#104 babaaabbba=aba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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