/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES After renaming modulo the bijection { a ↦ 0, b ↦ 1, c ↦ 2, d ↦ 3, e ↦ 4, f ↦ 5 }, it remains to prove termination of the 9-rule system { 0 1 ⟶ 1 2 0 , 1 2 ⟶ 2 1 1 , 0 2 ⟶ 2 0 1 , 0 0 ⟶ 0 3 3 3 , 3 0 ⟶ 3 3 2 , 0 3 3 2 ⟶ 0 0 0 3 , 4 4 5 5 ⟶ 5 5 5 4 4 , 4 ⟶ 0 , 1 3 ⟶ 3 3 } The system was reversed. After renaming modulo the bijection { 1 ↦ 0, 0 ↦ 1, 2 ↦ 2, 3 ↦ 3, 5 ↦ 4, 4 ↦ 5 }, it remains to prove termination of the 9-rule system { 0 1 ⟶ 1 2 0 , 2 0 ⟶ 0 0 2 , 2 1 ⟶ 0 1 2 , 1 1 ⟶ 3 3 3 1 , 1 3 ⟶ 2 3 3 , 2 3 3 1 ⟶ 3 1 1 1 , 4 4 5 5 ⟶ 5 5 4 4 4 , 5 ⟶ 1 , 3 0 ⟶ 3 3 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5 }, it remains to prove termination of the 8-rule system { 0 1 ⟶ 1 2 0 , 2 0 ⟶ 0 0 2 , 2 1 ⟶ 0 1 2 , 1 1 ⟶ 3 3 3 1 , 1 3 ⟶ 2 3 3 , 2 3 3 1 ⟶ 3 1 1 1 , 4 4 5 5 ⟶ 5 5 4 4 4 , 3 0 ⟶ 3 3 } Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols. After renaming modulo the bijection { (0,↑) ↦ 0, (1,↓) ↦ 1, (1,↑) ↦ 2, (2,↓) ↦ 3, (0,↓) ↦ 4, (2,↑) ↦ 5, (3,↑) ↦ 6, (3,↓) ↦ 7, (4,↑) ↦ 8, (4,↓) ↦ 9, (5,↓) ↦ 10 }, it remains to prove termination of the 30-rule system { 0 1 ⟶ 2 3 4 , 0 1 ⟶ 5 4 , 0 1 ⟶ 0 , 5 4 ⟶ 0 4 3 , 5 4 ⟶ 0 3 , 5 4 ⟶ 5 , 5 1 ⟶ 0 1 3 , 5 1 ⟶ 2 3 , 5 1 ⟶ 5 , 2 1 ⟶ 6 7 7 1 , 2 1 ⟶ 6 7 1 , 2 1 ⟶ 6 1 , 2 7 ⟶ 5 7 7 , 2 7 ⟶ 6 7 , 5 7 7 1 ⟶ 6 1 1 1 , 5 7 7 1 ⟶ 2 1 1 , 5 7 7 1 ⟶ 2 1 , 8 9 10 10 ⟶ 8 9 9 , 8 9 10 10 ⟶ 8 9 , 8 9 10 10 ⟶ 8 , 6 4 ⟶ 6 7 , 6 4 ⟶ 6 , 4 1 →= 1 3 4 , 3 4 →= 4 4 3 , 3 1 →= 4 1 3 , 1 1 →= 7 7 7 1 , 1 7 →= 3 7 7 , 3 7 7 1 →= 7 1 1 1 , 9 9 10 10 →= 10 10 9 9 9 , 7 4 →= 7 7 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 9 ↦ 8, 10 ↦ 9 }, it remains to prove termination of the 27-rule system { 0 1 ⟶ 2 3 4 , 0 1 ⟶ 5 4 , 0 1 ⟶ 0 , 5 4 ⟶ 0 4 3 , 5 4 ⟶ 0 3 , 5 4 ⟶ 5 , 5 1 ⟶ 0 1 3 , 5 1 ⟶ 2 3 , 5 1 ⟶ 5 , 2 1 ⟶ 6 7 7 1 , 2 1 ⟶ 6 7 1 , 2 1 ⟶ 6 1 , 2 7 ⟶ 5 7 7 , 2 7 ⟶ 6 7 , 5 7 7 1 ⟶ 6 1 1 1 , 5 7 7 1 ⟶ 2 1 1 , 5 7 7 1 ⟶ 2 1 , 6 4 ⟶ 6 7 , 6 4 ⟶ 6 , 4 1 →= 1 3 4 , 3 4 →= 4 4 3 , 3 1 →= 4 1 3 , 1 1 →= 7 7 7 1 , 1 7 →= 3 7 7 , 3 7 7 1 →= 7 1 1 1 , 8 8 9 9 →= 9 9 8 8 8 , 7 4 →= 7 7 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 7 ↦ 6, 6 ↦ 7, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 22-rule system { 0 1 ⟶ 2 3 4 , 0 1 ⟶ 5 4 , 0 1 ⟶ 0 , 5 4 ⟶ 0 4 3 , 5 4 ⟶ 0 3 , 5 4 ⟶ 5 , 5 1 ⟶ 0 1 3 , 5 1 ⟶ 2 3 , 5 1 ⟶ 5 , 2 6 ⟶ 5 6 6 , 5 6 6 1 ⟶ 2 1 1 , 5 6 6 1 ⟶ 2 1 , 7 4 ⟶ 7 6 , 7 4 ⟶ 7 , 4 1 →= 1 3 4 , 3 4 →= 4 4 3 , 3 1 →= 4 1 3 , 1 1 →= 6 6 6 1 , 1 6 →= 3 6 6 , 3 6 6 1 →= 6 1 1 1 , 8 8 9 9 →= 9 9 8 8 8 , 6 4 →= 6 6 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 3: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 1 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 1 2 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 1 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 1 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 8 ↦ 7, 9 ↦ 8 }, it remains to prove termination of the 20-rule system { 0 1 ⟶ 2 3 4 , 0 1 ⟶ 5 4 , 0 1 ⟶ 0 , 5 4 ⟶ 0 4 3 , 5 4 ⟶ 0 3 , 5 4 ⟶ 5 , 5 1 ⟶ 0 1 3 , 5 1 ⟶ 2 3 , 5 1 ⟶ 5 , 2 6 ⟶ 5 6 6 , 5 6 6 1 ⟶ 2 1 1 , 5 6 6 1 ⟶ 2 1 , 4 1 →= 1 3 4 , 3 4 →= 4 4 3 , 3 1 →= 4 1 3 , 1 1 →= 6 6 6 1 , 1 6 →= 3 6 6 , 3 6 6 1 →= 6 1 1 1 , 7 7 8 8 →= 8 8 7 7 7 , 6 4 →= 6 6 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 