#161 baaaababba=abaaa

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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