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#012 bababbba=aba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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