#010 baaaaabbba=abaa

Original presentation

\( \left\langle a, b \;\middle|\; b a^{5} b^{3} a = a b a^{2} \right\rangle \)

Modified presentation

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

Complete rewriting system

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

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