#034 baaaabbbaa=abaaa

Original presentation

\( \left\langle a, b \;\middle|\; b a^{4} b^{3} a^{2} = a b a^{3} \right\rangle \)

Modified presentation

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

Complete rewriting system

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

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