#328 abbabbbba=babba

Original presentation

\( \left\langle a, b \;\middle|\; a b^{2} a b^{4} a = b a b^{2} a \right\rangle \)

Modified presentation

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

Complete rewriting system

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

\( \begin{aligned} b^{2} &\rightarrow c \\ b a c a &\rightarrow a d c a \\ c a c &\rightarrow d \\ a d^{2} c &\rightarrow e \\ b c &\rightarrow c b \\ b a d &\rightarrow a d^{2} \\ e a c &\rightarrow a d^{3} \\ d a &\rightarrow e a \\ c a d &\rightarrow a d^{3} \\ b e &\rightarrow a d^{3} c \\ d e &\rightarrow e^{2} \\ b a e a &\rightarrow a e^{2} a \\ b a^{2} d^{3} &\rightarrow a e a d^{3} \\ c b a c &\rightarrow b d \\ c e &\rightarrow a d^{4} c \\ c a e a &\rightarrow a e^{3} a \\ c a^{2} d^{3} &\rightarrow e a d \\ a e^{2} a d^{3} &\rightarrow e a d \\ e a^{2} d^{3} &\rightarrow a e^{3} a d \\ b a e^{2} &\rightarrow a e^{3} \\ b a^{2} e^{4} &\rightarrow a e a e^{4} \\ b a^{2} e^{3} a &\rightarrow a e a e^{3} a \\ c a e^{2} &\rightarrow a e^{4} \\ c a^{2} e^{4} &\rightarrow e a e^{2} \\ c a^{2} e^{3} a &\rightarrow e a e a \\ d b a c &\rightarrow c a b d \\ e b a c &\rightarrow a d^{2} b d \\ a e^{3} a e^{2} &\rightarrow e a^{2} e^{4} \\ a e^{3} a e a &\rightarrow e a^{2} e^{3} a \\ a e^{2} a e^{3} a &\rightarrow e a e a \\ a e^{2} a e^{4} &\rightarrow e a e^{2} \\ a e^{3} a^{2} e^{4} &\rightarrow e a e a e^{2} \\ a e^{3} a^{2} e^{3} a &\rightarrow e a e a e a \\ e a^{2} e^{4} a e^{3} a &\rightarrow a e^{4} a e a \\ e a^{2} e^{5} a e^{2} &\rightarrow a e^{4} a^{2} e^{4} \\ e a^{2} e^{5} a e a &\rightarrow a e^{4} a^{2} e^{3} a \\ e a^{2} e^{4} a d^{3} &\rightarrow a e^{4} a d \\ e a^{2} e^{4} a e^{4} &\rightarrow a e^{4} a e^{2} \\ e a^{2} e^{5} a^{2} e^{4} &\rightarrow a e^{4} a e a e^{2} \\ e a^{2} e^{5} a^{2} e^{3} a &\rightarrow a e^{4} a e a e a \end{aligned} \)