#309 baaaaababa=abbaa

Original presentation

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

Modified presentation

\( \left\langle a, b, c, d \;\middle|\; b c a b a b a = a b^{2} a^{2},\; c = a^{4},\; d = b a b c \right\rangle \)

Complete rewriting system

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

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