#302 babaabbbba=aba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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