#187 babbababa=abba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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