#005 baaaaabbba=aaaaa

Original presentation

\( \left\langle a, b \;\middle|\; b a^{5} b^{3} a = a^{5} \right\rangle \)

Modified presentation

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

Complete rewriting system

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

\( \begin{aligned} c &\rightarrow a^{2} \\ b a^{4} d a &\rightarrow a^{5} \\ a b^{3} &\rightarrow d \\ a b^{2} a^{5} &\rightarrow d a^{4} d a \\ b a^{4} d^{2} &\rightarrow a^{4} d \\ a b^{2} a^{4} d &\rightarrow d a^{4} d^{2} \\ a b a^{4} d &\rightarrow d a^{4} d^{3} \\ a b a^{5} &\rightarrow d a^{4} d^{2} a \\ d a^{4} d^{4} &\rightarrow a^{5} d \\ d a^{4} d^{3} a &\rightarrow a^{6} \\ b a^{9} d &\rightarrow a^{8} d^{4} \\ b a^{10} &\rightarrow a^{8} d^{3} a \\ d a^{4} d^{2} a^{5} d &\rightarrow a^{9} d^{4} \\ d a^{4} d^{2} a^{6} &\rightarrow a^{9} d^{3} a \end{aligned} \)