#050 babaabaaba=aabbabaa

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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