TORPAcyc is applied to the string rewriting system 0 1 2 2 3 0 4 4 -> 0 4 5 0 3 4 0 4 0 2 2 2 0 5 2 5 4 -> 0 0 0 3 4 3 0 1 0 2 1 1 3 4 3 1 1 5 -> 1 0 2 3 1 0 5 1 5 5 4 0 4 3 3 1 2 5 3 0 -> 5 5 2 5 3 1 5 0 3 5 2 2 4 5 0 1 1 3 3 5 3 0 0 -> 4 4 2 2 1 0 4 0 1 3 2 0 4 4 3 0 3 1 5 3 5 1 3 1 5 3 -> 0 2 4 5 0 0 0 5 1 0 5 4 4 2 2 0 0 2 1 0 5 3 2 2 1 4 0 5 -> 0 0 3 5 3 0 4 3 1 3 0 2 5 5 3 0 0 4 2 5 5 1 3 0 2 3 3 5 1 4 5 -> 3 4 0 1 0 5 5 3 1 4 0 3 5 3 2 2 5 5 1 1 4 1 5 3 0 4 3 2 5 4 1 3 3 5 -> 5 0 1 0 4 0 2 4 5 1 5 4 1 5 3 3 5 0 1 4 4 3 2 0 4 1 4 3 4 4 1 5 3 4 4 -> 0 2 4 2 3 1 0 1 1 1 3 0 2 4 4 1 1 2 1 1 4 1 0 1 0 3 3 4 4 1 5 4 0 4 4 5 5 3 -> 1 3 0 3 2 2 4 4 2 0 3 3 4 0 3 0 4 3 4 0 2 0 3 3 3 4 1 1 0 4 4 0 3 3 3 0 0 1 5 3 -> 5 3 3 1 4 0 4 5 4 4 4 2 4 3 1 1 1 5 4 2 3 3 2 1 5 0 5 0 1 3 3 2 5 1 5 0 3 0 5 -> 3 4 0 2 5 5 2 4 2 4 3 1 1 4 4 5 5 3 5 3 3 2 2 3 3 4 0 0 0 2 5 0 5 3 0 0 1 1 4 -> 3 1 1 0 3 3 5 4 2 2 1 0 1 1 0 5 0 3 4 0 4 2 4 1 0 5 0 4 1 0 3 0 2 5 4 3 5 3 5 3 -> 4 2 5 4 4 2 4 5 0 0 3 1 5 0 2 0 2 2 1 interpretation of 0: 5 interpretation of 1: 6 interpretation of 2: 6 interpretation of 3: 7 interpretation of 4: 7 interpretation of 5: 6 remove rule 15 continue with: 0 1 2 2 3 0 4 4 -> 0 4 5 0 3 4 0 4 0 2 2 2 0 5 2 5 4 -> 0 0 0 3 4 3 0 1 0 2 1 1 3 4 3 1 1 5 -> 1 0 2 3 1 0 5 1 5 5 4 0 4 3 3 1 2 5 3 0 -> 5 5 2 5 3 1 5 0 3 5 2 2 4 5 0 1 1 3 3 5 3 0 0 -> 4 4 2 2 1 0 4 0 1 3 2 0 4 4 3 0 3 1 5 3 5 1 3 1 5 3 -> 0 2 4 5 0 0 0 5 1 0 5 4 4 2 2 0 0 2 1 0 5 3 2 2 1 4 0 5 -> 0 0 3 5 3 0 4 3 1 3 0 2 5 5 3 0 0 4 2 5 5 1 3 0 2 3 3 5 1 4 5 -> 3 4 0 1 0 5 5 3 1 4 0 3 5 3 2 2 5 5 1 1 4 1 5 3 0 4 3 2 5 4 1 3 3 5 -> 5 0 1 0 4 0 2 4 5 1 5 4 1 5 3 3 5 0 1 4 4 3 2 0 4 1 4 3 4 4 1 5 3 4 4 -> 0 2 4 2 3 1 0 1 1 1 3 0 2 4 4 1 1 2 1 1 4 1 0 1 0 3 3 4 4 1 5 4 0 4 4 5 5 3 -> 1 3 0 3 2 2 4 4 2 0 3 3 4 0 3 0 4 3 4 0 2 0 3 3 3 4 1 1 0 4 4 0 3 3 3 0 0 1 5 3 -> 5 3 3 1 4 0 4 5 4 4 4 2 4 3 1 1 1 5 4 2 3 3 2 1 5 0 5 0 1 3 3 2 5 1 5 0 3 0 5 -> 3 4 0 2 5 5 2 4 2 4 3 1 1 4 4 5 5 3 5 3 3 2 2 3 3 4 0 0 0 2 5 0 5 3 0 0 1 1 