/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 { a ↦ 0, b ↦ 1, c ↦ 2 }, it remains to prove termination of the 4-rule system { 0 ⟶ , 0 0 ⟶ 0 1 , 1 ⟶ , 2 1 ⟶ 0 1 2 2 } Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols. After renaming modulo the bijection { (0,↑) ↦ 0, (0,↓) ↦ 1, (1,↓) ↦ 2, (1,↑) ↦ 3, (2,↑) ↦ 4, (2,↓) ↦ 5 }, it remains to prove termination of the 10-rule system { 0 1 ⟶ 0 2 , 0 1 ⟶ 3 , 4 2 ⟶ 0 2 5 5 , 4 2 ⟶ 3 5 5 , 4 2 ⟶ 4 5 , 4 2 ⟶ 4 , 1 →= , 1 1 →= 1 2 , 2 →= , 5 2 →= 1 2 5 5 } 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 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5 }, it remains to prove termination of the 8-rule system { 0 1 ⟶ 0 2 , 0 1 ⟶ 3 , 4 2 ⟶ 4 5 , 4 2 ⟶ 4 , 1 →= , 1 1 →= 1 2 , 2 →= , 5 2 →= 1 2 5 5 } 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 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 4 ↦ 3, 5 ↦ 4 }, it remains to prove termination of the 7-rule system { 0 1 ⟶ 0 2 , 3 2 ⟶ 3 4 , 3 2 ⟶ 3 , 1 →= , 1 1 →= 1 2 , 2 →= , 4 2 →= 1 2 4 4 } 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, (2,1) ↦ 4, (1,2) ↦ 5, (2,2) ↦ 6, (1,4) ↦ 7, (2,4) ↦ 8, (1,6) ↦ 9, (2,6) ↦ 10, (5,3) ↦ 11, (3,2) ↦ 12, (3,4) ↦ 13, (4,1) ↦ 14, (4,2) ↦ 15, (4,4) ↦ 16, (4,6) ↦ 17, (3,1) ↦ 18, (3,6) ↦ 19, (0,4) ↦ 20, (0,6) ↦ 21, (5,1) ↦ 22, (5,2) ↦ 23, (5,4) ↦ 24, (5,6) ↦ 25 }, it remains to prove termination of the 108-rule system { 0 1 2 ⟶ 0 3 4 , 0 1 5 ⟶ 0 3 6 , 0 1 7 ⟶ 0 3 8 , 0 1 9 ⟶ 0 3 10 , 11 12 4 ⟶ 11 13 14 , 11 12 6 ⟶ 11 13 15 , 11 12 8 ⟶ 11 13 16 , 11 12 10 ⟶ 11 13 17 , 11 12 4 ⟶ 11 18 , 11 12 6 ⟶ 11 12 , 11 12 8 ⟶ 11 13 , 11 12 10 ⟶ 11 19 , 1 2 →= 1 , 1 5 →= 3 , 1 7 →= 20 , 1 9 →= 21 , 2 2 →= 2 , 2 5 →= 5 , 2 7 →= 7 , 2 9 →= 9 , 4 2 →= 4 , 4 5 →= 6 , 4 7 →= 8 , 4 9 →= 10 , 18 2 →= 18 , 18 5 →= 12 , 18 7 →= 13 , 18 9 →= 19 , 14 2 →= 14 , 14 5 →= 15 , 14 7 →= 16 , 14 9 →= 17 , 22 2 →= 22 , 22 5 →= 23 , 22 7 →= 24 , 22 9 →= 25 , 1 2 2 →= 1 5 4 , 1 2 5 →= 1 5 6 , 1 2 7 →= 1 5 8 , 1 2 9 →= 1 5 10 , 2 2 2 →= 2 5 4 , 2 2 5 →= 2 5 6 , 2 2 7 →= 2 5 8 , 2 2 9 →= 2 5 10 , 4 2 2 →= 4 5 4 , 4 2 5 →= 4 5 6 , 4 2 7 →= 4 5 8 , 4 2 9 →= 4 5 10 , 18 2 2 →= 18 5 4 , 18 2 5 →= 18 5 6 , 18 2 7 →= 18 5 8 , 18 2 9 →= 18 5 10 , 14 2 2 →= 14 5 4 , 14 2 5 →= 14 5 6 , 14 2 7 →= 14 5 8 , 14 2 9 →= 14 5 10 , 22 2 2 →= 22 5 4 , 22 2 5 →= 22 5 6 , 22 2 7 →= 22 5 8 , 22 2 9 →= 22 5 10 , 3 4 →= 1 , 3 6 →= 3 , 3 8 →= 20 , 3 10 →= 21 , 5 4 →= 2 , 5 6 →= 5 , 5 8 →= 7 , 5 10 →= 9 , 6 4 →= 4 , 6 6 →= 6 , 6 8 →= 8 , 6 10 →= 10 , 12 4 →= 18 , 12 6 →= 12 , 12 8 →= 13 , 12 10 →= 19 , 15 4 →= 14 , 15 6 →= 15 , 15 8 →= 16 , 15 10 →= 17 , 23 4 →= 22 , 23 6 →= 23 , 23 8 →= 24 , 23 10 →= 25 , 20 15 4 →= 1 5 8 16 14 , 20 15 6 →= 1 5 8 16 15 , 20 15 8 →= 1 5 8 16 16 , 20 15 10 →= 1 5 8 16 17 , 7 15 4 →= 2 5 8 16 14 , 7 15 6 →= 2 5 8 16 15 , 7 15 8 →= 2 5 8 16 16 , 7 15 10 →= 2 5 8 16 17 , 8 15 4 →= 4 5 8 16 14 , 8 15 6 →= 4 5 8 16 15 , 8 15 8 →= 4 5 8 16 16 , 8 15 10 →= 4 5 8 16 17 , 13 15 4 →= 18 5 8 16 14 , 13 15 6 →= 18 5 8 16 15 , 13 15 8 →= 18 5 8 16 16 , 13 15 10 →= 18 5 8 16 17 , 16 15 4 →= 14 5 8 16 14 , 16 15 6 →= 14 5 8 16 15 , 16 15 8 →= 14 5 8 16 16 , 16 15 10 →= 14 5 8 16 17 , 24 15 