#423 baaaaabbaa=abaaaaa

Original presentation

\( \left\langle a, b \;\middle|\; b a^{5} b^{2} a^{2} = a b a^{5} \right\rangle \)

Modified presentation

\( \left\langle a, b, c, d \;\middle|\; b c d a^{2} = a b c a^{2},\; c = a^{3},\; d = a^{2} b^{2} \right\rangle \)

Complete rewriting system

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

\( \begin{aligned} a^{3} &\rightarrow c \\ a^{2} b^{2} &\rightarrow d \\ a b c a^{2} &\rightarrow b c d a^{2} \\ a c &\rightarrow c a \\ a d &\rightarrow c b^{2} \\ a b c^{2} &\rightarrow b c d c \\ a b c d &\rightarrow b c d^{2} \\ c b c a^{2} &\rightarrow b c d^{3} a^{2} \\ c b c^{2} &\rightarrow b c d^{3} c \\ c b c d &\rightarrow b c d^{4} \\ c b^{2} c d^{5} a^{2} &\rightarrow b c d^{3} b c d^{2} a^{2} \\ c b^{2} c d^{5} c &\rightarrow b c d^{3} b c d^{2} c \\ c b^{2} c d^{6} &\rightarrow b c d^{3} b c d^{3} \\ a b^{2} c d^{6} &\rightarrow b c d b c d^{3} \\ a b^{2} c d^{5} c &\rightarrow b c d b c d^{2} c \\ a b^{2} c d^{5} a^{2} &\rightarrow b c d b c d^{2} a^{2} \\ b c d^{2} b c d^{2} c &\rightarrow d c d^{5} c \\ b c d^{2} b c d^{3} &\rightarrow d c d^{6} \\ b c d^{3} c d^{5} c &\rightarrow d c d^{5} b c d^{2} c \\ b c d^{3} c d^{6} &\rightarrow d c d^{5} b c d^{3} \\ c b^{2} c d^{5} b c d^{3} &\rightarrow b c d^{4} c d^{6} \\ c b^{2} c d^{5} b c d^{2} c &\rightarrow b c d^{4} c d^{5} c \\ a b^{2} c d^{5} b c d^{2} c &\rightarrow b c d^{2} c d^{5} c \\ a b^{2} c d^{5} b c d^{3} &\rightarrow b c d^{2} c d^{6} \\ a^{2} b d c d^{5} c &\rightarrow d c d^{2} b c d^{2} c \\ a^{2} b d c d^{6} &\rightarrow d c d^{2} b c d^{3} \\ a^{2} b d c d^{5} b c d^{2} c &\rightarrow d c d^{3} c d^{5} c \\ a^{2} b d c d^{5} b c d^{3} &\rightarrow d c d^{3} c d^{6} \\ b c d^{2} b c d^{2} a^{2} &\rightarrow d c d^{5} a^{2} \\ c d c d^{5} c &\rightarrow b c d^{5} b c d^{2} c \\ c d c d^{6} &\rightarrow b c d^{5} b c d^{3} \\ c d c d^{5} b c d^{2} a^{2} &\rightarrow b c d^{6} c d^{5} a^{2} \\ c d c d^{5} b c d^{2} c &\rightarrow b c d^{6} c d^{5} c \\ c d c d^{5} b c d^{3} &\rightarrow b c d^{6} c d^{6} \\ b c d^{3} c d^{5} b c d^{2} c &\rightarrow d c d^{6} c d^{5} c \\ b c d^{3} c d^{5} b c d^{3} &\rightarrow d c d^{6} c d^{6} \\ a^{2} b d c d^{5} b c d^{2} a^{2} &\rightarrow d c d^{3} c d^{5} a^{2} \\ b c d^{3} c d^{5} b c d^{2} a^{2} &\rightarrow d c d^{6} c d^{5} a^{2} \\ b c d^{3} c d^{5} a^{2} &\rightarrow d c d^{5} b c d^{2} a^{2} \\ a b^{2} c d^{5} b c d^{2} a^{2} &\rightarrow b c d^{2} c d^{5} a^{2} \\ c b^{2} c d^{5} b c d^{2} a^{2} &\rightarrow b c d^{4} c d^{5} a^{2} \\ a^{2} b d c d^{5} a^{2} &\rightarrow d c d^{2} b c d^{2} a^{2} \\ c d c d^{5} a^{2} &\rightarrow b c d^{5} b c d^{2} a^{2} \end{aligned} \)