#312 baaaabbaba=abaaa

Original presentation

\( \left\langle a, b \;\middle|\; b a^{4} b^{2} a b a = a b a^{3} \right\rangle \)

Modified presentation

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

Complete rewriting system

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

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