/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 2 , 1 0 1 ⟶ , 2 2 ⟶ 0 0 0 1 } The system was reversed. After renaming modulo the bijection { 0 ↦ 0, 2 ↦ 1, 1 ↦ 2 }, it remains to prove termination of the 3-rule system { 0 ⟶ 1 2 , 2 0 2 ⟶ , 1 1 ⟶ 2 0 0 0 } Applying sparse tiling TRFC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (0,0) ↦ 0, (0,1) ↦ 1, (1,2) ↦ 2, (2,0) ↦ 3, (2,1) ↦ 4, (0,2) ↦ 5, (2,2) ↦ 6, (0,4) ↦ 7, (2,4) ↦ 8, (1,0) ↦ 9, (1,1) ↦ 10, (3,0) ↦ 11, (3,1) ↦ 12, (1,4) ↦ 13, (3,2) ↦ 14, (3,4) ↦ 15 }, it remains to prove termination of the 48-rule system { 0 0 ⟶ 1 2 3 , 0 1 ⟶ 1 2 4 , 0 5 ⟶ 1 2 6 , 0 7 ⟶ 1 2 8 , 9 0 ⟶ 10 2 3 , 9 1 ⟶ 10 2 4 , 9 5 ⟶ 10 2 6 , 9 7 ⟶ 10 2 8 , 3 0 ⟶ 4 2 3 , 3 1 ⟶ 4 2 4 , 3 5 ⟶ 4 2 6 , 3 7 ⟶ 4 2 8 , 11 0 ⟶ 12 2 3 , 11 1 ⟶ 12 2 4 , 11 5 ⟶ 12 2 6 , 11 7 ⟶ 12 2 8 , 5 3 5 3 ⟶ 0 , 5 3 5 4 ⟶ 1 , 5 3 5 6 ⟶ 5 , 5 3 5 8 ⟶ 7 , 2 3 5 3 ⟶ 9 , 2 3 5 4 ⟶ 10 , 2 3 5 6 ⟶ 2 , 2 3 5 8 ⟶ 13 , 6 3 5 3 ⟶ 3 , 6 3 5 4 ⟶ 4 , 6 3 5 6 ⟶ 6 , 6 3 5 8 ⟶ 8 , 14 3 5 3 ⟶ 11 , 14 3 5 4 ⟶ 12 , 14 3 5 6 ⟶ 14 , 14 3 5 8 ⟶ 15 , 1 10 9 ⟶ 5 3 0 0 0 , 1 10 10 ⟶ 5 3 0 0 1 , 1 10 2 ⟶ 5 3 0 0 5 , 1 10 13 ⟶ 5 3 0 0 7 , 10 10 9 ⟶ 2 3 0 0 0 , 10 10 10 ⟶ 2 3 0 0 1 , 10 10 2 ⟶ 2 3 0 0 5 , 10 10 13 ⟶ 2 3 0 0 7 , 4 10 9 ⟶ 6 3 0 0 0 , 4 10 10 ⟶ 6 3 0 0 1 , 4 10 2 ⟶ 6 3 0 0 5 , 4 10 13 ⟶ 6 3 0 0 7 , 12 10 9 ⟶ 14 3 0 0 0 , 12 10 10 ⟶ 14 3 0 0 1 , 12 10 2 ⟶ 14 3 0 0 5 , 12 10 13 ⟶ 14 3 0 0 7 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 9 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 6 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 5 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 5 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 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 }, it remains to prove termination of the 13-rule system { 0 0 ⟶ 1 2 3 , 0 1 ⟶ 1 2 4 , 0 5 ⟶ 1 2 6 , 7 0 ⟶ 8 2 3 , 7 1 ⟶ 8 2 4 , 7 5 ⟶ 8 2 6 , 3 0 ⟶ 4 2 3 , 3 1 ⟶ 4 2 4 , 3 5 ⟶ 4 2 6 , 2 3 5 3 ⟶ 7 , 2 3 5 4 ⟶ 8 , 1 8 2 ⟶ 5 3 0 0 5 , 4 8 2 ⟶ 6 3 0 0 5 } Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols. After renaming modulo the bijection { (0,↑) ↦ 0, (0,↓) ↦ 1, (1,↑) ↦ 2, (2,↓) ↦ 3, (3,↓) ↦ 4, (2,↑) ↦ 5, (3,↑) ↦ 6, (1,↓) ↦ 7, (4,↓) ↦ 8, (4,↑) ↦ 9, (5,↓) ↦ 10, (6,↓) ↦ 11, (7,↑) ↦ 12, (8,↓) ↦ 13, (7,↓) ↦ 14 }, it remains to prove termination of the 41-rule system { 0 1 ⟶ 2 3 4 , 0 1 ⟶ 5 4 , 0 1 ⟶ 6 , 0 7 ⟶ 2 3 8 , 0 7 ⟶ 5 8 , 0 7 ⟶ 9 , 0 10 ⟶ 2 3 11 , 0 10 ⟶ 5 11 , 12 1 ⟶ 5 4 , 12 1 ⟶ 6 , 12 7 ⟶ 5 8 , 12 7 ⟶ 9 , 12 10 ⟶ 5 11 , 6 1 ⟶ 9 3 4 , 6 1 ⟶ 5 4 , 6 1 ⟶ 6 , 6 7 ⟶ 9 3 8 , 6 7 ⟶ 5 8 , 6 7 ⟶ 9 , 6 10 ⟶ 9 3 11 , 6 10 ⟶ 5 11 , 5 4 10 4 ⟶ 12 , 2 13 3 ⟶ 6 1 1 10 , 2 13 3 ⟶ 0 1 10 , 2 13 3 ⟶ 0 10 , 9 13 3 ⟶ 6 1 1 10 , 9 13 3 ⟶ 0 1 10 , 9 13 3 ⟶ 0 10 , 1 1 →= 7 3 4 , 1 7 →= 7 3 8 , 1 10 →= 7 3 11 , 14 1 →= 13 3 4 , 14 7 →= 13 3 8 , 14 10 →= 13 3 11 , 4 1 →= 8 3 4 , 4 7 →= 8 3 8 , 4 10 →= 8 3 11 , 3 4 10 4 →= 14 , 3 4 10 8 →= 13 , 7 13 3 →= 10 4 1 1 10 , 8 13 3 →= 11 4 1 1 10 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 6 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 6 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 6 ↦ 0, 1 ↦ 1, 9 ↦ 2, 3 ↦ 3, 4 ↦ 4, 7 ↦ 5, 8 ↦ 6, 10 ↦ 7, 11 ↦ 8, 13 ↦ 9, 14 ↦ 10 }, it remains to prove termination of the 17-rule system { 0 1 ⟶ 2 3 4 , 0 5 ⟶ 2 3 6 , 0 7 ⟶ 2 3 8 , 2 9 3 ⟶ 0 1 1 7 , 1 1 →= 5 3 4 , 1 5 →= 5 3 6 , 1 7 →= 5 3 8 , 10 1 →= 9 3 4 , 10 5 →= 9 3 6 , 10 7 →= 9 3 8 , 4 1 →= 6 3 4 , 4 5 →= 6 3 6 , 4 7 →= 6 3 8 , 3 4 7 4 →= 10 , 3 4 7 6 →= 9 , 5 9 3 →= 7 4 1 1 7 , 6 9 3 →= 8 4 1 1 7 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 8: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 1 ⎟ ⎜ 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 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 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 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 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 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 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 ⎟ ⎜ 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 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 ⎟ ⎜ 0 0 0 0 0 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 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 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 ⎟ ⎜ 0 0 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 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 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 ⎟ ⎜ 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 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 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 ⎟ ⎜ 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 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 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 ⎟ ⎜ 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 10 ↦ 9, 9 ↦ 10 }, it remains to prove termination of the 16-rule system { 0 1 ⟶ 2 3 4 , 0 5 ⟶ 2 3 6 , 0 7 ⟶ 2 3 8 , 1 1 →= 5 3 4 , 1 5 →= 5 3 6 , 1 7 →= 5 3 8 , 9 1 →= 10 3 4 , 9 5 →= 10 3 6 , 9 7 →= 10 3 8 , 4 1 →= 6 3 4 , 4 5 →= 6 3 6 , 4 7 →= 6 3 8 , 3 4 7 4 →= 9 , 3 4 7 6 →= 10 , 5 10 3 →= 7 4 1 1 7 , 6 10 3 →= 8 4 1 1 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 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 1 ↦ 0, 5 ↦ 1, 3 ↦ 2, 4 ↦ 3, 6 ↦ 4, 7 ↦ 5, 8 ↦ 6, 9 ↦ 7, 10 ↦ 8 }, it remains to prove termination of the 13-rule system { 0 0 →= 1 2 3 , 0 1 →= 1 2 4 , 0 5 →= 1 2 6 , 7 0 →= 8 2 3 , 7 1 →= 8 2 4 , 7 5 →= 8 2 6 , 3 0 →= 4 2 3 , 3 1 →= 4 2 4 , 3 5 →= 4 2 6 , 2 3 5 3 →= 7 , 2 3 5 4 →= 8 , 1 8 2 →= 5 3 0 0 5 , 4 8 2 →= 6 3 0 0 5 } The system is trivially terminating.