#266 ababbabbaa=baba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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