/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 { r ↦ 0, s ↦ 1, n ↦ 2, b ↦ 3, u ↦ 4, t ↦ 5, c ↦ 6 }, it remains to prove termination of the 15-rule system { 0 0 ⟶ 1 0 , 0 1 ⟶ 1 0 , 0 2 ⟶ 1 0 , 0 3 ⟶ 4 1 3 , 0 4 ⟶ 4 0 , 1 4 ⟶ 4 1 , 2 4 ⟶ 4 2 , 5 0 4 ⟶ 5 6 0 , 5 1 4 ⟶ 5 6 0 , 5 2 4 ⟶ 5 6 0 , 6 4 ⟶ 4 6 , 6 1 ⟶ 1 6 , 6 0 ⟶ 0 6 , 6 2 ⟶ 2 6 , 6 2 ⟶ 2 } The system was reversed. After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6 }, it remains to prove termination of the 15-rule system { 0 0 ⟶ 0 1 , 1 0 ⟶ 0 1 , 2 0 ⟶ 0 1 , 3 0 ⟶ 3 1 4 , 4 0 ⟶ 0 4 , 4 1 ⟶ 1 4 , 4 2 ⟶ 2 4 , 4 0 5 ⟶ 0 6 5 , 4 1 5 ⟶ 0 6 5 , 4 2 5 ⟶ 0 6 5 , 4 6 ⟶ 6 4 , 1 6 ⟶ 6 1 , 0 6 ⟶ 6 0 , 2 6 ⟶ 6 2 , 2 6 ⟶ 2 } 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 3 ↦ 2, 4 ↦ 3, 2 ↦ 4, 5 ↦ 5, 6 ↦ 6 }, it remains to prove termination of the 13-rule system { 0 0 ⟶ 0 1 , 1 0 ⟶ 0 1 , 2 0 ⟶ 2 1 3 , 3 0 ⟶ 0 3 , 3 1 ⟶ 1 3 , 3 4 ⟶ 4 3 , 3 0 5 ⟶ 0 6 5 , 3 1 5 ⟶ 0 6 5 , 3 6 ⟶ 6 3 , 1 6 ⟶ 6 1 , 0 6 ⟶ 6 0 , 4 6 ⟶ 6 4 , 4 6 ⟶ 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 1 ↦ 0, 0 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6 }, it remains to prove termination of the 11-rule system { 0 1 ⟶ 1 0 , 2 1 ⟶ 2 0 3 , 3 1 ⟶ 1 3 , 3 0 ⟶ 0 3 , 3 4 ⟶ 4 3 , 3 0 5 ⟶ 1 6 5 , 3 6 ⟶ 6 3 , 0 6 ⟶ 6 0 , 1 6 ⟶ 6 1 , 4 6 ⟶ 6 4 , 4 6 ⟶ 4 } Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols. After renaming modulo the bijection { (0,↑) ↦ 0, (1,↓) ↦ 1, (1,↑) ↦ 2, (0,↓) ↦ 3, (2,↑) ↦ 4, (3,↓) ↦ 5, (3,↑) ↦ 6, (4,↓) ↦ 7, (4,↑) ↦ 8, (5,↓) ↦ 9, (6,↓) ↦ 10, (2,↓) ↦ 11 }, it remains to prove termination of the 27-rule system { 0 1 ⟶ 2 3 , 0 1 ⟶ 0 , 4 1 ⟶ 4 3 5 , 4 1 ⟶ 0 5 , 4 1 ⟶ 6 , 6 1 ⟶ 2 5 , 6 1 ⟶ 6 , 6 3 ⟶ 0 5 , 6 3 ⟶ 6 , 6 7 ⟶ 8 5 , 6 7 ⟶ 6 , 6 3 9 ⟶ 2 10 9 , 6 10 ⟶ 6 , 0 10 ⟶ 0 , 2 10 ⟶ 2 , 8 10 ⟶ 8 , 3 1 →= 1 3 , 11 1 →= 11 3 5 , 5 1 →= 1 5 , 5 3 →= 3 5 , 5 7 →= 7 5 , 5 3 9 →= 1 10 9 , 5 10 →= 10 5 , 3 10 →= 10 3 , 1 10 →= 10 1 , 7 10 →= 10 7 , 7 10 →= 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 9 ↦ 7, 10 ↦ 8, 8 ↦ 9, 11 ↦ 10, 7 ↦ 11 }, it remains to prove termination of the 23-rule system { 0 1 ⟶ 2 3 , 0 1 ⟶ 0 , 4 1 ⟶ 4 3 5 , 6 1 ⟶ 2 5 , 6 1 ⟶ 6 , 6 3 ⟶ 0 5 , 6 3 ⟶ 6 , 6 3 7 ⟶ 2 8 7 , 6 8 ⟶ 6 , 0 8 ⟶ 0 , 2 8 ⟶ 2 , 9 8 ⟶ 9 , 3 1 →= 1 3 , 10 1 →= 10 3 5 , 5 1 →= 1 5 , 5 3 →= 3 5 , 5 11 →= 11 5 , 5 3 7 →= 1 8 7 , 5 8 →= 8 5 , 3 8 →= 8 3 , 1 8 →= 8 1 , 11 8 →= 8 11 , 11 8 →= 11 } 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 8 ↦ 7, 9 ↦ 8, 10 ↦ 9, 11 ↦ 10, 7 ↦ 11 }, it remains to prove termination of the 20-rule system { 0 1 ⟶ 2 3 , 0 1 ⟶ 0 , 4 1 ⟶ 4 3 5 , 6 1 ⟶ 6 , 6 3 ⟶ 6 , 6 7 ⟶ 