#004 baabbaaaba=aabba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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