/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/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 27-rule system { [a, a] [a, a] -> [a, a] , [a, a] [a, b] [b, a] -> [a, c] [c, b] [b, c] [c, b] [b, c] [c, a] [a, a] , [a, c] [c, c] [c, a] -> [a, a] [a, a] , [a, a] [a, b] -> [a, b] , [a, a] [a, b] [b, b] -> [a, c] [c, b] [b, c] [c, b] [b, c] [c, a] [a, b] , [a, c] [c, c] [c, b] -> [a, a] [a, b] , [a, a] [a, c] -> [a, c] , [a, a] [a, b] [b, c] -> [a, c] [c, b] [b, c] [c, b] [b, c] [c, a] [a, c] , [a, c] [c, c] [c, c] -> [a, a] [a, c] , [b, a] [a, a] -> [b, a] , [b, a] [a, b] [b, a] -> [b, c] [c, b] [b, c] [c, b] [b, c] [c, a] [a, a] , [b, c] [c, c] [c, a] -> [b, a] [a, a] , [b, a] [a, b] -> [b, b] , [b, a] [a, b] [b, b] -> [b, c] [c, b] [b, c] [c, b] [b, c] [c, a] [a, b] , [b, c] [c, c] [c, b] -> [b, a] [a, b] , [b, a] [a, c] -> [b, c] , [b, a] [a, b] [b, c] -> [b, c] [c, b] [b, c] [c, b] [b, c] [c, a] [a, c] , [b, c] [c, c] [c, c] -> [b, a] [a, c] , [c, a] [a, a] -> [c, a] , [c, a] [a, b] [b, a] -> [c, c] [c, b] [b, c] [c, b] [b, c] [c, a] [a, a] , [c, c] [c, c] [c, a] -> [c, a] [a, a] , [c, a] [a, b] -> [c, b] , [c, a] [a, b] [b, b] -> [c, c] [c, b] [b, c] [c, b] [b, c] [c, a] [a, b] , [c, c] [c, c] [c, b] -> [c, a] [a, b] , [c, a] [a, c] -> [c, c] , [c, a] [a, b] [b, c] -> [c, c] [c, b] [b, c] [c, b] [b, c] [c, a] [a, c] , [c, c] [c, c] [c, c] -> [c, a] [a, 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, a] is interpreted by / \ | 1 0 | | 0 1 | \ / [a, c] is interpreted by / \ | 1 0 | | 0 1 | \ / [c, b] is interpreted by / \ | 1 0 | | 0 1 | \ / [b, c] is interpreted by / \ | 1 0 | | 0 1 | \ / [c, a] is interpreted by / \ | 1 1 | | 0 1 | \ / [c, c] is interpreted by / \ | 1 1 | | 0 1 | \ / [b, b] is interpreted by / \ | 1 1 | | 0 1 | \ / Remains to prove termination of the 20-rule system { [a, a] [a, a] -> [a, a] , [a, a] [a, b] [b, a] -> [a, c] [c, b] [b, c] [c, b] [b, c] [c, a] [a, a] , [a, a] [a, b] -> [a, b] , [a, a] [a, b] [b, b] -> [a, c] [c, b] [b, c] [c, b] [b, c] [c, a] [a, b] , [a, c] [c, c] [c, b] -> [a, a] [a, b] , [a, a] [a, c] -> [a, c] , [a, a] [a, b] [b, c] -> [a, c] [c, b] [b, c] [c, b] [b, c] [c, a] [a, c] , [b, a] [a, a] -> [b, a] , [b, a] [a, b] [b, a] -> [b, c] [c, b] [b, c] [c, b] [b, c] [c, a] [a, a] , [b, a] [a, b] -> [b, b] , [b, a] [a, b] [b, b] -> [b, c] [c, b] [b, c] [c, b] [b, c] [c, a] [a, b] , [b, c] [c, c] [c, b] -> [b, a] [a, b] , [b, a] [a, c] -> [b, c] , [b, a] [a, b] [b, c] -> [b, c] [c, b] [b, c] [c, b] [b, c] [c, a] [a, c] , [c, a] [a, a] -> [c, a] , [c, a] [a, b] [b, a] -> [c, c] [c, b] [b, c] [c, b] [b, c] [c, a] [a, a] , [c, a] [a, b] [b, b] -> [c, c] [c, b] [b, c] [c, b] [b, c] [c, a] [a, b] , [c, c] [c, c] [c, b] -> [c, a] [a, b] , [c, a] [a, c] -> [c, c] , [c, a] [a, b] [b, c] -> [c, c] [c, b] [b, c] [c, b] [b, c] [c, a] [a, c] } The system was reversed. Remains to prove termination of the 20-rule system { [a, a] [a, a] -> [a, a] , [b, a] [a, b] [a, a] -> [a, a] [c, a] [b, c] [c, b] [b, c] [c, b] [a, c] , [a, b] [a, a] -> [a, b] , [b, b] [a, b] [a, a] -> [a, b] [c, a] [b, c] [c, b] [b, c] [c, b] [a, c] , [c, b] [c, c] [a, c] -> [a, b] [a, a] , [a, c] [a, a] -> [a, c] , [b, c] [a, b] [a, a] -> [a, c] [c, a] [b, c] [c, b] [b, c] [c, b] [a, c] , [a, a] [b, a] -> [b, a] , [b, a] [a, b] [b, a] -> [a, a] [c, a] [b, c] [c, b] [b, c] [c, b] [b, c] , [a, b] [b, a] -> [b, b] , [b, b] [a, b] [b, a] -> [a, b] [c, a] [b, c] [c, b] [b, c] [c, b] [b, c] , [c, b] [c, c] [b, c] -> [a, b] [b, a] , [a, c] [b, a] -> [b, c] , [b, c] [a, b] [b, a] -> [a, c] [c, a] [b, c] [c, b] [b, c] [c, b] [b, c] , [a, a] [c, a] -> [c, a] , [b, a] [a, b] [c, a] -> [a, a] [c, a] [b, c] [c, b] [b, c] [c, b] [c, c] , [b, b] [a, b] [c, a] -> [a, b] [c, a] [b, c] [c, b] [b, c] [c, b] [c, c] , [c, b] [c, c] [c, c] -> [a, b] [c, a] , [a, c] [c, a] -> [c, c] , [b, c] [a, b] [c, a] -> [a, c] [c, a] [b, c] [c, b] [b, c] [c, b] [c, c] } The dependency pairs transformation was applied. Remains to prove termination of the 82-rule system { ([a, a],true) ([a, a],false) -> ([a, a],true) , ([b, a],true) ([a, b],false) ([a, a],false) -> ([a, a],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([b, a],true) ([a, b],false) ([a, a],false) -> ([b, c],true) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([b, a],true) ([a, b],false) ([a, a],false) -> ([c, b],true) ([b, c],false) ([c, b],false) ([a, c],false) , ([b, a],true) ([a, b],false) ([a, a],false) -> ([b, c],true) ([c, b],false) ([a, c],false) , ([b, a],true) ([a, b],false) ([a, a],false) -> ([c, b],true) ([a, c],false) , ([b, a],true) ([a, b],false) ([a, a],false) -> ([a, c],true) , ([a, b],true) ([a, a],false) -> ([a, b],true) , ([b, b],true) ([a, b],false) ([a, a],false) -> ([a, b],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([b, b],true) ([a, b],false) ([a, a],false) -> ([b, c],true) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([b, b],true) ([a, b],false) ([a, a],false) -> ([c, b],true) ([b, c],false) ([c, b],false) ([a, c],false) , ([b, b],true) ([a, b],false) ([a, a],false) -> ([b, c],true) ([c, b],false) ([a, c],false) , ([b, b],true) ([a, b],false) ([a, a],false) -> ([c, b],true) ([a, c],false) , ([b, b],true) ([a, b],false) ([a, a],false) -> ([a, c],true) , ([c, b],true) ([c, c],false) ([a, c],false) -> ([a, b],true) ([a, a],false) , ([c, b],true) ([c, c],false) ([a, c],false) -> ([a, a],true) , ([a, c],true) ([a, a],false) -> ([a, c],true) , ([b, c],true) ([a, b],false) ([a, a],false) -> ([a, c],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([b, c],true) ([a, b],false) ([a, a],false) -> ([b, c],true) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([b, c],true) ([a, b],false) ([a, a],false) -> ([c, b],true) ([b, c],false) ([c, b],false) ([a, c],false) , ([b, c],true) ([a, b],false) ([a, a],false) -> ([b, c],true) ([