#100 aaabababaa=baaaaba

Original presentation

\( \left\langle a, b \;\middle|\; a^{3} b a b a b a^{2} = b a^{4} b a \right\rangle \)

Modified presentation

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

Complete rewriting system

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

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