/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Applying context closure of depth 1 in the following form: System R over Sigma maps to { fold(xly) -> fold(xry) | l -> r in R, x,y in Sigma } over Sigma^2, where fold(a_1,...,a_n) = (a_1,a_2)...(a_{n-1}a_{n}) Remains to prove termination of the 45-rule system { [a, a] [a, b] [b, c] [c, a] -> [a, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, a] , [a, c] [c, b] [b, a] -> [a, a] [a, a] [a, a] [a, a] , [a, a] [a, a] -> [a, a] , [a, b] [b, a] -> [a, a] , [a, c] [c, a] -> [a, a] , [a, a] [a, b] [b, c] [c, b] -> [a, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, b] , [a, c] [c, b] [b, b] -> [a, a] [a, a] [a, a] [a, b] , [a, a] [a, b] -> [a, b] , [a, b] [b, b] -> [a, b] , [a, c] [c, b] -> [a, b] , [a, a] [a, b] [b, c] [c, c] -> [a, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, c] , [a, c] [c, b] [b, c] -> [a, a] [a, a] [a, a] [a, c] , [a, a] [a, c] -> [a, c] , [a, b] [b, c] -> [a, c] , [a, c] [c, c] -> [a, c] , [b, a] [a, b] [b, c] [c, a] -> [b, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, a] , [b, c] [c, b] [b, a] -> [b, a] [a, a] [a, a] [a, a] , [b, a] [a, a] -> [b, a] , [b, b] [b, a] -> [b, a] , [b, c] [c, a] -> [b, a] , [b, a] [a, b] [b, c] [c, b] -> [b, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, b] , [b, c] [c, b] [b, b] -> [b, a] [a, a] [a, a] [a, b] , [b, a] [a, b] -> [b, b] , [b, b] [b, b] -> [b, b] , [b, c] [c, b] -> [b, b] , [b, a] [a, b] [b, c] [c, c] -> [b, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, c] , [b, c] [c, b] [b, c] -> [b, a] [a, a] [a, a] [a, c] , [b, a] [a, c] -> [b, c] , [b, b] [b, c] -> [b, c] , [b, c] [c, c] -> [b, c] , [c, a] [a, b] [b, c] [c, a] -> [c, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, a] , [c, c] [c, b] [b, a] -> [c, a] [a, a] [a, a] [a, a] , [c, a] [a, a] -> [c, a] , [c, b] [b, a] -> [c, a] , [c, c] [c, a] -> [c, a] , [c, a] [a, b] [b, c] [c, b] -> [c, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, b] , [c, c] [c, b] [b, b] -> [c, a] [a, a] [a, a] [a, b] , [c, a] [a, b] -> [c, b] , [c, b] [b, b] -> [c, b] , [c, c] [c, b] -> [c, b] , [c, a] [a, b] [b, c] [c, c] -> [c, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, c] , [c, c] [c, b] [b, c] -> [c, a] [a, a] [a, a] [a, c] , [c, a] [a, c] -> [c, c] , [c, b] [b, c] -> [c, c] , [c, c] [c, c] -> [c, c] } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: [a, a] is interpreted by / \ | 1 0 | | 0 1 | \ / [a, b] is interpreted by / \ | 1 1 | | 0 1 | \ / [b, c] is interpreted by / \ | 1 0 | | 0 1 | \ / [c, a] is interpreted by / \ | 1 0 | | 0 1 | \ / [a, c] is interpreted by / \ | 1 0 | | 0 1 | \ / [c, c] is interpreted by / \ | 1 0 | | 0 1 | \ / [c, b] is interpreted by / \ | 1 1 | | 0 1 | \ / [b, b] is interpreted by / \ | 1 0 | | 0 1 | \ / [b, a] is interpreted by / \ | 1 0 | | 0 1 | \ / Remains to prove termination of the 33-rule system { [a, a] [a, b] [b, c] [c, a] -> [a, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, a] , [a, a] [a, a] -> [a, a] , [a, c] [c, a] -> [a, a] , [a, a] [a, b] [b, c] [c, b] -> [a, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, b] , [a, c] [c, b] [b, b] -> [a, a] [a, a] [a, a] [a, b] , [a, a] [a, b] -> [a, b] , [a, b] [b, b] -> [a, b] , [a, c] [c, b] -> [a, b] , [a, a] [a, b] [b, c] [c, c] -> [a, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, c] , [a, a] [a, c] -> [a, c] , [a, c] [c, c] -> [a, c] , [b, a] [a, b] [b, c] [c, a] -> [b, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, a] , [b, a] [a, a] -> [b, a] , [b, b] [b, a] -> [b, a] , [b, c] [c, a] -> [b, a] , [b, a] [a, b] [b, c] [c, b] -> [b, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, b] , [b, c] [c, b] [b, b] -> [b, a] [a, a] [a, a] [a, b] , [b, b] [b, b] -> [b, b] , [b, a] [a, b] [b, c] [c, c] -> [b, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, c] , [b, a] [a, c] -> [b, c] , [b, b] [b, c] -> [b, c] , [b, c] [c, c] -> [b, c] , [c, a] [a, b] [b, c] [c, a] -> [c, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, a] , [c, a] [a, a] -> [c, a] , [c, c] [c, a] -> [c, a] , [c, a] [a, b] [b, c] [c, b] -> [c, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, b] , [c, c] [c, b] [b, b] -> [c, a] [a, a] [a, a] [a, b] , [c, a] [a, b] -> [c, b] , [c, b] [b, b] -> [c, b] , [c, c] [c, b] -> [c, b] , [c, a] [a, b] [b, c] [c, c] -> [c, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, c] , [c, a] [a, c] -> [c, c] , [c, c] [c, c] -> [c, c] } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 3: [a, a] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [a, b] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [b, c] is interpreted by / \ | 3 0 3 | | 0 1 0 | | 0 0 0 | \ / [c, a] is interpreted by / \ | 1 1 0 | | 0 1 0 | | 0 1 0 | \ / [a, c] is interpreted by / \ | 1 0 1 | | 0 1 0 | | 0 0 0 | \ / [c, c] is interpreted by / \ | 1 0 1 | | 0 1 0 | | 0 1 0 | \ / [c, b] is interpreted by / \ | 1 1 0 | | 0 1 0 | | 0 0 0 | \ / [b, b] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [b, a] is interpreted by / \ | 3 0 0 | | 0 1 0 | | 0 0 0 | \ / Remains to prove termination of the 18-rule system { [a, a] [a, a] -> [a, a] , [a, a] [a, b] [b, c] [c, b] -> [a, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, b] , [a, a] [a, b] -> [a, b] , [a, b] [b, b] -> [a, b] , [a, a] [a, b] [b, c] [c, c] -> [a, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, c] , [a, a] [a, c] -> [a, c] , [b, a] [a, a] -> [b, a] , [b, b] [b, a] -> [b, a] , [b, a] [a, b] [b, c] [c, b] -> [b, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, b] , [b, b] [b, b] -> [b, b] , [b, a] [a, b] [b, c] [c, c] -> [b, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, c] , [b, a] [a, c] -> [b, c] , [b, b] [b, c] -> [b, c] , [c, a] [a, a] -> [c, a] , [c, c] [c, b] [b, b] -> [c, a] [a, a] [a, a] [a, b] , [c, a] [a, b] -> [c, b] , [c, b] [b, b] -> [c, b] , [c, c] [c, b] -> [c, b] } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 3: [a, a] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 1 | \ / [a, b] is interpreted by / \ | 3 0 0 | | 0 1 0 | | 0 1 0 | \ / [b, c] is interpreted by / \ | 1 1 0 | | 0 1 0 | | 0 0 0 | \ / [c, a] is interpreted by / \ | 1 0 1 | | 0 1 0 | | 0 0 0 | \ / [a, c] is interpreted by / \ | 1 1 0 | | 0 1 0 | | 0 0 0 | \ / [c, c] is interpreted by / \ | 1 1 0 | | 0 1 0 | | 0 0 0 | \ / [c, b] is interpreted by / \ | 3 0 0 | | 0 1 0 | | 0 0 0 | \ / [b, b] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [b, a] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / Remains to prove termination of the 16-rule system { [a, a] [a, a] -> [a, a] , [a, a] [a, b] [b, c] [c, b] -> [a, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, b] , [a, a] [a, b] -> [a, b] , [a, b] [b, b] -> [a, b] , [a, a] [a, b] [b, c] [c, c] -> [a, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, c] , [a, a] [a, c] -> [a, c] , [b, a] [a, a] -> [b, a] , [b, b] [b, a] -> [b, a] , [b, a] [a, b] [b, c] [c, b] -> [b, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, b] , [b, b] [b, b] -> [b, b] , [b, a] [a, b] [b, c] [c, c] -> [b, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, c] , [b, a] [a, c] -> [b, c] , [b, b] [b, c] -> [b, c] , [c, a] [a, a] -> [c, a] , [c, c] [c, b] [b, b] -> [c, a] [a, a] [a, a] [a, b] , [c, b] [b, b] -> [c, b] } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 3: [a, a] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 1 1 | \ / [a, b] is interpreted by / \ | 3 0 0 | | 0 1 0 | | 0 0 0 | \ / [b, c] is interpreted by / \ | 1 1 0 | | 0 1 0 | | 0 0 0 | \ / [c, a] is interpreted by / \ | 1 0 1 | | 0 1 0 | | 0 0 0 | \ / [a, c] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [c, c] is interpreted by / \ | 1 1 0 | | 0 1 0 | | 0 0 0 | \ / [c, b] is interpreted by / \ | 3 1 0 | | 0 1 0 | | 0 0 0 | \ / [b, b] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [b, a] is interpreted by / \ | 1 1 0 | | 0 1 0 | | 0 0 0 | \ / Remains to prove termination of the 15-rule system { [a, a] [a, a] -> [a, a] , [a, a] [a, b] [b, c] [c, b] -> [a, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, b] , [a, a] [a, b] -> [a, b] , [a, b] [b, b] -> [a, b] , [a, a] [a, b] [b, c] [c, c] -> [a, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, c] , [a, a] [a, c] -> [a, c] , [b, a] [a, a] -> [b, a] , [b, b] [b, a] -> [b, a] , [b, a] [a, b] [b, c] [c, b] -> [b, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, b] , [b, b] [b, b] -> [b, b] , [b, a] [a, b] [b, c] [c, c] -> [b, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, c] , [b, a] [a, c] -> [b, c] , [b, b] [b, c] -> [b, c] , [c, c] [c, b] [b, b] -> [c, a] [a, a] [a, a] [a, b] , [c, b] [b, b] -> [c, b] } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 3: [a, a] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [a, b] is interpreted by / \ | 3 0 0 | | 0 1 0 | | 0 0 0 | \ / [b, c] is interpreted by / \ | 2 2 0 | | 0 1 0 | | 0 0 0 | \ / [c, a] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [a, c] is interpreted by / \ | 1 1 0 | | 0 1 0 | | 0 0 0 | \ / [c, c] is interpreted by / \ | 1 1 0 | | 0 1 0 | | 0 0 0 | \ / [c, b] is interpreted by / \ | 3 0 0 | | 0 1 0 | | 0 0 0 | \ / [b, b] is interpreted by / \ | 1 0 1 | | 0 1 0 | | 0 1 0 | \ / [b, a] is interpreted by / \ | 2 0 0 | | 0 1 0 | | 0 0 0 | \ / Remains to prove termination of the 13-rule system { [a, a] [a, a] -> [a, a] , [a, a] [a, b] [b, c] [c, b] -> [a, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, b] , [a, a] [a, b] -> [a, b] , [a, b] [b, b] -> [a, b] , [a, a] [a, b] [b, c] [c, c] -> [a, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, c] , [a, a] [a, c] -> [a, c] , [b, a] [a, a] -> [b, a] , [b, b] [b, a] -> [b, a] , [b, a] [a, b] [b, c] [c, b] -> [b, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, b] , [b, a] [a, b] [b, c] [c, c] -> [b, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, c] , [b, a] [a, c] -> [b, c] , [b, b] [b, c] -> [b, c] , [c, b] [b, b] -> [c, b] } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: [a, a] is interpreted by / \ | 1 0 | | 0 1 | \ / [a, b] is interpreted by / \ | 1 1 | | 0 1 | \ / [b, c] is interpreted by / \ | 1 0 | | 0 1 | \ / [c, a] is interpreted by / \ | 1 0 | | 0 1 | \ / [a, c] is interpreted by / \ | 1 0 | | 0 1 | \ / [c, c] is interpreted by / \ | 1 0 | | 0 1 | \ / [c, b] is interpreted by / \ | 1 0 | | 0 1 | \ / [b, b] is interpreted by / \ | 1 0 | | 0 1 | \ / [b, a] is interpreted by / \ | 1 0 | | 0 1 | \ / Remains to prove termination of the 11-rule system { [a, a] [a, a] -> [a, a] , [a, a] [a, b] [b, c] [c, b] -> [a, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, b] , [a, a] [a, b] -> [a, b] , [a, b] [b, b] -> [a, b] , [a, a] [a, c] -> [a, c] , [b, a] [a, a] -> [b, a] , [b, b] [b, a] -> [b, a] , [b, a] [a, b] [b, c] [c, b] -> [b, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, b] , [b, a] [a, c] -> [b, c] , [b, b] [b, c] -> [b, c] , [c, b] [b, b] -> [c, b] } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 3: [a, a] is interpreted by / \ | 1 0 1 | | 0 1 0 | | 0 1 0 | \ / [a, b] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [b, c] is interpreted by / \ | 1 1 0 | | 0 1 0 | | 0 0 0 | \ / [c, a] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [a, c] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [c, c] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [c, b] is interpreted by / \ | 1 0 0 | | 0 2 0 | | 0 0 0 | \ / [b, b] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [b, a] is interpreted by / \ | 1 1 0 | | 0 1 0 | | 0 0 0 | \ / Remains to prove termination of the 10-rule system { [a, a] [a, b] [b, c] [c, b] -> [a, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, b] , [a, a] [a, b] -> [a, b] , [a, b] [b, b] -> [a, b] , [a, a] [a, c] -> [a, c] , [b, a] [a, a] -> [b, a] , [b, b] [b, a] -> [b, a] , [b, a] [a, b] [b, c] [c, b] -> [b, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, b] , [b, a] [a, c] -> [b, c] , [b, b] [b, c] -> [b, c] , [c, b] [b, b] -> [c, b] } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 3: [a, a] is interpreted by / \ | 1 0 1 | | 0 1 0 | | 0 1 0 | \ / [a, b] is interpreted by / \ | 3 0 0 | | 0 1 0 | | 0 1 0 | \ / [b, c] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [c, a] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [a, c] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [c, c] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [c, b] is interpreted by / \ | 1 1 0 | | 0 1 0 | | 0 0 0 | \ / [b, b] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [b, a] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / Remains to prove termination of the 8-rule system { [a, b] [b, b] -> [a, b] , [a, a] [a, c] -> [a, c] , [b, a] [a, a] -> [b, a] , [b, b] [b, a] -> [b, a] , [b, a] [a, b] [b, c] [c, b] -> [b, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, b] , [b, a] [a, c] -> [b, c] , [b, b] [b, c] -> [b, c] , [c, b] [b, b] -> [c, b] } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: [a, a] is interpreted by / \ | 1 0 | | 0 1 | \ / [a, b] is interpreted by / \ | 1 0 | | 0 1 | \ / [b, c] is interpreted by / \ | 1 0 | | 0 1 | \ / [c, a] is interpreted by / \ | 1 0 | | 0 1 | \ / [a, c] is interpreted by / \ | 1 1 | | 0 1 | \ / [c, c] is interpreted by / \ | 1 0 | | 0 1 | \ / [c, b] is interpreted by / \ | 1 0 | | 0 1 | \ / [b, b] is interpreted by / \ | 1 0 | | 0 1 | \ / [b, a] is interpreted by / \ | 1 0 | | 0 1 | \ / Remains to prove termination of the 7-rule system { [a, b] [b, b] -> [a, b] , [a, a] [a, c] -> [a, c] , [b, a] [a, a] -> [b, a] , [b, b] [b, a] -> [b, a] , [b, a] [a, b] [b, c] [c, b] -> [b, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, b] , [b, b] [b, c] -> [b, c] , [c, b] [b, b] -> [c, b] } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 3: [a, a] is interpreted by / \ | 1 0 1 | | 0 1 0 | | 0 0 1 | \ / [a, b] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [b, c] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [c, a] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [a, c] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 1 0 | \ / [c, c] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [c, b] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [b, b] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [b, a] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / Remains to prove termination of the 6-rule system { [a, b] [b, b] -> [a, b] , [b, a] [a, a] -> [b, a] , [b, b] [b, a] -> [b, a] , [b, a] [a, b] [b, c] [c, b] -> [b, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, b] , [b, b] [b, c] -> [b, c] , [c, b] [b, b] -> [c, b] } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 3: [a, a] is interpreted by / \ | 1 0 1 | | 0 1 0 | | 0 1 0 | \ / [a, b] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 1 0 | \ / [b, c] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [c, a] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [a, c] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [c, c] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [c, b] is interpreted by / \ | 1 0 0 | | 0 2 0 | | 0 0 0 | \ / [b, b] is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / [b, a] is interpreted by / \ | 1 1 1 | | 0 1 0 | | 0 0 0 | \ / Remains to prove termination of the 5-rule system { [a, b] [b, b] -> [a, b] , [b, b] [b, a] -> [b, a] , [b, a] [a, b] [b, c] [c, b] -> [b, c] [c, c] [c, c] [c, b] [b, b] [b, b] [b, a] [a, a] [a, a] [a, b] , [b, b] [b, c] -> [b, c] , [c, b] [b, b] -> [c, b] } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 5: [a, a] is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / [a, b] is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 1 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / [b, c] is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 1 | | 0 0 0 0 0 | \ / [c, a] is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / [a, c] is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / [c, c] is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / [c, b] is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | | 0 1 0 0 0 | \ / [b, b] is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 1 0 | | 0 0 0 0 0 | \ / [b, a] is interpreted by / \ | 1 0 1 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / Remains to prove termination of the 4-rule system { [a, b] [b, b] -> [a, b] , [b, b] [b, a] -> [b, a] , [b, b] [b, c] -> [b, c] , [c, b] [b, b] -> [c, b] } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: [a, a] is interpreted by / \ | 1 0 | | 0 1 | \ / [a, b] is interpreted by / \ | 1 0 | | 0 1 | \ / [b, c] is interpreted by / \ | 1 0 | | 0 1 | \ / [c, a] is interpreted by / \ | 1 0 | | 0 1 | \ / [a, c] is interpreted by / \ | 1 0 | | 0 1 | \ / [c, c] is interpreted by / \ | 1 0 | | 0 1 | \ / [c, b] is interpreted by / \ | 1 0 | | 0 1 | \ / [b, b] is interpreted by / \ | 1 1 | | 0 1 | \ / [b, a] is interpreted by / \ | 1 0 | | 0 1 | \ / Remains to prove termination of the 0-rule system { } The system is trivially terminating.