c, b],false) ([a, c],false) , ([b, c],true) ([a, b],false) ([a, a],false) -> ([c, b],true) ([a, c],false) , ([b, c],true) ([a, b],false) ([a, a],false) -> ([a, c],true) , ([a, a],true) ([b, a],false) -> ([b, a],true) , ([b, a],true) ([a, b],false) ([b, a],false) -> ([a, a],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([b, a],true) ([a, b],false) ([b, a],false) -> ([b, c],true) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([b, a],true) ([a, b],false) ([b, a],false) -> ([c, b],true) ([b, c],false) ([c, b],false) ([b, c],false) , ([b, a],true) ([a, b],false) ([b, a],false) -> ([b, c],true) ([c, b],false) ([b, c],false) , ([b, a],true) ([a, b],false) ([b, a],false) -> ([c, b],true) ([b, c],false) , ([b, a],true) ([a, b],false) ([b, a],false) -> ([b, c],true) , ([a, b],true) ([b, a],false) -> ([b, b],true) , ([b, b],true) ([a, b],false) ([b, a],false) -> ([a, b],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([b, b],true) ([a, b],false) ([b, a],false) -> ([b, c],true) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([b, b],true) ([a, b],false) ([b, a],false) -> ([c, b],true) ([b, c],false) ([c, b],false) ([b, c],false) , ([b, b],true) ([a, b],false) ([b, a],false) -> ([b, c],true) ([c, b],false) ([b, c],false) , ([b, b],true) ([a, b],false) ([b, a],false) -> ([c, b],true) ([b, c],false) , ([b, b],true) ([a, b],false) ([b, a],false) -> ([b, c],true) , ([c, b],true) ([c, c],false) ([b, c],false) -> ([a, b],true) ([b, a],false) , ([c, b],true) ([c, c],false) ([b, c],false) -> ([b, a],true) , ([a, c],true) ([b, a],false) -> ([b, c],true) , ([b, c],true) ([a, b],false) ([b, a],false) -> ([a, c],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([b, c],true) ([a, b],false) ([b, a],false) -> ([b, c],true) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([b, c],true) ([a, b],false) ([b, a],false) -> ([c, b],true) ([b, c],false) ([c, b],false) ([b, c],false) , ([b, c],true) ([a, b],false) ([b, a],false) -> ([b, c],true) ([c, b],false) ([b, c],false) , ([b, c],true) ([a, b],false) ([b, a],false) -> ([c, b],true) ([b, c],false) , ([b, c],true) ([a, b],false) ([b, a],false) -> ([b, c],true) , ([b, a],true) ([a, b],false) ([c, a],false) -> ([a, a],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, a],true) ([a, b],false) ([c, a],false) -> ([b, c],true) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, a],true) ([a, b],false) ([c, a],false) -> ([c, b],true) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, a],true) ([a, b],false) ([c, a],false) -> ([b, c],true) ([c, b],false) ([c, c],false) , ([b, a],true) ([a, b],false) ([c, a],false) -> ([c, b],true) ([c, c],false) , ([b, b],true) ([a, b],false) ([c, a],false) -> ([a, b],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, b],true) ([a, b],false) ([c, a],false) -> ([b, c],true) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, b],true) ([a, b],false) ([c, a],false) -> ([c, b],true) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, b],true) ([a, b],false) ([c, a],false) -> ([b, c],true) ([c, b],false) ([c, c],false) , ([b, b],true) ([a, b],false) ([c, a],false) -> ([c, b],true) ([c, c],false) , ([c, b],true) ([c, c],false) ([c, c],false) -> ([a, b],true) ([c, a],false) , ([b, c],true) ([a, b],false) ([c, a],false) -> ([a, c],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, c],true) ([a, b],false) ([c, a],false) -> ([b, c],true) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, c],true) ([a, b],false) ([c, a],false) -> ([c, b],true) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, c],true) ([a, b],false) ([c, a],false) -> ([b, c],true) ([c, b],false) ([c, c],false) , ([b, c],true) ([a, b],false) ([c, a],false) -> ([c, b],true) ([c, c],false) , ([a, a],false) ([a, a],false) ->= ([a, a],false) , ([b, a],false) ([a, b],false) ([a, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, b],false) ([a, a],false) ->= ([a, b],false) , ([b, b],false) ([a, b],false) ([a, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([c, b],false) ([c, c],false) ([a, c],false) ->= ([a, b],false) ([a, a],false) , ([a, c],false) ([a, a],false) ->= ([a, c],false) , ([b, c],false) ([a, b],false) ([a, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, a],false) ([b, a],false) ->= ([b, a],false) , ([b, a],false) ([a, b],false) ([b, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, b],false) ([b, a],false) ->= ([b, b],false) , ([b, b],false) ([a, b],false) ([b, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([c, b],false) ([c, c],false) ([b, c],false) ->= ([a, b],false) ([b, a],false) , ([a, c],false) ([b, a],false) ->= ([b, c],false) , ([b, c],false) ([a, b],false) ([b, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, a],false) ([c, a],false) ->= ([c, a],false) , ([b, a],false) ([a, b],false) ([c, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, b],false) ([a, b],false) ([c, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([c, b],false) ([c, c],false) ([c, c],false) ->= ([a, b],false) ([c, a],false) , ([a, c],false) ([c, a],false) ->= ([c, c],false) , ([b, c],false) ([a, b],false) ([c, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: ([a, a],true) is interpreted by / \ | 1 1 | | 0 1 | \ / ([a, a],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, a],true) is interpreted by / \ | 1 1 | | 0 1 | \ / ([a, b],false) is interpreted by / \ | 1 1 | | 0 1 | \ / ([c, a],false) is interpreted by / \ | 1 1 | | 0 1 | \ / ([b, c],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([c, b],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([a, c],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, c],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([c, b],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([a, c],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([a, b],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, b],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([c, c],false) is