\rightarrow c a^{2} d^{2} a^{2} \\ c b a^{2} d a d a c a^{2} d &\rightarrow b a^{4} d a c a \\ c b a^{2} d a d c a^{2} d &\rightarrow b a^{4} d c a \end{aligned} \)