/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 }, it remains to prove termination of the 3-rule system { 0 1 ⟶ , 0 2 ⟶ 1 2 0 1 2 0 , 1 2 ⟶ } The system was reversed. After renaming modulo the bijection { 1 ↦ 0, 0 ↦ 1, 2 ↦ 2 }, it remains to prove termination of the 3-rule system { 0 1 ⟶ , 2 1 ⟶ 1 2 0 1 2 0 , 2 0 ⟶ } Applying sparse tiling TRFC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (2,0) ↦ 0, (0,1) ↦ 1, (1,1) ↦ 2, (2,1) ↦ 3, (1,2) ↦ 4, (2,2) ↦ 5, (1,4) ↦ 6, (2,4) ↦ 7, (0,2) ↦ 8, (0,4) ↦ 9 }, it remains to prove termination of the 21-rule system { 0 1 2 ⟶ 3 , 0 1 4 ⟶ 5 , 0 1 6 ⟶ 7 , 8 3 2 ⟶ 1 4 0 1 4 0 1 , 8 3 4 ⟶ 1 4 0 1 4 0 8 , 8 3 6 ⟶ 1 4 0 1 4 0 9 , 4 3 2 ⟶ 2 4 0 1 4 0 1 , 4 3 4 ⟶ 2 4 0 1 4 0 8 , 4 3 6 ⟶ 2 4 0 1 4 0 9 , 5 3 2 ⟶ 3 4 0 1 4 0 1 , 5 3 4 ⟶ 3 4 0 1 4 0 8 , 5 3 6 ⟶ 3 4 0 1 4 0 9 , 8 0 1 ⟶ 1 , 8 0 8 ⟶ 8 , 8 0 9 ⟶ 9 , 4 0 1 ⟶ 2 , 4 0 8 ⟶ 4 , 4 0 9 ⟶ 6 , 5 0 1 ⟶ 3 , 5 0 8 ⟶ 5 , 5 0 9 ⟶ 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 8 ↦ 6, 6 ↦ 7, 9 ↦ 8 }, it remains to prove termination of the 19-rule system { 0 1 2 ⟶ 3 , 0 1 4 ⟶ 5 , 6 3 2 ⟶ 1 4 0 1 4 0 1 , 6 3 4 ⟶ 1 4 0 1 4 0 6 , 6 3 7 ⟶ 1 4 0 1 4 0 8 , 4 3 2 ⟶ 2 4 0 1 4 0 1 , 4 3 4 ⟶ 2 4 0 1 4 0 6 , 4 3 7 ⟶ 2 4 0 1 4 0 8 , 5 3 2 ⟶ 3 4 0 1 4 0 1 , 5 3 4 ⟶ 3 4 0 1 4 0 6 , 5 3 7 ⟶ 3 4 0 1 4 0 8 , 6 0 1 ⟶ 1 , 6 0 6 ⟶ 6 , 6 0 8 ⟶ 8 , 4 0 1 ⟶ 2 , 4 0 6 ⟶ 4 , 4 0 8 ⟶ 7 , 5 0 1 ⟶ 3 , 5 0 6 ⟶ 5 } Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols. After renaming modulo the bijection { (0,↑) ↦ 0, (1,↓) ↦ 1, (4,↓) ↦ 2, (5,↑) ↦ 3, (6,↑) ↦ 4, (3,↓) ↦ 5, (2,↓) ↦ 6, (4,↑) ↦ 7, (0,↓) ↦ 8, (6,↓) ↦ 9, (7,↓) ↦ 10, (8,↓) ↦ 11, (5,↓) ↦ 12 }, it remains to prove termination of the 61-rule system { 0 1 2 ⟶ 3 , 4 5 6 ⟶ 7 8 1 2 8 1 , 4 5 6 ⟶ 0 1 2 8 1 , 4 5 6 ⟶ 7 8 1 , 4 5 6 ⟶ 0 1 , 4 5 2 ⟶ 7 8 1 2 8 9 , 4 5 2 ⟶ 0 1 2 8 9 , 4 5 2 ⟶ 7 8 9 , 4 5 2 ⟶ 0 9 , 4 5 2 ⟶ 4 , 4 5 10 ⟶ 7 8 1 2 8 11 , 4 5 10 ⟶ 0 1 2 8 11 , 4 5 10 ⟶ 7 8 11 , 4 5 10 ⟶ 0 11 , 7 5 6 ⟶ 7 8 1 2 8 1 , 7 5 6 ⟶ 0 1 2 8 1 , 7 5 6 ⟶ 7 8 1 , 7 5 6 ⟶ 0 1 , 7 5 2 ⟶ 7 8 1 2 8 9 , 7 5 2 ⟶ 0 1 2 8 9 , 7 5 2 ⟶ 7 8 9 , 7 5 2 ⟶ 0 9 , 7 5 2 ⟶ 4 , 7 5 10 ⟶ 7 8 1 2 8 11 , 7 5 10 ⟶ 0 1 2 8 11 , 7 5 10 ⟶ 7 8 11 , 7 5 10 ⟶ 0 11 , 3 5 6 ⟶ 7 8 1 2 8 1 , 3 5 6 ⟶ 0 1 2 8 1 , 3 5 6 ⟶ 7 8 1 , 3 5 6 ⟶ 0 1 , 3 5 2 ⟶ 7 8 1 2 8 9 , 3 5 2 ⟶ 0 1 2 8 9 , 3 5 2 ⟶ 7 8 9 , 3 5 2 ⟶ 0 9 , 3 5 2 ⟶ 4 , 3 5 10 ⟶ 7 8 1 2 8 11 , 3 5 10 ⟶ 0 1 2 8 11 , 3 5 10 ⟶ 7 8 11 , 3 5 10 ⟶ 0 11 , 7 8 