#160 baaabbabba=abbbaa

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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