#433 baaaabaaba=abbaa

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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