4 -> 3 1 1 0 3 3 5 4 2 2 1 0 1 1 0 5 0 3 4 0 interpretation of 0: 5 interpretation of 1: 6 interpretation of 2: 6 interpretation of 3: 7 interpretation of 4: 7 interpretation of 5: 6 remove rule 10 continue with: 0 1 2 2 3 0 4 4 -> 0 4 5 0 3 4 0 4 0 2 2 2 0 5 2 5 4 -> 0 0 0 3 4 3 0 1 0 2 1 1 3 4 3 1 1 5 -> 1 0 2 3 1 0 5 1 5 5 4 0 4 3 3 1 2 5 3 0 -> 5 5 2 5 3 1 5 0 3 5 2 2 4 5 0 1 1 3 3 5 3 0 0 -> 4 4 2 2 1 0 4 0 1 3 2 0 4 4 3 0 3 1 5 3 5 1 3 1 5 3 -> 0 2 4 5 0 0 0 5 1 0 5 4 4 2 2 0 0 2 1 0 5 3 2 2 1 4 0 5 -> 0 0 3 5 3 0 4 3 1 3 0 2 5 5 3 0 0 4 2 5 5 1 3 0 2 3 3 5 1 4 5 -> 3 4 0 1 0 5 5 3 1 4 0 3 5 3 2 2 5 5 1 1 4 1 5 3 0 4 3 2 5 4 1 3 3 5 -> 5 0 1 0 4 0 2 4 5 1 5 4 1 5 3 3 5 1 1 4 1 0 1 0 3 3 4 4 1 5 4 0 4 4 5 5 3 -> 1 3 0 3 2 2 4 4 2 0 3 3 4 0 3 0 4 3 4 0 2 0 3 3 3 4 1 1 0 4 4 0 3 3 3 0 0 1 5 3 -> 5 3 3 1 4 0 4 5 4 4 4 2 4 3 1 1 1 5 4 2 3 3 2 1 5 0 5 0 1 3 3 2 5 1 5 0 3 0 5 -> 3 4 0 2 5 5 2 4 2 4 3 1 1 4 4 5 5 3 5 3 3 2 2 3 3 4 0 0 0 2 5 0 5 3 0 0 1 1 4 -> 3 1 1 0 3 3 5 4 2 2 1 0 1 1 0 5 0 3 4 0 interpretation of 0: 1 interpretation of 1: 1 interpretation of 2: 1 interpretation of 3: 1 interpretation of 4: 1 interpretation of 5: 1 remove rule 6 continue with: 0 1 2 2 3 0 4 4 -> 0 4 5 0 3 4 0 4 0 2 2 2 0 5 2 5 4 -> 0 0 0 3 4 3 0 1 0 2 1 1 3 4 3 1 1 5 -> 1 0 2 3 1 0 5 1 5 5 4 0 4 3 3 1 2 5 3 0 -> 5 5 2 5 3 1 5 0 3 5 2 2 4 5 0 1 1 3 3 5 3 0 0 -> 4 4 2 2 1 0 4 0 1 3 2 0 2 2 0 0 2 1 0 5 3 2 2 1 4 0 5 -> 0 0 3 5 3 0 4 3 1 3 0 2 5 5 3 0 0 4 2 5 5 1 3 0 2 3 3 5 1 4 5 -> 3 4 0 1 0 5 5 3 1 4 0 3 5 3 2 2 5 5 1 1 4 1 5 3 0 4 3 2 5 4 1 3 3 5 -> 5 0 1 0 4 0 2 4 5 1 5 4 1 5 3 3 5 1 1 4 1 0 1 0 3 3 4 4 1 5 4 0 4 4 5 5 3 -> 1 3 0 3 2 2 4 4 2 0 3 3 4 0 3 0 4 3 4 0 2 0 3 3 3 4 1 1 0 4 4 0 3 3 3 0 0 1 5 3 -> 5 3 3 1 4 0 4 5 4 4 4 2 4 3 1 1 1 5 4 2 3 3 2 1 5 0 5 0 1 3 3 2 5 1 5 0 3 0 5 -> 3 4 0 2 5 5 2 4 2 4 3 1 1 4 4 5 5 3 5 3 3 2 2 3 3 4 0 0 0 2 5 0 5 3 0 0 1 1 4 -> 3 1 1 0 3 3 5 4 2 2 1 0 1 1 0 5 0 3 4 0 interpretation of 0: 1 interpretation of 1: 1 interpretation of 2: 1 interpretation