#394 baaaababa=abbaa

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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