#136 abaaababa=baaaba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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