/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES After renaming modulo the bijection { r1 ↦ 0, a ↦ 1, r2 ↦ 2, l1 ↦ 3, l2 ↦ 4, b ↦ 5 }, it remains to prove termination of the 9-rule system { 0 1 ⟶ 1 1 1 0 , 2 1 ⟶ 1 1 1 2 , 1 3 ⟶ 3 1 1 1 , 1 1 4 ⟶ 4 1 1 , 0 5 ⟶ 3 5 , 2 5 ⟶ 4 1 5 , 5 3 ⟶ 5 2 , 5 4 ⟶ 5 0 , 1 1 ⟶ } 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, (2,↓) ↦ 5, (3,↓) ↦ 6, (4,↓) ↦ 7, (5,↓) ↦ 8, (5,↑) ↦ 9 }, it remains to prove termination of the 27-rule system { 0 1 ⟶ 2 1 1 3 , 0 1 ⟶ 2 1 3 , 0 1 ⟶ 2 3 , 0 1 ⟶ 0 , 4 1 ⟶ 2 1 1 5 , 4 1 ⟶ 2 1 5 , 4 1 ⟶ 2 5 , 4 1 ⟶ 4 , 2 6 ⟶ 2 1 1 , 2 6 ⟶ 2 1 , 2 6 ⟶ 2 , 2 1 7 ⟶ 2 1 , 2 1 7 ⟶ 2 , 4 8 ⟶ 2 8 , 9 6 ⟶ 9 5 , 9 6 ⟶ 4 , 9 7 ⟶ 9 3 , 9 7 ⟶ 0 , 3 1 →= 1 1 1 3 , 5 1 →= 1 1 1 5 , 1 6 →= 6 1 1 1 , 1 1 7 →= 7 1 1 , 3 8 →= 6 8 , 5 8 →= 7 1 8 , 8 6 →= 8 5 , 8 7 →= 8 3 , 1 1 →= } 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 25-rule system { 0 1 ⟶ 2 1 1 3 , 0 1 ⟶ 2 1 3 , 0 1 ⟶ 2 3 , 0 1 ⟶ 0 , 4 1 ⟶ 2 1 1 5 , 4 1 ⟶ 2 1 5 , 4 1 ⟶ 2 5 , 4 1 ⟶ 4 , 2 6 ⟶ 2 1 1 , 2 6 ⟶ 2 1 , 2 6 ⟶ 2 , 2 1 7 ⟶ 2 1 , 2 1 7 ⟶ 2 , 4 8 ⟶ 2 8 , 9 6 ⟶ 9 5 , 9 7 ⟶ 9 3 , 3 1 →= 1 1 1 3 , 5 1 →= 1 1 1 5 , 1 6 →= 6 1 1 1 , 1 1 7 →= 7 1 1 , 3 8 →= 6 8 , 5 8 →= 7 1 8 , 8 6 →= 8 5 , 8 7 →= 8 3 , 1 1 →= } 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 4 ↦ 2, 2 ↦ 3, 6 ↦ 4, 7 ↦ 5, 9 ↦ 6, 5 ↦ 7, 3 ↦ 8, 8 ↦ 9 }, it remains to prove termination of the 18-rule system { 0 1 ⟶ 0 , 2 1 ⟶ 2 , 3 4 ⟶ 3 1 1 , 3 4 ⟶ 3 1 , 3 4 ⟶ 3 , 3 1 5 ⟶ 3 1 , 3 1 5 ⟶ 3 , 6 4 ⟶ 6 7 , 6 5 ⟶ 6 8 , 8 1 →= 1 1 1 8 , 7 1 →= 1 1 1 7 , 1 4 →= 4 1 1 1 , 1 1 5 →= 5 1 1 , 8 9 →= 4 9 , 7 9 →= 5 1 9 , 9 4 →= 9 7 , 9 5 →= 9 8 , 1 1 →= } 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 6 ↦ 3, 4 ↦ 4, 7 ↦ 5, 5 ↦ 6, 8 ↦ 7, 9 ↦ 8 }, it remains to prove termination of the 13-rule system { 0 1 ⟶ 0 , 2 1 ⟶ 2 , 3 4 ⟶ 3 5 , 3 6 ⟶ 3 7 , 7 1 →= 1 1 1 7 , 5 1 →= 1 1 1 5 , 1 4 →= 4 1 1 1 , 1 1 6 →= 6 1 1 , 7 8 →= 4 8 , 5 8 →= 6 1 8 , 8 4 →= 8 5 , 8 6 →= 8 7 , 1 1 →= } 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 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 1 ⎟ ⎜ 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 3 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 3 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 3 ↦ 2, 4 ↦ 3, 5 ↦ 4, 6 ↦ 5, 7 ↦ 6, 8 ↦ 7 }, it remains to prove termination of the 12-rule system { 0 1 ⟶ 0 , 2 3 ⟶ 2 4 , 2 5 ⟶ 2 6 , 6 1 →= 1 1 1 6 , 4 1 →= 1 1 1 4 , 1 3 →= 3 1 1 1 , 1 1 5 →= 5 1 1 , 6 7 →= 3 7 , 4 7 →= 5 1 7 , 7 3 →= 7 4 , 7 5 →= 7 6 , 1 1 →= } 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 0 3 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 3 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 2 ↦ 0, 3 ↦ 1, 4 ↦ 2, 5 ↦ 3, 6 ↦ 4, 1 ↦ 5, 7 ↦ 6 }, it remains to prove termination of the 11-rule system { 0 1 ⟶ 0 2 , 0 3 ⟶ 0 4 , 4 5 →= 5 5 5 4 , 2 5 →= 5 5 5 2 , 5 1 →= 1 5 5 5 , 5 5 3 →= 3 5 5 , 4 6 →= 1 6 , 2 6 →= 3 5 6 , 6 1 →= 6 2 , 6 3 →= 6 4 , 5 5 →= } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 6: 0 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 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 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 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 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 1 0 ⎟ ⎜ 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 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 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 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 ⎟ ⎜ 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 1 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 4 ↦ 3, 5 ↦ 4, 3 ↦ 5, 6 ↦ 6 }, it remains to prove termination of the 10-rule system { 0 1 ⟶ 0 2 , 3 4 →= 4 4 4 3 , 2 4 →= 4 4 4 2 , 4 1 →= 1 4 4 4 , 4 4 5 →= 5 4 4 , 3 6 →= 1 6 , 2 6 →= 5 4 6 , 6 1 →= 6 2 , 6 5 →= 6 3 , 4 4 →= } 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 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 1 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 3 ↦ 0, 4 ↦ 1, 2 ↦ 2, 1 ↦ 3, 5 ↦ 4, 6 ↦ 5 }, it remains to prove termination of the 9-rule system { 0 1 →= 1 1 1 0 , 2 1 →= 1 1 1 2 , 1 3 →= 3 1 1 1 , 1 1 4 →= 4 1 1 , 0 5 →= 3 5 , 2 5 →= 4 1 5 , 5 3 →= 5 2 , 5 4 →= 5 0 , 1 1 →= } The system is trivially terminating.