#024 baaabaaaba=abbaa

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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