#242 abbabbbaaa=bababba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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