#092 abbaababba=bababbaa

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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