interpreted by / \ | 1 1 | | 0 1 | \ / ([b, a],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, b],false) is interpreted by / \ | 1 1 | | 0 1 | \ / Remains to prove termination of the 37-rule system { ([a, a],true) ([a, a],false) -> ([a, a],true) , ([b, a],true) ([a, b],false) ([a, a],false) -> ([a, a],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, b],true) ([a, a],false) -> ([a, b],true) , ([b, b],true) ([a, b],false) ([a, a],false) -> ([a, b],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([c, b],true) ([c, c],false) ([a, c],false) -> ([a, a],true) , ([a, c],true) ([a, a],false) -> ([a, c],true) , ([b, c],true) ([a, b],false) ([a, a],false) -> ([a, c],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, a],true) ([b, a],false) -> ([b, a],true) , ([b, a],true) ([a, b],false) ([b, a],false) -> ([a, a],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, b],true) ([b, a],false) -> ([b, b],true) , ([b, b],true) ([a, b],false) ([b, a],false) -> ([a, b],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([c, b],true) ([c, c],false) ([b, c],false) -> ([b, a],true) , ([a, c],true) ([b, a],false) -> ([b, c],true) , ([b, c],true) ([a, b],false) ([b, a],false) -> ([a, c],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([b, a],true) ([a, b],false) ([c, a],false) -> ([a, a],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, b],true) ([a, b],false) ([c, a],false) -> ([a, b],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, c],true) ([a, b],false) ([c, a],false) -> ([a, c],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([a, a],false) ([a, a],false) ->= ([a, a],false) , ([b, a],false) ([a, b],false) ([a, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, b],false) ([a, a],false) ->= ([a, b],false) , ([b, b],false) ([a, b],false) ([a, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([c, b],false) ([c, c],false) ([a, c],false) ->= ([a, b],false) ([a, a],false) , ([a, c],false) ([a, a],false) ->= ([a, c],false) , ([b, c],false) ([a, b],false) ([a, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, a],false) ([b, a],false) ->= ([b, a],false) , ([b, a],false) ([a, b],false) ([b, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, b],false) ([b, a],false) ->= ([b, b],false) , ([b, b],false) ([a, b],false) ([b, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([c, b],false) ([c, c],false) ([b, c],false) ->= ([a, b],false) ([b, a],false) , ([a, c],false) ([b, a],false) ->= ([b, c],false) , ([b, c],false) ([a, b],false) ([b, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, a],false) ([c, a],false) ->= ([c, a],false) , ([b, a],false) ([a, b],false) ([c, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, b],false) ([a, b],false) ([c, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([c, b],false) ([c, c],false) ([c, c],false) ->= ([a, b],false) ([c, a],false) , ([a, c],false) ([c, a],false) ->= ([c, c],false) , ([b, c],false) ([a, b],false) ([c, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: ([a, a],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([a, a],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, a],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([a, b],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([c, a],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, c],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([c, b],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([a, c],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, c],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([c, b],true) is interpreted by / \ | 1 1 | | 0 1 | \ / ([a, c],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([a, b],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, b],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([c, c],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, a],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, b],false) is interpreted by / \ | 1 0 | | 0 1 | \ / Remains to prove termination of the 35-rule system { ([a, a],true) ([a, a],false) -> ([a, a],true) , ([b, a],true) ([a, b],false) ([a, a],false) -> ([a, a],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, b],true) ([a, a],false) -> ([a, b],true) , ([b, b],true) ([a, b],false) ([a, a],false) -> ([a, b],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, c],true) ([a, a],false) -> ([a, c],true) , ([b, c],true) ([a, b],false) ([a, a],false) -> ([a, c],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, a],true) ([b, a],false) -> ([b, a],true) , ([b, a],true) ([a, b],false) ([b, a],false) -> ([a, a],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, b],true) ([b, a],false) -> ([b, b],true) , ([b, b],true) ([a, b],false) ([b, a],false) -> ([a, b],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, c],true) ([b, a],false) -> ([b, c],true) , ([b, c],true) ([a, b],false) ([b, a],false) -> ([a, c],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([b, a],true) ([a, b],false) ([c, a],false) -> ([a, a],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, b],true) ([a, b],false) ([c, a],false) -> ([a, b],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, c],true) ([a, b],false) ([c, a],false) -> ([a, c],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([a, a],false) ([a, a],false) ->= ([a, a],false) , ([b, a],false) ([a, b],false) ([a, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, b],false) ([a, a],false) ->= ([a, b],false) , ([b, b],false) ([a, b],false) ([a, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([c, b],false) ([c, c],false) ([a, c],false) ->= ([a, b],false) ([a, a],false) , ([a, c],false) ([a, a],false) ->= ([a, c],false) , ([b, c],false) ([a, b],false) ([a, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, a],