#058 baabbaaaba=abbaa

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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