#166 abbabaaaba=baba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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