6 , 0 7 ⟶ 0 , 2 7 ⟶ 2 , 8 7 ⟶ 8 , 3 1 →= 1 3 , 9 1 →= 9 3 5 , 5 1 →= 1 5 , 5 3 →= 3 5 , 5 10 →= 10 5 , 5 3 11 →= 1 7 11 , 5 7 →= 7 5 , 3 7 →= 7 3 , 1 7 →= 7 1 , 10 7 →= 7 10 , 10 7 →= 10 } 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 4 ↦ 2, 3 ↦ 3, 5 ↦ 4, 6 ↦ 5, 7 ↦ 6, 2 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11 }, it remains to prove termination of the 19-rule system { 0 1 ⟶ 0 , 2 1 ⟶ 2 3 4 , 5 1 ⟶ 5 , 5 3 ⟶ 5 , 5 6 ⟶ 5 , 0 6 ⟶ 0 , 7 6 ⟶ 7 , 8 6 ⟶ 8 , 3 1 →= 1 3 , 9 1 →= 9 3 4 , 4 1 →= 1 4 , 4 3 →= 3 4 , 4 10 →= 10 4 , 4 3 11 →= 1 6 11 , 4 6 →= 6 4 , 3 6 →= 6 3 , 1 6 →= 6 1 , 10 6 →= 6 10 , 10 6 →= 10 } 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 2 ↦ 0, 1 ↦ 1, 3 ↦ 2, 4 ↦ 3, 5 ↦ 4, 6 ↦ 5, 0 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11 }, it remains to prove termination of the 16-rule system { 0 1 ⟶ 0 2 3 , 4 5 ⟶ 4 , 6 5 ⟶ 6 , 7 5 ⟶ 7 , 8 5 ⟶ 8 , 2 1 →= 1 2 , 9 1 →= 9 2 3 , 3 1 →= 1 3 , 3 2 →= 2 3 , 3 10 →= 10 3 , 3 2 11 →= 1 5 11 , 3 5 →= 5 3 , 2 5 →= 5 2 , 1 5 →= 5 1 , 10 5 →= 5 10 , 10 5 →= 10 } 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 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 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 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 1 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 4 ↦ 0, 5 ↦ 1, 6 ↦ 2, 7 ↦ 3, 8 ↦ 4, 2 ↦ 5, 1 ↦ 6, 9 ↦ 7, 3 ↦ 8, 10 ↦ 9, 11 ↦ 10 }, it remains to prove termination of the 15-rule system { 0 1 ⟶ 0 , 2 1 ⟶ 2 , 3 1 ⟶ 3 , 4 1 ⟶ 4 , 5 6 →= 6 5 , 7 6 →= 7 5 8 , 8 6 →= 6 8 , 8 5 →= 5 8 , 8 9 →= 9 8 , 8 5 10 →= 6 1 10 , 8 1 →= 1 8 , 5 1 →= 1 5 , 6 1 →= 1 6 , 9 1 →= 1 9 , 9 1 →= 9 } 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 0 0 ⎟ ⎝ ⎠ 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 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 1 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 1 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 1 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 8 ↦ 7, 9 ↦ 8, 10 ↦ 9 }, it remains to prove termination of the 14-rule system { 0 1 ⟶ 0 , 2 1 ⟶ 2 , 3 1 ⟶ 3 , 4 1 ⟶ 4 , 5 6 →= 6 5 , 7 6 →= 6 7 , 7 5 →= 5 7 , 7 8 →= 8 7 , 7 5 9 →= 6 1 9 , 7 1 →= 1 7 , 5 1 →= 1 5 , 6 1 →= 1 6 , 8 1 →= 1 8 , 8 1 →= 8 } 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 1 ⎟ ⎜ 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 5 ↦ 0, 6 ↦ 1, 7 ↦ 2, 8 ↦ 3, 1 ↦ 4 }, it remains to prove termination of the 8-rule system { 0 1 →= 1 0 , 2 1 →= 1 2 , 2 0 →= 0 2 , 2 3 →= 3 2 , 2 4 →= 4 2 , 0 4 →= 4 0 , 1 4 →= 4 1 , 3 4 →= 4 3 } The system is trivially terminating.