of 3: 1 interpretation of 4: 1 interpretation of 5: 1 remove rule 6 continue with: 0 1 2 2 3 0 4 4 -> 0 4 5 0 3 4 0 4 0 2 2 2 0 5 2 5 4 -> 0 0 0 3 4 3 0 1 0 2 1 1 3 4 3 1 1 5 -> 1 0 2 3 1 0 5 1 5 5 4 0 4 3 3 1 2 5 3 0 -> 5 5 2 5 3 1 5 0 3 5 2 2 4 5 0 1 1 3 3 5 3 0 0 -> 4 4 2 2 1 0 4 0 1 3 2 0 3 0 0 4 2 5 5 1 3 0 2 3 3 5 1 4 5 -> 3 4 0 1 0 5 5 3 1 4 0 3 5 3 2 2 5 5 1 1 4 1 5 3 0 4 3 2 5 4 1 3 3 5 -> 5 0 1 0 4 0 2 4 5 1 5 4 1 5 3 3 5 1 1 4 1 0 1 0 3 3 4 4 1 5 4 0 4 4 5 5 3 -> 1 3 0 3 2 2 4 4 2 0 3 3 4 0 3 0 4 3 4 0 2 0 3 3 3 4 1 1 0 4 4 0 3 3 3 0 0 1 5 3 -> 5 3 3 1 4 0 4 5 4 4 4 2 4 3 1 1 1 5 4 2 3 3 2 1 5 0 5 0 1 3 3 2 5 1 5 0 3 0 5 -> 3 4 0 2 5 5 2 4 2 4 3 1 1 4 4 5 5 3 5 3 3 2 2 3 3 4 0 0 0 2 5 0 5 3 0 0 1 1 4 -> 3 1 1 0 3 3 5 4 2 2 1 0 1 1 0 5 0 3 4 0 interpretation of 0: 1 interpretation of 1: 1 interpretation of 2: 1 interpretation of 3: 1 interpretation of 4: 1 interpretation of 5: 1 remove rule 9 continue with: 0 1 2 2 3 0 4 4 -> 0 4 5 0 3 4 0 4 0 2 2 2 0 5 2 5 4 -> 0 0 0 3 4 3 0 1 0 2 1 1 3 4 3 1 1 5 -> 1 0 2 3 1 0 5 1 5 5 4 0 4 3 3 1 2 5 3 0 -> 5 5 2 5 3 1 5 0 3 5 2 2 4 5 0 1 1 3 3 5 3 0 0 -> 4 4 2 2 1 0 4 0 1 3 2 0 3 0 0 4 2 5 5 1 3 0 2 3 3 5 1 4 5 -> 3 4 0 1 0 5 5 3 1 4 0 3 5 3 2 2 5 5 1 1 4 1 5 3 0 4 3 2 5 4 1 3 3 5 -> 5 0 1 0 4 0 2 4 5 1 5 4 1 5 3 3 5 1 1 4 1 0 1 0 3 3 4 4 1 5 4 0 4 4 5 5 3 -> 1 3 0 3 2 2 4 4 2 0 3 3 4 0 3 0 4 3 4 0 2 3 3 2 1 5 0 5 0 1 3 3 2 5 1 5 0 3 0 5 -> 3 4 0 2 5 5 2 4 2 4 3 1 1 4 4 5 5 3 5 3 3 2 2 3 3 4 0 0 0 2 5 0 5 3 0 0 1 1 4 -> 3 1 1 0 3 3 5 4 2 2 1 0 1 1 0 5 0 3 4 0 interpretation of 0: 1 interpretation of 1: 1 interpretation of 2: 1 interpretation of 3: 1 interpretation of 4: 1 interpretation of 5: 1 remove rule 9 continue with: 0 1 2 2 3 0 4 4 -> 0 4 5 0 3 4 0 4 0 2 2 2 0 5 2 5 4 -> 0 0 0 3 4 3 0 1 0 2 1 1 3 4 3 1 1 5 -> 1 0 2 3 1 0 5 1 5 5 4 0 4 3 3 1 2 5 3 0 -> 5 5 2 5 3 1 5 0 3 5 2 2 4 5 0 1 1 3 3 5 3 0 0 -> 4 4 2 2 1 0 4 0 1 3 2 0 3 0 0 4 2 5 5 1 3 0 2 3 3 5 1 4 5 -> 3 4 0 1 0 5 5 3 1 4 0 3 5 3 2 2 5 5 1 1 4 1 5 3 0 4 