#122 babaababaa=abaabba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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