#440 ababbabba=baba

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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