3 2 5 4 1 3 3 5 -> 5 0 1 0 4 0 2 4 5 1 5 4 1 5 3 3 5 1 1 4 1 0 1 0 3 3 4 4 1 5 4 0 4 4 5 5 3 -> 1 3 0 3 2 2 4 4 2 0 3 3 4 0 3 0 4 3 4 0 3 3 2 2 3 3 4 0 0 0 2 5 0 5 3 0 0 1 1 4 -> 3 1 1 0 3 3 5 4 2 2 1 0 1 1 0 5 0 3 4 0 interpretation of 0: 0 interpretation of 1: 4 interpretation of 2: 5 interpretation of 3: 8 interpretation of 4: 8 interpretation of 5: 5 remove rule 7 continue with: 0 1 2 2 3 0 4 4 -> 0 4 5 0 3 4 0 4 0 2 2 2 0 5 2 5 4 -> 0 0 0 3 4 3 0 1 0 2 1 1 3 4 3 1 1 5 -> 1 0 2 3 1 0 5 1 5 5 4 0 4 3 3 1 2 5 3 0 -> 5 5 2 5 3 1 5 0 3 5 2 2 4 5 0 1 1 3 3 5 3 0 0 -> 4 4 2 2 1 0 4 0 1 3 2 0 3 0 0 4 2 5 5 1 3 0 2 3 3 5 1 4 5 -> 3 4 0 1 0 5 5 3 1 4 0 3 5 3 2 2 5 1 1 4 1 0 1 0 3 3 4 4 1 5 4 0 4 4 5 5 3 -> 1 3 0 3 2 2 4 4 2 0 3 3 4 0 3 0 4 3 4 0 3 3 2 2 3 3 4 0 0 0 2 5 0 5 3 0 0 1 1 4 -> 3 1 1 0 3 3 5 4 2 2 1 0 1 1 0 5 0 3 4 0 interpretation of 0: 0 interpretation of 1: 2 interpretation of 2: 3 interpretation of 3: 4 interpretation of 4: 4 interpretation of 5: 3 remove rule 3 continue with: 0 1 2 2 3 0 4 4 -> 0 4 5 0 3 4 0 4 0 2 2 2 0 5 2 5 4 -> 0 0 0 3 4 3 0 1 0 5 4 0 4 3 3 1 2 5 3 0 -> 5 5 2 5 3 1 5 0 3 5 2 2 4 5 0 1 1 3 3 5 3 0 0 -> 4 4 2 2 1 0 4 0 1 3 2 0 3 0 0 4 2 5 5 1 3 0 2 3 3 5 1 4 5 -> 3 4 0 1 0 5 5 3 1 4 0 3 5 3 2 2 5 1 1 4 1 0 1 0 3 3 4 4 1 5 4 0 4 4 5 5 3 -> 1 3 0 3 2 2 4 4 2 0 3 3 4 0 3 0 4 3 4 0 3 3 2 2 3 3 4 0 0 0 2 5 0 5 3 0 0 1 1 4 -> 3 1 1 0 3 3 5 4 2 2 1 0 1 1 0 5 0 3 4 0 interpretation of 0: 1 interpretation of 1: 3 interpretation of 2: 4 interpretation of 3: 5 interpretation of 4: 5 interpretation of 5: 4 remove rule 2 continue with: 0 1 2 2 3 0 4 4 -> 0 4 5 0 3 4 0 4 5 4 0 4 3 3 1 2 5 3 0 -> 5 5 2 5 3 1 5 0 3 5 2 2 4 5 0 1 1 3 3 5 3 0 0 -> 4 4 2 2 1 0 4 0 1 3 2 0 3 0 0 4 2 5 5 1 3 0 2 3 3 5 1 4 5 -> 3 4 0 1 0 5 5 3 1 4 0 3 5 3 2 2 5 1 1 4 1 0 1 0 3 3 4 4 1 5 4 0 4 4 5 5 3 -> 1 3 0 3 2 2 4 4 2 0 3 3 4 0 3 0 4 3 4 0 3 3 2 2 3 3 4 0 0 0 2 5 0 5 3 0 0 1 1 4 -> 3 1 1 0 3 3 5 4 2 2 1 0 1 1 0 5 0 3 4 0 interpretation of 0: 0 interpretation of 1: 1 interpretation