9 ⟶ 7 , 3 8 9 ⟶ 3 , 8 1 6 →= 5 , 8 1 2 →= 12 , 9 5 6 →= 1 2 8 1 2 8 1 , 9 5 2 →= 1 2 8 1 2 8 9 , 9 5 10 →= 1 2 8 1 2 8 11 , 2 5 6 →= 6 2 8 1 2 8 1 , 2 5 2 →= 6 2 8 1 2 8 9 , 2 5 10 →= 6 2 8 1 2 8 11 , 12 5 6 →= 5 2 8 1 2 8 1 , 12 5 2 →= 5 2 8 1 2 8 9 , 12 5 10 →= 5 2 8 1 2 8 11 , 9 8 1 →= 1 , 9 8 9 →= 9 , 9 8 11 →= 11 , 2 8 1 →= 6 , 2 8 9 →= 2 , 2 8 11 →= 10 , 12 8 1 →= 5 , 12 8 9 →= 12 } 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 7 ↦ 0, 8 ↦ 1, 9 ↦ 2, 3 ↦ 3, 1 ↦ 4, 6 ↦ 5, 5 ↦ 6, 2 ↦ 7, 12 ↦ 8, 10 ↦ 9, 11 ↦ 10 }, it remains to prove termination of the 21-rule system { 0 1 2 ⟶ 0 , 3 1 2 ⟶ 3 , 1 4 5 →= 6 , 1 4 7 →= 8 , 2 6 5 →= 4 7 1 4 7 1 4 , 2 6 7 →= 4 7 1 4 7 1 2 , 2 6 9 →= 4 7 1 4 7 1 10 , 7 6 5 →= 5 7 1 4 7 1 4 , 7 6 7 →= 5 7 1 4 7 1 2 , 7 6 9 →= 5 7 1 4 7 1 10 , 8 6 5 →= 6 7 1 4 7 1 4 , 8 6 7 →= 6 7 1 4 7 1 2 , 8 6 9 →= 6 7 1 4 7 1 10 , 2 1 4 →= 4 , 2 1 2 →= 2 , 2 1 10 →= 10 , 7 1 4 →= 5 , 7 1 2 →= 7 , 7 1 10 →= 9 , 8 1 4 →= 6 , 8 1 2 →= 8 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 4: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 1 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 1 1 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 3 ↦ 0, 1 ↦ 1, 2 ↦ 2, 4 ↦ 3, 5 ↦ 4, 6 ↦ 5, 7 ↦ 6, 8 ↦ 7, 9 ↦ 8, 10 ↦ 9 }, it remains to prove termination of the 20-rule system { 0 1 2 ⟶ 0 , 1 3 4 →= 5 , 1 3 6 →= 7 , 2 5 4 →= 3 6 1 3 6 1 3 , 2 5 6 →= 3 6 1 3 6 1 2 , 2 5 8 →= 3 6 1 3 6 1 9 , 6 5 4 →= 4 6 1 3 6 1 3 , 6 5 6 →= 4 6 1 3 6 1 2 , 6 5 8 →= 4 6 1 3 6 1 9 , 7 5 4 →= 5 6 1 3 6 1 3 , 7 5 6 →= 5 6 1 3 6 1 2 , 7 5 8 →= 5 6 1 3 6 1 9 , 2 1 3 →= 3 , 2 1 2 →= 2 , 2 1 9 →= 9 , 6 1 3 →= 4 , 6 1 2 →= 6 , 6 1 9 →= 8 , 7 1 3 →= 5 , 7 1 2 →= 7 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 4: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 1 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 1 1 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 1 ↦ 0, 3 ↦ 1, 4 ↦ 2, 5 ↦ 3, 6 ↦ 4, 7 ↦ 5, 2 ↦ 6, 8 ↦ 7, 9 ↦ 8 }, it remains to prove termination of the 19-rule system { 0 1 2 →= 3 , 0 1 4 →= 5 , 6 3 2 →= 1 4 0 1 4 0 1 , 6 3 4 →= 1 4 0 1 4 0 6 , 6 3 7 →= 1 4 0 1 4 0 8 , 4 3 2 →= 2 4 0 1 4 0 1 , 4 3 4 →= 2 4 0 1 4 0 6 , 4 3 7 →= 2 4 0 1 4 0 8 , 5 3 2 →= 3 4 0 1 4 0 1 , 5 3 4 →= 3 4 0 1 4 0 6 , 5 3 7 →= 3 4 0 1 4 0 8 , 6 0 1 →= 1 , 6 0 6 →= 6 , 6 0 8 →= 8 , 4 0 1 →= 2 , 4 0 6 →= 4 , 4 0 8 →= 7 , 5 0 1 →= 3 , 5 0 6 →= 5 } The system is trivially terminating.