#014 babbaababa=abba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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