#013 ababaaabba=baaababa

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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