false) ([b, a],false) ->= ([b, a],false) , ([b, a],false) ([a, b],false) ([b, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, b],false) ([b, a],false) ->= ([b, b],false) , ([b, b],false) ([a, b],false) ([b, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([c, b],false) ([c, c],false) ([b, c],false) ->= ([a, b],false) ([b, a],false) , ([a, c],false) ([b, a],false) ->= ([b, c],false) , ([b, c],false) ([a, b],false) ([b, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, a],false) ([c, a],false) ->= ([c, a],false) , ([b, a],false) ([a, b],false) ([c, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, b],false) ([a, b],false) ([c, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([c, b],false) ([c, c],false) ([c, c],false) ->= ([a, b],false) ([c, a],false) , ([a, c],false) ([c, a],false) ->= ([c, c],false) , ([b, c],false) ([a, b],false) ([c, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 3: ([a, a],true) is interpreted by / \ | 1 0 1 | | 0 1 0 | | 0 0 0 | \ / ([a, a],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 1 | \ / ([b, a],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, b],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, a],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([b, c],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, b],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, c],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([b, c],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, b],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, c],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, b],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([b, b],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, c],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([b, a],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 1 0 | \ / ([b, b],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / Remains to prove termination of the 34-rule system { ([a, a],true) ([a, a],false) -> ([a, a],true) , ([b, a],true) ([a, b],false) ([a, a],false) -> ([a, a],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, b],true) ([a, a],false) -> ([a, b],true) , ([b, b],true) ([a, b],false) ([a, a],false) -> ([a, b],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, c],true) ([a, a],false) -> ([a, c],true) , ([b, c],true) ([a, b],false) ([a, a],false) -> ([a, c],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([b, a],true) ([a, b],false) ([b, a],false) -> ([a, a],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, b],true) ([b, a],false) -> ([b, b],true) , ([b, b],true) ([a, b],false) ([b, a],false) -> ([a, b],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, c],true) ([b, a],false) -> ([b, c],true) , ([b, c],true) ([a, b],false) ([b, a],false) -> ([a, c],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([b, a],true) ([a, b],false) ([c, a],false) -> ([a, a],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, b],true) ([a, b],false) ([c, a],false) -> ([a, b],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, c],true) ([a, b],false) ([c, a],false) -> ([a, c],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([a, a],false) ([a, a],false) ->= ([a, a],false) , ([b, a],false) ([a, b],false) ([a, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, b],false) ([a, a],false) ->= ([a, b],false) , ([b, b],false) ([a, b],false) ([a, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([c, b],false) ([c, c],false) ([a, c],false) ->= ([a, b],false) ([a, a],false) , ([a, c],false) ([a, a],false) ->= ([a, c],false) , ([b, c],false) ([a, b],false) ([a, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, a],false) ([b, a],false) ->= ([b, a],false) , ([b, a],false) ([a, b],false) ([b, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, b],false) ([b, a],false) ->= ([b, b],false) , ([b, b],false) ([a, b],false) ([b, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([c, b],false) ([c, c],false) ([b, c],false) ->= ([a, b],false) ([b, a],false) , ([a, c],false) ([b, a],false) ->= ([b, c],false) , ([b, c],false) ([a, b],false) ([b, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, a],false) ([c, a],false) ->= ([c, a],false) , ([b, a],false) ([a, b],false) ([c, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, b],false) ([a, b],false) ([c, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([c, b],false) ([c, c],false) ([c, c],false) ->= ([a, b],false) ([c, a],false) , ([a, c],false) ([c, a],false) ->= ([c, c],false) , ([b, c],false) ([a, b],false) ([c, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: ([a, a],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([a, a],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, a],true) is interpreted by / \ | 1 1 | | 0 1 | \ / ([a, b],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([c, a],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, c],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([c, b],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([a, c],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, c],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([c, b],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([a, c],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([a, b],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, b],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([c, c],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, a],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, b],false) is interpreted by / \ | 1 0 | | 0 1 | \ / Remains to prove termination of the 31-rule system { ([a, a],true) ([a, a],false) -> ([a, a],true) , ([a, b],true) ([a, a],false) -> ([a, b],true) , ([b, b],true) ([a, b],false) ([a, a],false) -> ([a, b],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, c],true) ([a, a],false) -> ([a, c],true) , ([b, c],true) ([a, b],false) ([a, a],false) -> ([a, c],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, b],true) ([b, a],false) -> ([b, b],true) , ([b, b],true) ([a, b],false) ([b, a],false) -> ([a, b],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, c],true) ([b, a],false) -> ([b, c],true) , ([b, c],true) ([a, b],false) ([b, a],false) -> ([a, c],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([b, b],true) ([a, b],false) ([c, a],false) -> ([a, b],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, c],true) ([a, b],false) ([c, a],false) -> ([a, c],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([a, a],false) ([a, a],false) ->= ([a, a],false) , ([b, a],false) ([a, b],false) ([a, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, b],false) ([a, a],false) ->= ([a, b],false) , ([b, b],false) ([a, b],false) ([a, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([c, b],false) ([c, c],false) ([a, c],false) ->= ([a, b],false) ([a, a],false) , ([a, c],false) ([a, a],false) ->= ([a, c],false) , ([b, c],false) ([a, b],false) ([a, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, a],false) ([b, a],false) ->= ([b, a],false) , ([b, a],false) ([a, b],false) ([b, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, b],false) ([b, a],false) ->= ([b, b],false) , ([b, b],false) ([a, b],false) ([b, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([c, b],false) ([c, c],false) ([b, c],false) ->= ([a, b],false) ([b, a],false) , ([a, c],false) ([b, a],false) ->= ([b, c],false) , ([b, c],false) ([a, b],false) ([b, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, a],false) ([c, a],false) ->= ([c, a],false) , ([b, a],false) ([a, b],false) ([c, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, b],false) ([a, b],false) ([c, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([c, b],false) ([c, c],false) ([c, c],false) ->= ([a, b],false) ([c, a],false) , ([a, c],false) ([c, a],false) ->= ([c, c],false) , ([b, c],false) ([a, b],false) ([c, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 3: ([a, a],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, a],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 1 | \ / ([b, a],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, b],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, a],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([b, c],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, b],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, c],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([b, c],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, b],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, c],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, b],true) is interpreted by / \ | 1 0 1 | | 0 1 0 | | 0 0 0 | \ / ([b, b],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, c],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([b, a],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 1 0 | \ / ([b, b],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / Remains to prove termination of the 30-rule system { ([a, a],true) ([a, a],false) -> ([a, a],true) , ([a, b],true) ([a, a],false) -> ([a, b],true) , ([b, b],true) ([a, b],false) ([a, a],false) -> ([a, b],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, c],true) ([a, a],false) -> ([a, c],true) , ([b, c],true) ([a, b],false) ([a, a],false) -> ([a, c],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([b, b],true) ([a, b],false) ([b, a],false) -> ([a, b],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, c],true) ([b, a],false) -> ([b, c],true) , ([b, c],true) ([a, b],false) ([b, a],false) -> ([a, c],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([b, b],true) ([a, b],false) ([c, a],false) -> ([a, b],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, c],true) ([a, b],false) ([c, a],false) -> ([a, c],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([a, a],false) ([a, a],false) ->= ([a, a],false) , ([b, a],false) ([a, b],false) ([a, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, b],false) ([a, a],false) ->= ([a, b],false) , ([b, b],false) ([a, b],false) ([a, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([c, b],false) ([c, c],false) ([a, c],false) ->= ([a, b],false) ([a, a],false) , ([a, c],false) ([a, a],false) ->= ([a, c],false) , ([b, c],false) ([a, b],false) ([a, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, a],false) ([b, a],false) ->= ([b, a],false) , ([b, a],false) ([a, b],false) ([b, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, b],false) ([b, a],false) ->= ([b, b],false) , ([b, b],false) ([a, b],false) ([b, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([c, b],false) ([c, c],false) ([b, c],false) ->= ([a, b],false) ([b, a],false) , ([a, c],false) ([b, a],false) ->= ([b, c],false) , ([b, c],false) ([a, b],false) ([b, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, a],false) ([c, a],false) ->= ([c, a],false) , ([b, a],false) ([a, b],false) ([c, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, b],false) ([a, b],false) ([c, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([c, b],false) ([c, c],false) ([c, c],false) ->= ([a, b],false) ([c, a],false) , ([a, c],false) ([c, a],false) ->= ([c, c],false) , ([b, c],false) ([a, b],false) ([c, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: ([a, a],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([a, a],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, a],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([a, b],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([c, a],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, c],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([c, b],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([a, c],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, c],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([c, b],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([a, c],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([a, b],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, b],true) is