#422 aabbbaaba=baabbaa

Original presentation

\( \left\langle a, b \;\middle|\; a^{2} b^{3} a^{2} b a = b a^{2} b^{2} a^{2} \right\rangle \)

Modified presentation

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

Complete rewriting system

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

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