#357 abbabbaaa=babba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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