#458 baaabbbaa=aaabbba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

\( \begin{aligned} b^{3} a &\rightarrow c \\ a^{3} c &\rightarrow d \\ c a^{2} c d &\rightarrow d c \\ b c a^{2} c a &\rightarrow c a^{2} c \\ c a^{2} c a d &\rightarrow d a c \\ b d &\rightarrow c a^{2} c a^{2} \\ a^{3} d c &\rightarrow d a^{2} c d \\ c a^{2} c a^{2} d &\rightarrow d a^{2} c \\ c a^{2} c a^{3} &\rightarrow d \\ b c a^{2} d &\rightarrow c a^{2} c a c a^{3} \\ a^{3} d a^{2} c &\rightarrow d a^{2} c a^{2} d \\ a^{3} d a c &\rightarrow d a^{2} c a d \\ d a^{2} c a^{3} &\rightarrow a^{3} d \\ d a^{2} c d a^{2} c a^{2} d &\rightarrow a^{3} d^{2} a^{2} c \\ d a^{2} c d a^{2} c a d &\rightarrow a^{3} d^{2} a c \\ d a^{2} c d a^{2} c d &\rightarrow a^{3} d^{2} c \\ b a^{3} d &\rightarrow d a c a^{3} \\ b c a^{5} d &\rightarrow c a^{2} c a c a^{2} d a^{3} \\ a^{6} d^{2} c &\rightarrow d a^{2} c a^{2} d^{2} a^{2} c d \\ d a^{2} c a^{2} d a^{3} &\rightarrow a^{6} d \\ d a^{2} c a^{2} d a^{2} c a^{2} d &\rightarrow a^{3} d a^{2} d a^{2} c \\ d a^{2} c a^{2} d a^{2} c a d &\rightarrow a^{3} d a^{2} d a c \\ d a^{2} c a^{2} d a^{2} c d &\rightarrow a^{3} d a^{2} d c \\ d a^{2} c a^{2} d^{2} a^{2} c a^{2} d &\rightarrow a^{6} d^{2} a^{2} c \\ d a^{2} c a^{2} d^{2} a^{2} c a d &\rightarrow a^{6} d^{2} a c \\ d a^{2} c a d a^{2} c a^{2} d &\rightarrow a^{3} d a d a^{2} c \\ d a^{2} c a d a^{2} c a d &\rightarrow a^{3} d a d a c \\ d a^{2} c a d a^{2} c d &\rightarrow a^{3} d a d c \\ b a^{6} d &\rightarrow d a c a^{2} d a^{3} \\ b c a^{8} d &\rightarrow c a^{2} c a c a^{2} d a^{2} d a^{3} \\ a^{9} d &\rightarrow d a^{2} c a^{2} d a^{2} d a^{3} \\ a^{6} d a^{2} d c &\rightarrow d a^{2} c a^{2} d a^{2} d a^{2} c d \\ a^{6} d a d c &\rightarrow d a^{2} c a^{2} d a d a^{2} c d \\ d a^{2} c a^{2} d a^{2} d a^{2} c a^{2} d &\rightarrow a^{6} d a^{2} d a^{2} c \\ d a^{2} c a^{2} d a^{2} d a^{2} c a d &\rightarrow a^{6} d a^{2} d a c \\ d a^{2} c a^{2} d a d a^{2} c a^{2} d &\rightarrow a^{6} d a d a^{2} c \\ d a^{2} c a^{2} d a d a^{2} c a d &\rightarrow a^{6} d a d a c \end{aligned} \)