#372 abbabbaaba=babba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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