4 →= 22 5 8 16 14 , 24 15 6 →= 22 5 8 16 15 , 24 15 8 →= 22 5 8 16 16 , 24 15 10 →= 22 5 8 16 17 } 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 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 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 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 23 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 24 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 25 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 5 ↦ 2, 3 ↦ 3, 6 ↦ 4, 4 ↦ 5, 14 ↦ 6, 15 ↦ 7, 2 ↦ 8, 18 ↦ 9, 22 ↦ 10, 8 ↦ 11, 20 ↦ 12, 7 ↦ 13, 16 ↦ 14 }, it remains to prove termination of the 25-rule system { 0 1 2 ⟶ 0 3 4 , 5 2 →= 4 , 6 2 →= 7 , 1 8 2 →= 1 2 4 , 8 8 2 →= 8 2 4 , 5 8 2 →= 5 2 4 , 9 8 2 →= 9 2 4 , 6 8 2 →= 6 2 4 , 10 8 2 →= 10 2 4 , 3 5 →= 1 , 3 11 →= 12 , 2 5 →= 8 , 2 11 →= 13 , 12 7 5 →= 1 2 11 14 6 , 12 7 4 →= 1 2 11 14 7 , 12 7 11 →= 1 2 11 14 14 , 13 7 5 →= 8 2 11 14 6 , 13 7 4 →= 8 2 11 14 7 , 13 7 11 →= 8 2 11 14 14 , 11 7 5 →= 5 2 11 14 6 , 11 7 4 →= 5 2 11 14 7 , 11 7 11 →= 5 2 11 14 14 , 14 7 5 →= 6 2 11 14 6 , 14 7 4 →= 6 2 11 14 7 , 14 7 11 →= 6 2 11 14 14 } 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 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 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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, 13 ↦ 12, 12 ↦ 13, 14 ↦ 14 }, it remains to prove termination of the 24-rule system { 0 1 2 ⟶ 0 3 4 , 5 2 →= 4 , 6 2 →= 7 , 1 8 2 →= 1 2 4 , 8 8 2 →= 8 2 4 , 5 8 2 →= 5 2 4 , 9 8 2 →= 9 2 4 , 6 8 2 →= 6 2 4 , 10 8 2 →= 10 2 4 , 3 5 →= 1 , 2 5 →= 8 , 2 11 →= 12 , 13 7 5 →= 1 2 11 14 6 , 13 7 4 →= 1 2 11 14 7 , 13 7 11 →= 1 2 11 14 14 , 12 7 5 →= 8 2 11 14 6 , 12 7 4 →= 8 2 11 14 7 , 12 7 11 →= 8 2 11 14 14 , 11 7 5 →= 5 2 11 14 6 , 11 7 4 →= 5 2 11 14 7 , 11 7 11 →= 5 2 11 14 14 , 14 7 5 →= 6 2 11 14 6 , 14 7 4 →= 6 2 11 14 7 , 14 7 11 →= 6 2 11 14 14 } 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 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, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 14 ↦ 13 }, it remains to prove termination of the 21-rule system { 0 1 2 ⟶ 0 3 4 , 5 2 →= 4 , 6 2 →= 7 , 1 8 2 →= 1 2 4 , 8 8 2 →= 8 2 4 , 5 8 2 →= 5 2 4 , 9 8 2 →= 9 2 4 , 6 8 2 →= 6 2 4 , 10 8 2 →= 10 2 4 , 3 5 →= 1 , 2 5 →= 8 , 2 11 →= 12 , 12 7 5 →= 8 2 11 13 6 , 12 7 4 →= 8 2 11 13 7 , 12 7 11 →= 8 2 11 13 13 , 11 7 5 →= 5 2 11 13 6 , 11 7 4 →= 5 2 11 13 7 , 11 7 11 →= 5 2 11 13 13 , 13 7 5 →= 6 2 11 13 6 , 13 7 4 →= 6 2 11 13 7 , 13 7 11 →= 6 2 11 13 13 } Applying sparse untiling TROCU(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { 5 ↦ 0, 2 ↦ 1, 4 ↦ 2, 6 ↦ 3, 7 ↦ 4, 1 ↦ 5, 8 ↦ 6, 9 ↦ 7, 10 ↦ 8, 11 ↦ 9, 12 ↦ 10, 13 ↦ 11 }, it remains to prove termination of the 19-rule system { 0 1 →= 2 , 3 1 →= 4 , 5 6 1 →= 5 1 2 , 6 6 1 →= 6 1 2 , 0 6 1 →= 0 1 2 , 7 6 1 →= 7 1 2 , 3 6 1 →= 3 1 2 , 8 6 1 →= 8 1 2 , 1 0 →= 6 , 1 9 →= 10 , 10 4 0 →= 6 1 9 11 3 , 10 4 2 →= 6 1 9 11 4 , 10 4 9 →= 6 1 9 11 11 , 9 4 0 →= 0 1 9 11 3 , 9 4 2 →= 0 1 9 11 4 , 9 4 9 →= 0 1 9 11 11 , 11 4 0 →= 3 1 9 11 3 , 11 4 2 →= 3 1 9 11 4 , 11 4 9 →= 3 1 9 11 11 } The system is trivially terminating.