#255 baabababaa=abbaaba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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