#329 babaaabbba=aabbbaba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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