#140 abaaabaaba=baaaba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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