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#421 abaababa=baaba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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