#029 abbabbaa=baba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

\( \begin{aligned} b a b a &\rightarrow d a \\ c b^{2} a &\rightarrow d \\ a b^{2} a &\rightarrow c \\ b a b c &\rightarrow d c \\ b a d a &\rightarrow d a b a \\ c b d a &\rightarrow d b a \\ a b^{2} c &\rightarrow d \\ a b d a &\rightarrow c b a \\ b a d c &\rightarrow d a b c \\ d b^{2} a &\rightarrow a b^{2} d \\ a b d^{2} &\rightarrow c b d \\ a b d c &\rightarrow c b c \\ c b d c &\rightarrow d b c \\ d b^{2} c &\rightarrow c b^{2} d \\ c b d^{2} &\rightarrow d b d \\ b a b d &\rightarrow d^{2} \\ b a d^{2} &\rightarrow d a b d \\ c b^{2} c &\rightarrow a b^{2} d \\ b c b c &\rightarrow d^{2} c \\ b c b a &\rightarrow d^{2} a \\ b c b d &\rightarrow d^{3} \\ b c d c &\rightarrow d^{2} a b c \\ b c d a &\rightarrow d^{2} a b a \\ b c d^{2} &\rightarrow d^{2} a b d \\ b d b c &\rightarrow d^{3} c \\ b d b a &\rightarrow d^{3} a \\ b d b d &\rightarrow d^{4} \\ b d^{2} c &\rightarrow d^{3} a b c \\ b d^{2} a &\rightarrow d^{3} a b a \\ b d^{3} &\rightarrow d^{3} a b d \\ d b d^{2} &\rightarrow a d^{3} c b d \\ a d^{3} a b c &\rightarrow d b c \\ a d^{3} a b a &\rightarrow d b a \\ a d^{3} a b d &\rightarrow d b d \\ a d^{4} b c &\rightarrow d^{4} a b c \\ a d^{4} b a &\rightarrow d^{4} a b a \\ a d^{4} b d &\rightarrow d^{4} a b d \\ c d^{3} a b d &\rightarrow a d^{3} c b d \\ c d^{3} c b d &\rightarrow d^{4} a b d \\ c d^{3} a d^{2} &\rightarrow a d^{3} c d^{2} \\ c d^{3} a d a &\rightarrow a d^{3} c d a \\ c d^{3} a d c &\rightarrow a d^{3} c d c \\ c d^{5} c &\rightarrow d^{4} c d c \\ c d^{5} a &\rightarrow d^{4} c d a \\ c d^{6} &\rightarrow d^{4} c d^{2} \\ c d^{4} b c &\rightarrow d^{4} c b c \\ c d^{4} b a &\rightarrow d^{4} c b a \\ a d^{3} a d c &\rightarrow d^{2} c \\ a d^{3} a d a &\rightarrow d^{2} a \\ a d^{3} a d^{2} &\rightarrow d^{3} \\ c d^{3} a b c &\rightarrow a d^{3} c b c \\ c d^{3} a b a &\rightarrow a d^{3} c b a \\ a d^{5} c &\rightarrow d^{4} a d c \\ a d^{5} a &\rightarrow d^{4} a d a \\ a d^{6} &\rightarrow d^{4} a d^{2} \\ d b d a &\rightarrow a d^{3} c b a \\ d b d c &\rightarrow a d^{3} c b c \\ c d^{3} c d^{2} &\rightarrow d^{4} a d^{2} \\ c d^{3} c d a &\rightarrow d^{4} a d a \\ c d^{3} c d c &\rightarrow d^{4} a d c \\ c d^{4} b d &\rightarrow d^{4} c b d \\ c d^{3} c b c &\rightarrow d^{4} a b c \\ c d^{3} c b a &\rightarrow d^{4} a b a \end{aligned} \)