of 2: 1 interpretation of 3: 2 interpretation of 4: 2 interpretation of 5: 1 remove rule 2 continue with: 0 1 2 2 3 0 4 4 -> 0 4 5 0 3 4 0 4 2 4 5 0 1 1 3 3 5 3 0 0 -> 4 4 2 2 1 0 4 0 1 3 2 0 3 0 0 4 2 5 5 1 3 0 2 3 3 5 1 4 5 -> 3 4 0 1 0 5 5 3 1 4 0 3 5 3 2 2 5 1 1 4 1 0 1 0 3 3 4 4 1 5 4 0 4 4 5 5 3 -> 1 3 0 3 2 2 4 4 2 0 3 3 4 0 3 0 4 3 4 0 3 3 2 2 3 3 4 0 0 0 2 5 0 5 3 0 0 1 1 4 -> 3 1 1 0 3 3 5 4 2 2 1 0 1 1 0 5 0 3 4 0 interpretation of 0: 0 interpretation of 1: 0 interpretation of 2: 1 interpretation of 3: 0 interpretation of 4: 0 interpretation of 5: 1 remove rule 5 continue with: 0 1 2 2 3 0 4 4 -> 0 4 5 0 3 4 0 4 2 4 5 0 1 1 3 3 5 3 0 0 -> 4 4 2 2 1 0 4 0 1 3 2 0 3 0 0 4 2 5 5 1 3 0 2 3 3 5 1 4 5 -> 3 4 0 1 0 5 5 3 1 4 0 3 5 3 2 2 5 1 1 4 1 0 1 0 3 3 4 4 1 5 4 0 4 4 5 5 3 -> 1 3 0 3 2 2 4 4 2 0 3 3 4 0 3 0 4 3 4 0 interpretation of 0: 0 interpretation of 1: 1 interpretation of 2: 1 interpretation of 3: 0 interpretation of 4: 0 interpretation of 5: 2 remove rule 2 continue with: 0 1 2 2 3 0 4 4 -> 0 4 5 0 3 4 0 4 3 0 0 4 2 5 5 1 3 0 2 3 3 5 1 4 5 -> 3 4 0 1 0 5 5 3 1 4 0 3 5 3 2 2 5 1 1 4 1 0 1 0 3 3 4 4 1 5 4 0 4 4 5 5 3 -> 1 3 0 3 2 2 4 4 2 0 3 3 4 0 3 0 4 3 4 0 interpretation of 0: 1 interpretation of 1: 3 interpretation of 2: 0 interpretation of 3: 0 interpretation of 4: 0 interpretation of 5: 1 remove rule 3 continue with: 0 1 2 2 3 0 4 4 -> 0 4 5 0 3 4 0 4 3 0 0 4 2 5 5 1 3 0 2 3 3 5 1 4 5 -> 3 4 0 1 0 5 5 3 1 4 0 3 5 3 2 2 5 interpretation of 0: 0 interpretation of 1: 2 interpretation of 2: 0 interpretation of 3: 0 interpretation of 4: 0 interpretation of 5: 1 remove rule 1 continue with: 3 0 0 4 2 5 5 1 3 0 2 3 3 5 1 4 5 -> 3 4 0 1 0 5 5 3 1 4 0 3 5 3 2 2 5 Tropical matrix interpretation interpretation of 0: / 1 3 \ \ 3 0 / interpretation of 1: / 3 0 \ \ - 3 / interpretation of 2: / 0 - \ \ 0 - / interpretation of 3: / 0 - \ \ - - / interpretation of 4: / 2 - \ \ 3 0 / interpretation of 5: / 0 - \ \ 1 - / remove rule 1 termination has been proved