#444 abbababaa=bababa

Original presentation

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

Modified presentation

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

Complete rewriting system

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

\( \begin{aligned} b a &\rightarrow c \\ c a e a &\rightarrow a e \\ b c^{2} a e &\rightarrow e c^{3} \\ b c a e &\rightarrow e c^{3} a \\ d &\rightarrow a e a^{3} \\ a e a^{3} c^{2} a e &\rightarrow e a^{2} e a \\ c e &\rightarrow e c^{3} a^{2} \\ a e a^{4} e &\rightarrow e a^{2} e a^{3} \\ a e a^{3} c^{3} &\rightarrow e a \\ a^{2} e a^{4} &\rightarrow c^{3} \\ a e a^{5} e a^{3} &\rightarrow e \\ b c^{3} &\rightarrow a e a^{3} \\ a e a c^{3} &\rightarrow e c^{3} a^{2} c^{3} a^{3} \\ b e &\rightarrow e c^{3} a^{7} e a^{3} \\ a e a^{3} e &\rightarrow e c^{3} a^{4} e a^{3} \\ a e a^{2} c^{3} &\rightarrow e c^{3} a^{3} \\ a e c^{3} &\rightarrow e c^{3} a^{2} c^{3} a^{2} c^{3} a^{3} \\ a e a^{2} c^{2} a e &\rightarrow e c^{3} a^{4} e a \\ a e^{2} &\rightarrow e c^{3} a^{2} c^{3} a^{2} c^{3} a^{2} c^{3} a^{4} e a^{3} \\ a e a c^{2} a e &\rightarrow e c^{3} a^{2} c^{3} a^{4} e a \\ a e a^{2} e &\rightarrow e c^{3} a^{2} c^{3} a^{4} e a^{3} \\ a e a^{3} c a e &\rightarrow e a^{2} e a^{2} \\ a e a e &\rightarrow e c^{3} a^{2} c^{3} a^{2} c^{3} a^{4} e a^{3} \\ a e c^{2} a e &\rightarrow e c^{3} a^{2} c^{3} a^{2} c^{3} a^{4} e a \\ a e a c a e &\rightarrow e c^{3} a^{2} c^{3} a^{4} e a^{2} \\ a e a^{2} c a e &\rightarrow e c^{3} a^{4} e a^{2} \\ a e c a e &\rightarrow e c^{3} a^{2} c^{3} a^{2} c^{3} a^{4} e a^{2} \end{aligned} \)