#378 baabaaaba=aaabbaa

Original presentation

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

Modified presentation

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

Complete rewriting system

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

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