/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 4-rule system { 0 ⟶ , 0 0 ⟶ 0 1 2 , 1 ⟶ , 2 1 ⟶ 1 0 2 } Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols. After renaming modulo the bijection { (0,↑) ↦ 0, (0,↓) ↦ 1, (1,↓) ↦ 2, (2,↓) ↦ 3, (1,↑) ↦ 4, (2,↑) ↦ 5 }, it remains to prove termination of the 10-rule system { 0 1 ⟶ 0 2 3 , 0 1 ⟶ 4 3 , 0 1 ⟶ 5 , 5 2 ⟶ 4 1 3 , 5 2 ⟶ 0 3 , 5 2 ⟶ 5 , 1 →= , 1 1 →= 1 2 3 , 2 →= , 3 2 →= 2 1 3 } 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 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 5 ↦ 4 }, it remains to prove termination of the 8-rule system { 0 1 ⟶ 0 2 3 , 0 1 ⟶ 4 , 4 2 ⟶ 0 3 , 4 2 ⟶ 4 , 1 →= , 1 1 →= 1 2 3 , 2 →= , 3 2 →= 2 1 3 } Applying sparse tiling TROC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (5,0) ↦ 0, (0,1) ↦ 1, (1,1) ↦ 2, (0,2) ↦ 3, (2,3) ↦ 4, (3,1) ↦ 5, (1,2) ↦ 6, (3,2) ↦ 7, (1,3) ↦ 8, (3,3) ↦ 9, (5,4) ↦ 10, (4,1) ↦ 11, (4,2) ↦ 12, (4,3) ↦ 13, (2,1) ↦ 14, (0,3) ↦ 15, (2,2) ↦ 16, (5,1) ↦ 17, (5,2) ↦ 18, (5,3) ↦ 19 }, it remains to prove termination of the 84-rule system { 0 1 2 ⟶ 0 3 4 5 , 0 1 6 ⟶ 0 3 4 7 , 0 1 8 ⟶ 0 3 4 9 , 0 1 2 ⟶ 10 11 , 0 1 6 ⟶ 10 12 , 0 1 8 ⟶ 10 13 , 10 12 14 ⟶ 0 15 5 , 10 12 16 ⟶ 0 15 7 , 10 12 4 ⟶ 0 15 9 , 10 12 14 ⟶ 10 11 , 10 12 16 ⟶ 10 12 , 10 12 4 ⟶ 10 13 , 1 2 →= 1 , 1 6 →= 3 , 1 8 →= 15 , 2 2 →= 2 , 2 6 →= 6 , 2 8 →= 8 , 14 2 →= 14 , 14 6 →= 16 , 14 8 →= 4 , 5 2 →= 5 , 5 6 →= 7 , 5 8 →= 9 , 11 2 →= 11 , 11 6 →= 12 , 11 8 →= 13 , 17 2 →= 17 , 17 6 →= 18 , 17 8 →= 19 , 1 2 2 →= 1 6 4 5 , 1 2 6 →= 1 6 4 7 , 1 2 8 →= 1 6 4 9 , 2 2 2 →= 2 6 4 5 , 2 2 6 →= 2 6 4 7 , 2 2 8 →= 2 6 4 9 , 14 2 2 →= 14 6 4 5 , 14 2 6 →= 14 6 4 7 , 14 2 8 →= 14 6 4 9 , 5 2 2 →= 5 6 4 5 , 5 2 6 →= 5 6 4 7 , 5 2 8 →= 5 6 4 9 , 11 2 2 →= 11 6 4 5 , 11 2 6 →= 11 6 4 7 , 11 2 8 →= 11 6 4 9 , 17 2 2 →= 17 6 4 5 , 17 2 6 →= 17 6 4 7 , 17 2 8 →= 17 6 4 9 , 3 14 →= 1 , 3 16 →= 3 , 3 4 →= 15 , 6 14 →= 2 , 6 16 →= 6 , 6 4 →= 8 , 16 14 →= 14 , 16 16 →= 16 , 16 4 →= 4 , 7 14 →= 5 , 7 16 →= 7 , 7 4 →= 9 , 12 14 →= 11 , 12 16 →= 12 , 12 4 →= 13 , 18 14 →= 17 , 18 16 →= 18 , 18 4 →= 19 , 15 7 14 →= 3 14 8 5 , 15 7 16 →= 3 14 8 7 , 15 7 4 →= 3 14 8 9 , 8 7 14 →= 6 14 8 5 , 8 7 16 →= 6 14 8 7 , 8 7 4 →= 6 14 8 9 , 4 7 14 →= 16 14 8 5 , 4 7 16 →= 16 14 8 7 , 4 7 4 →= 16 14 8 9 , 9 7 14 →= 7 14 8 5 , 9 7 16 →= 7 14 8 7 , 9 7 4 →= 7 14 8 9 , 13 7 14 →= 12 14 8 5 , 13 7 16 →= 12 14 8 7 , 13 7 4 →= 12 14 8 9 , 19 7 14 →= 18 14 8 5 , 19 7 16 →= 