#141 ababaaabba=baba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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