#118 ababaaaaba=baba

Original presentation

\( \left\langle a, b \;\middle|\; a b a b a^{4} b a = b a b a \right\rangle \)

Modified presentation

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

Complete rewriting system

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

\( \begin{aligned} b a &\rightarrow c \\ c a^{3} &\rightarrow d \\ a c d c &\rightarrow c^{2} \\ a c d^{2} &\rightarrow c d \\ c a^{2} c^{2} &\rightarrow d c d c \\ c a^{2} c d &\rightarrow d c d^{2} \\ c a c^{2} &\rightarrow d c d^{2} c \\ c a c d &\rightarrow d c d^{3} \\ c^{3} &\rightarrow d c d^{3} c \\ c^{2} d &\rightarrow d c d^{4} \\ b c d &\rightarrow d c d^{5} \\ b c^{2} &\rightarrow d c d^{4} c \\ b d c d^{3} c &\rightarrow d c d^{4} c^{2} \\ b d c d^{4} &\rightarrow d c d^{4} c d \\ c d c d^{3} c &\rightarrow d c d^{3} c^{2} \\ c d c d^{4} &\rightarrow d c d^{3} c d \\ c a d c d^{3} c &\rightarrow d c d^{2} c^{2} \\ c a d c d^{4} &\rightarrow d c d^{2} c d \\ c a^{2} d c d^{3} c &\rightarrow d c d c^{2} \\ c a^{2} d c d^{4} &\rightarrow d c d c d \\ a d c d^{3} c^{2} &\rightarrow d c d^{6} c \\ a d c d^{3} c d &\rightarrow d c d^{7} \\ a d c d^{4} c d^{3} c &\rightarrow d c d^{6} c^{2} \\ a d c d^{4} c d^{4} &\rightarrow d c d^{6} c d \end{aligned} \)