3: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 1 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 1 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 5 ↦ 0, 4 ↦ 1, 0 ↦ 2, 3 ↦ 3, 1 ↦ 4, 2 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8 }, it remains to prove termination of the 15-rule system { 0 1 ⟶ 2 1 3 , 0 1 ⟶ 2 3 , 0 1 ⟶ 0 , 0 4 ⟶ 2 4 3 , 5 6 ⟶ 0 6 6 , 0 6 6 4 ⟶ 5 4 4 , 0 6 6 4 ⟶ 5 4 , 1 4 →= 4 3 1 , 3 1 →= 1 1 3 , 3 4 →= 1 4 3 , 4 4 →= 6 6 6 4 , 4 6 →= 3 6 6 , 3 6 6 4 →= 6 4 4 4 , 7 7 8 8 →= 8 8 7 7 7 , 6 1 →= 6 6 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 5 ↦ 2, 6 ↦ 3, 4 ↦ 4, 3 ↦ 5, 7 ↦ 6, 8 ↦ 7 }, it remains to prove termination of the 12-rule system { 0 1 ⟶ 0 , 2 3 ⟶ 0 3 3 , 0 3 3 4 ⟶ 2 4 4 , 0 3 3 4 ⟶ 2 4 , 1 4 →= 4 5 1 , 5 1 →= 1 1 5 , 5 4 →= 1 4 5 , 4 4 →= 3 3 3 4 , 4 3 →= 5 3 3 , 5 3 3 4 →= 3 4 4 4 , 6 6 7 7 →= 7 7 6 6 6 , 3 1 →= 3 3 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 3: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 1 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 1 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 2 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 2 ↦ 0, 3 ↦ 1, 0 ↦ 2, 4 ↦ 3, 1 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7 }, it remains to prove termination of the 11-rule system { 0 1 ⟶ 2 1 1 , 2 1 1 3 ⟶ 0 3 3 , 2 1 1 3 ⟶ 0 3 , 4 3 →= 3 5 4 , 5 4 →= 4 4 5 , 5 3 →= 4 3 5 , 3 3 →= 1 1 1 3 , 3 1 →= 5 1 1 , 5 1 1 3 →= 1 3 3 3 , 6 6 7 7 →= 7 7 6 6 6 , 1 4 →= 1 1 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 5: 0 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 1 0 ⎟ ⎜ 0 0 1 0 0 ⎟ ⎜ 0 0 1 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 2 ⎟ ⎜ 0 1 1 1 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5 }, it remains to prove termination of the 10-rule system { 0 1 ⟶ 2 1 1 , 2 1 1 3 ⟶ 0 3 3 , 2 1 1 3 ⟶ 0 3 , 4 3 →= 3 5 4 , 5 4 →= 4 4 5 , 5 3 →= 4 3 5 , 3 3 →= 1 1 1 3 , 3 1 →= 5 1 1 , 5 1 1 3 →= 1 3 3 3 , 1 4 →= 1 1 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 4: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 1 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 1 1 0 ⎟ ⎜ 0 0 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 2 1 ⎟ ⎜ 0 1 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5 }, it remains to prove termination of the 9-rule system { 0 1 ⟶ 2 1 1 , 2 1 1 3 ⟶ 0 3 3 , 2 1 1 3 ⟶ 0 3 , 4 3 →= 3 5 4 , 5 4 →= 4 4 5 , 5 3 →= 4 3 5 , 3 3 →= 1 1 1 3 , 3 1 →= 5 1 1 , 5 1 1 3 →= 1 3 3 3 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 6: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 1 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎜ 0 1 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 1 0 0 0 ⎟ ⎜ 0 0 1 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 1 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 4 ↦ 3, 3 ↦ 4, 5 ↦ 5 }, it remains to prove termination of the 7-rule system { 0 1 ⟶ 2 1 1 , 3 4 →= 4 5 3 , 5 3 →= 3 3 5 , 5 4 →= 3 4 5 , 4 4 →= 1 1 1 4 , 4 1 →= 5 1 1 , 5 1 1 4 →= 1 4 4 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 3 ↦ 0, 4 ↦ 1, 5 ↦ 2, 1 ↦ 3 }, it remains to prove termination of the 6-rule system { 0 1 →= 1 2 0 , 2 0 →= 0 0 2 , 2 1 →= 0 1 2 , 1 1 →= 3 3 3 1 , 1 3 →= 2 3 3 , 2 3 3 1 →= 3 1 1 1 } The system is trivially terminating.