interpreted by / \ | 1 1 | | 0 1 | \ / ([c, c],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, a],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, b],false) is interpreted by / \ | 1 0 | | 0 1 | \ / Remains to prove termination of the 27-rule system { ([a, a],true) ([a, a],false) -> ([a, a],true) , ([a, b],true) ([a, a],false) -> ([a, b],true) , ([a, c],true) ([a, a],false) -> ([a, c],true) , ([b, c],true) ([a, b],false) ([a, a],false) -> ([a, c],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, c],true) ([b, a],false) -> ([b, c],true) , ([b, c],true) ([a, b],false) ([b, a],false) -> ([a, c],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([b, c],true) ([a, b],false) ([c, a],false) -> ([a, c],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([a, a],false) ([a, a],false) ->= ([a, a],false) , ([b, a],false) ([a, b],false) ([a, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, b],false) ([a, a],false) ->= ([a, b],false) , ([b, b],false) ([a, b],false) ([a, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([c, b],false) ([c, c],false) ([a, c],false) ->= ([a, b],false) ([a, a],false) , ([a, c],false) ([a, a],false) ->= ([a, c],false) , ([b, c],false) ([a, b],false) ([a, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, a],false) ([b, a],false) ->= ([b, a],false) , ([b, a],false) ([a, b],false) ([b, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, b],false) ([b, a],false) ->= ([b, b],false) , ([b, b],false) ([a, b],false) ([b, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([c, b],false) ([c, c],false) ([b, c],false) ->= ([a, b],false) ([b, a],false) , ([a, c],false) ([b, a],false) ->= ([b, c],false) , ([b, c],false) ([a, b],false) ([b, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, a],false) ([c, a],false) ->= ([c, a],false) , ([b, a],false) ([a, b],false) ([c, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, b],false) ([a, b],false) ([c, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([c, b],false) ([c, c],false) ([c, c],false) ->= ([a, b],false) ([c, a],false) , ([a, c],false) ([c, a],false) ->= ([c, c],false) , ([b, c],false) ([a, b],false) ([c, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 3: ([a, a],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, a],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 1 | \ / ([b, a],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, b],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, a],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([b, c],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, b],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, c],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([b, c],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, b],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, c],true) is interpreted by / \ | 1 0 1 | | 0 1 0 | | 0 0 0 | \ / ([a, b],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([b, b],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, c],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([b, a],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 1 0 | \ / ([b, b],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / Remains to prove termination of the 26-rule system { ([a, a],true) ([a, a],false) -> ([a, a],true) , ([a, b],true) ([a, a],false) -> ([a, b],true) , ([a, c],true) ([a, a],false) -> ([a, c],true) , ([b, c],true) ([a, b],false) ([a, a],false) -> ([a, c],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([b, c],true) ([a, b],false) ([b, a],false) -> ([a, c],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([b, c],true) ([a, b],false) ([c, a],false) -> ([a, c],true) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([a, a],false) ([a, a],false) ->= ([a, a],false) , ([b, a],false) ([a, b],false) ([a, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, b],false) ([a, a],false) ->= ([a, b],false) , ([b, b],false) ([a, b],false) ([a, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([c, b],false) ([c, c],false) ([a, c],false) ->= ([a, b],false) ([a, a],false) , ([a, c],false) ([a, a],false) ->= ([a, c],false) , ([b, c],false) ([a, b],false) ([a, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, a],false) ([b, a],false) ->= ([b, a],false) , ([b, a],false) ([a, b],false) ([b, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, b],false) ([b, a],false) ->= ([b, b],false) , ([b, b],false) ([a, b],false) ([b, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([c, b],false) ([c, c],false) ([b, c],false) ->= ([a, b],false) ([b, a],false) , ([a, c],false) ([b, a],false) ->= ([b, c],false) , ([b, c],false) ([a, b],false) ([b, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, a],false) ([c, a],false) ->= ([c, a],false) , ([b, a],false) ([a, b],false) ([c, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, b],false) ([a, b],false) ([c, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([c, b],false) ([c, c],false) ([c, c],false) ->= ([a, b],false) ([c, a],false) , ([a, c],false) ([c, a],false) ->= ([c, c],false) , ([b, c],false) ([a, b],false) ([c, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: ([a, a],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([a, a],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, a],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([a, b],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([c, a],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, c],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([c, b],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([a, c],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, c],true) is