18 14 8 7 , 19 7 4 →= 18 14 8 9 } 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 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 6 ↦ 2, 3 ↦ 3, 4 ↦ 4, 7 ↦ 5, 8 ↦ 6, 9 ↦ 7, 14 ↦ 8, 16 ↦ 9, 5 ↦ 10, 2 ↦ 11, 11 ↦ 12, 17 ↦ 13 }, it remains to prove termination of the 29-rule system { 0 1 2 ⟶ 0 3 4 5 , 0 1 6 ⟶ 0 3 4 7 , 8 2 →= 9 , 8 6 →= 4 , 10 2 →= 5 , 10 6 →= 7 , 1 11 2 →= 1 2 4 5 , 1 11 6 →= 1 2 4 7 , 11 11 2 →= 11 2 4 5 , 11 11 6 →= 11 2 4 7 , 8 11 2 →= 8 2 4 5 , 8 11 6 →= 8 2 4 7 , 10 11 2 →= 10 2 4 5 , 10 11 6 →= 10 2 4 7 , 12 11 2 →= 12 2 4 5 , 12 11 6 →= 12 2 4 7 , 13 11 2 →= 13 2 4 5 , 13 11 6 →= 13 2 4 7 , 3 8 →= 1 , 2 8 →= 11 , 6 5 8 →= 2 8 6 10 , 6 5 9 →= 2 8 6 5 , 6 5 4 →= 2 8 6 7 , 4 5 8 →= 9 8 6 10 , 4 5 9 →= 9 8 6 5 , 4 5 4 →= 9 8 6 7 , 7 5 8 →= 5 8 6 10 , 7 5 9 →= 5 8 6 5 , 7 5 4 →= 5 8 6 7 } 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 1 ⎟ ⎜ 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 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 1 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 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 28-rule system { 0 1 2 ⟶ 0 3 4 5 , 0 1 6 ⟶ 0 3 4 7 , 8 2 →= 9 , 8 6 →= 4 , 10 2 →= 5 , 10 6 →= 7 , 1 11 2 →= 1 2 4 5 , 1 11 6 →= 1 2 4 7 , 11 11 2 →= 11 2 4 5 , 11 11 6 →= 11 2 4 7 , 8 11 2 →= 8 2 4 5 , 8 11 6 →= 8 2 4 7 , 10 11 2 →= 10 2 4 5 , 10 11 6 →= 10 2 4 7 , 12 11 2 →= 12 2 4 5 , 12 11 6 →= 12 2 4 7 , 13 11 2 →= 13 2 4 5 , 13 11 6 →= 13 2 4 7 , 2 8 →= 11 , 6 5 8 →= 2 8 6 10 , 6 5 9 →= 2 8 6 5 , 6 5 4 →= 2 8 6 7 , 4 5 8 →= 9 8 6 10 , 4 5 9 →= 9 8 6 5 , 4 5 4 →= 9 8 6 7 , 7 5 8 →= 5 8 6 10 , 7 5 9 →= 5 8 6 5 , 7 5 4 →= 5 8 6 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 1 ⎟ ⎜ 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 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 8 ↦ 0, 2 ↦ 1, 9 ↦ 2, 6 ↦ 3, 4 ↦ 4, 10 ↦ 5, 5 ↦ 6, 7 ↦ 7, 1 ↦ 8, 11 ↦ 9, 12 ↦ 10, 13 ↦ 11 }, it remains to prove termination of the 26-rule system { 0 1 →= 2 , 0 3 →= 4 , 5 1 →= 6 , 5 3 →= 7 , 8 9 1 →= 8 1 4 6 , 8 9 3 →= 8 1 4 7 , 9 9 1 →= 9 1 4 6 , 9 9 3 →= 9 1 4 7 , 0 9 1 →= 0 1 4 6 , 0 9 3 →= 0 1 4 7 , 5 9 1 →= 5 1 4 6 , 5 9 3 →= 5 1 4 7 , 10 9 1 →= 10 1 4 6 , 10 9 3 →= 10 1 4 7 , 11 9 1 →= 11 1 4 6 , 11 9 3 →= 11 1 4 7 , 1 0 →= 9 , 3 6 0 →= 1 0 3 5 , 3 6 2 →= 1 0 3 6 , 3 6 4 →= 1 0 3 7 , 4 6 0 →= 2 0 3 5 , 4 6 2 →= 2 0 3 6 , 4 6 4 →= 2 0 3 7 , 7 6 0 →= 6 0 3 5 , 7 6 2 →= 6 0 3 6 , 7 6 4 →= 6 0 3 7 } The system is trivially terminating.