#096 baabaabbaa=abababa

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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