interpreted by / \ | 1 1 | | 0 1 | \ / ([c, b],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([a, c],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([a, b],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, b],true) is interpreted by / \ | 1 0 | | 0 1 | \ / ([c, c],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, a],false) is interpreted by / \ | 1 0 | | 0 1 | \ / ([b, b],false) is interpreted by / \ | 1 0 | | 0 1 | \ / Remains to prove termination of the 23-rule system { ([a, a],true) ([a, a],false) -> ([a, a],true) , ([a, b],true) ([a, a],false) -> ([a, b],true) , ([a, c],true) ([a, a],false) -> ([a, c],true) , ([a, a],false) ([a, a],false) ->= ([a, a],false) , ([b, a],false) ([a, b],false) ([a, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, b],false) ([a, a],false) ->= ([a, b],false) , ([b, b],false) ([a, b],false) ([a, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([c, b],false) ([c, c],false) ([a, c],false) ->= ([a, b],false) ([a, a],false) , ([a, c],false) ([a, a],false) ->= ([a, c],false) , ([b, c],false) ([a, b],false) ([a, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, a],false) ([b, a],false) ->= ([b, a],false) , ([b, a],false) ([a, b],false) ([b, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, b],false) ([b, a],false) ->= ([b, b],false) , ([b, b],false) ([a, b],false) ([b, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([c, b],false) ([c, c],false) ([b, c],false) ->= ([a, b],false) ([b, a],false) , ([a, c],false) ([b, a],false) ->= ([b, c],false) , ([b, c],false) ([a, b],false) ([b, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, a],false) ([c, a],false) ->= ([c, a],false) , ([b, a],false) ([a, b],false) ([c, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, b],false) ([a, b],false) ([c, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([c, b],false) ([c, c],false) ([c, c],false) ->= ([a, b],false) ([c, a],false) , ([a, c],false) ([c, a],false) ->= ([c, c],false) , ([b, c],false) ([a, b],false) ([c, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 3: ([a, a],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, a],false) is interpreted by / \ | 1 0 1 | | 0 1 0 | | 0 1 0 | \ / ([b, a],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, b],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, a],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([b, c],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, b],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, c],false) is interpreted by / \ | 1 0 1 | | 0 1 0 | | 0 0 0 | \ / ([b, c],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, b],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, c],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, b],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([b, b],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, c],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([b, a],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 1 0 | \ / ([b, b],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / Remains to prove termination of the 19-rule system { ([a, a],true) ([a, a],false) -> ([a, a],true) , ([a, b],true) ([a, a],false) -> ([a, b],true) , ([a, c],true) ([a, a],false) -> ([a, c],true) , ([b, a],false) ([a, b],false) ([a, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, b],false) ([a, a],false) ->= ([a, b],false) , ([b, b],false) ([a, b],false) ([a, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([c, b],false) ([c, c],false) ([a, c],false) ->= ([a, b],false) ([a, a],false) , ([b, c],false) ([a, b],false) ([a, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([b, a],false) ([a, b],false) ([b, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, b],false) ([b, a],false) ->= ([b, b],false) , ([b, b],false) ([a, b],false) ([b, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([c, b],false) ([c, c],false) ([b, c],false) ->= ([a, b],false) ([b, a],false) , ([b, c],false) ([a, b],false) ([b, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, a],false) ([c, a],false) ->= ([c, a],false) , ([b, a],false) ([a, b],false) ([c, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, b],false) ([a, b],false) ([c, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([c, b],false) ([c, c],false) ([c, c],false) ->= ([a, b],false) ([c, a],false) , ([a, c],false) ([c, a],false) ->= ([c, c],false) , ([b, c],false) ([a, b],false) ([c, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 3: ([a, a],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, a],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 1 1 | \ / ([b, a],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, b],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, a],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([b, c],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, b],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, c],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([b, c],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, b],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, c],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, b],true) is interpreted by / \ | 1 0 1 | | 0 1 0 | | 0 0 0 | \ / ([b, b],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, c],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([b, a],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 1 0 | \ / ([b, b],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / Remains to prove termination of the 18-rule system { ([a, a],true) ([a, a],false) -> ([a, a],true) , ([a, c],true) ([a, a],false) -> ([a, c],true) , ([b, a],false) ([a, b],false) ([a, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, b],false) ([a, a],false) ->= ([a, b],false) , ([b, b],false) ([a, b],false) ([a, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([c, b],false) ([c, c],false) ([a, c],false) ->= ([a, b],false) ([a, a],false) , ([b, c],false) ([a, b],false) ([a, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([b, a],false) ([a, b],false) ([b, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, b],false) ([b, a],false) ->= ([b, b],false) , ([b, b],false) ([a, b],false) ([b, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([c, b],false) ([c, c],false) ([b, c],false) ->= ([a, b],false) ([b, a],false) , ([b, c],false) ([a, b],false) ([b, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, a],false) ([c, a],false) ->= ([c, a],false) , ([b, a],false) ([a, b],false) ([c, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, b],false) ([a, b],false) ([c, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([c, b],false) ([c, c],false) ([c, c],false) ->= ([a, b],false) ([c, a],false) , ([a, c],false) ([c, a],false) ->= ([c, c],false) , ([b, c],false) ([a, b],false) ([c, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 3: ([a, a],true) is interpreted by / \ | 1 0 1 | | 0 1 0 | | 0 0 0 | \ / ([a, a],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 1 1 | \ / ([b, a],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, b],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, a],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([b, c],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, b],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, c],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([b, c],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, b],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, c],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, b],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([b, b],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, c],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([b, a],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 1 0 | \ / ([b, b],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / Remains to prove termination of the 17-rule system { ([a, c],true) ([a, a],false) -> ([a, c],true) , ([b, a],false) ([a, b],false) ([a, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, b],false) ([a, a],false) ->= ([a, b],false) , ([b, b],false) ([a, b],false) ([a, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([c, b],false) ([c, c],false) ([a, c],false) ->= ([a, b],false) ([a, a],false) , ([b, c],false) ([a, b],false) ([a, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([b, a],false) ([a, b],false) ([b, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, b],false) ([b, a],false) ->= ([b, b],false) , ([b, b],false) ([a, b],false) ([b, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([c, b],false) ([c, c],false) ([b, c],false) ->= ([a, b],false) ([b, a],false) , ([b, c],false) ([a, b],false) ([b, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, a],false) ([c, a],false) ->= ([c, a],false) , ([b, a],false) ([a, b],false) ([c, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, b],false) ([a, b],false) ([c, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([c, b],false) ([c, c],false) ([c, c],false) ->= ([a, b],false) ([c, a],false) , ([a, c],false) ([c, a],false) ->= ([c, c],false) , ([b, c],false) ([a, b],false) ([c, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 3: ([a, a],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, a],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 1 1 | \ / ([b, a],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, b],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, a],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([b, c],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, b],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, c],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([b, c],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, b],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([a, c],true) is interpreted by / \ | 1 0 1 | | 0 1 0 | | 0 0 0 | \ / ([a, b],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([b, b],true) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([c, c],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / ([b, a],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 1 0 | \ / ([b, b],false) is interpreted by / \ | 1 0 0 | | 0 1 0 | | 0 0 0 | \ / Remains to prove termination of the 16-rule system { ([b, a],false) ([a, b],false) ([a, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([a, b],false) ([a, a],false) ->= ([a, b],false) , ([b, b],false) ([a, b],false) ([a, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([c, b],false) ([c, c],false) ([a, c],false) ->= ([a, b],false) ([a, a],false) , ([b, c],false) ([a, b],false) ([a, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([a, c],false) , ([b, a],false) ([a, b],false) ([b, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, b],false) ([b, a],false) ->= ([b, b],false) , ([b, b],false) ([a, b],false) ([b, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([c, b],false) ([c, c],false) ([b, c],false) ->= ([a, b],false) ([b, a],false) , ([b, c],false) ([a, b],false) ([b, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([b, c],false) , ([a, a],false) ([c, a],false) ->= ([c, a],false) , ([b, a],false) ([a, b],false) ([c, a],false) ->= ([a, a],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([b, b],false) ([a, b],false) ([c, a],false) ->= ([a, b],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) , ([c, b],false) ([c, c],false) ([c, c],false) ->= ([a, b],false) ([c, a],false) , ([a, c],false) ([c, a],false) ->= ([c, c],false) , ([b, c],false) ([a, b],false) ([c, a],false) ->= ([a, c],false) ([c, a],false) ([b, c],false) ([c, b],false) ([b, c],false) ([c, b],false) ([c, c],false) } The system is trivially terminating.