/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 1 ⟶ 1 2 0 , 2 ⟶ 1 , 2 2 ⟶ 0 2 } The system was reversed. After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2 }, it remains to prove termination of the 4-rule system { 0 ⟶ , 1 0 ⟶ 0 2 1 , 2 ⟶ 1 , 2 2 ⟶ 2 0 } Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols. After renaming modulo the bijection { (1,↑) ↦ 0, (0,↓) ↦ 1, (0,↑) ↦ 2, (2,↓) ↦ 3, (1,↓) ↦ 4, (2,↑) ↦ 5 }, it remains to prove termination of the 10-rule system { 0 1 ⟶ 2 3 4 , 0 1 ⟶ 5 4 , 0 1 ⟶ 0 , 5 ⟶ 0 , 5 3 ⟶ 5 1 , 5 3 ⟶ 2 , 1 →= , 4 1 →= 1 3 4 , 3 →= 4 , 3 3 →= 3 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 5 ↦ 2, 4 ↦ 3, 3 ↦ 4 }, it remains to prove termination of the 8-rule system { 0 1 ⟶ 2 3 , 0 1 ⟶ 0 , 2 ⟶ 0 , 2 4 ⟶ 2 1 , 1 →= , 3 1 →= 1 4 3 , 4 →= 3 , 4 4 →= 4 1 } Applying sparse tiling TROC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (5,0) ↦ 0, (0,1) ↦ 1, (1,1) ↦ 2, (5,2) ↦ 3, (2,3) ↦ 4, (3,1) ↦ 5, (1,3) ↦ 6, (3,3) ↦ 7, (1,4) ↦ 8, (3,4) ↦ 9, (1,6) ↦ 10, (3,6) ↦ 11, (0,3) ↦ 12, (0,4) ↦ 13, (0,6) ↦ 14, (2,1) ↦ 15, (2,4) ↦ 16, (2,6) ↦ 17, (4,1) ↦ 18, (4,3) ↦ 19, (4,4) ↦ 20, (4,6) ↦ 21, (5,1) ↦ 22, (5,3) ↦ 23, (5,4) ↦ 24, (5,6) ↦ 25 }, it remains to prove termination of the 112-rule system { 0 1 2 ⟶ 3 4 5 , 0 1 6 ⟶ 3 4 7 , 0 1 8 ⟶ 3 4 9 , 0 1 10 ⟶ 3 4 11 , 0 1 2 ⟶ 0 1 , 0 1 6 ⟶ 0 12 , 0 1 8 ⟶ 0 13 , 0 1 10 ⟶ 0 14 , 3 15 ⟶ 0 1 , 3 4 ⟶ 0 12 , 3 16 ⟶ 0 13 , 3 17 ⟶ 0 14 , 3 16 18 ⟶ 3 15 2 , 3 16 19 ⟶ 3 15 6 , 3 16 20 ⟶ 3 15 8 , 3 16 21 ⟶ 3 15 10 , 1 2 →= 1 , 1 6 →= 12 , 1 8 →= 13 , 1 10 →= 14 , 2 2 →= 2 , 2 6 →= 6 , 2 8 →= 8 , 2 10 →= 10 , 15 2 →= 15 , 15 6 →= 4 , 15 8 →= 16 , 15 10 →= 17 , 5 2 →= 5 , 5 6 →= 7 , 5 8 →= 9 , 5 10 →= 11 , 18 2 →= 18 , 18 6 →= 19 , 18 8 →= 20 , 18 10 →= 21 , 22 2 →= 22 , 22 6 →= 23 , 22 8 →= 24 , 22 10 →= 25 , 12 5 2 →= 1 8 19 5 , 12 5 6 →= 1 8 19 7 , 12 5 8 →= 1 8 19 9 , 12 5 10 →= 1 8 19 11 , 6 5 2 →= 2 8 19 5 , 6 5 6 →= 2 8 19 7 , 6 5 8 →= 2 8 19 9 , 6 5 10 →= 2 8 19 11 , 4 5 2 →= 15 8 19 5 , 4 5 6 →= 15 8 19 7 , 4 5 8 →= 15 8 19 9 , 4 5 10 →= 15 8 19 11 , 7 5 2 →= 5 8 19 5 , 7 5 6 →= 5 8 19 7 , 7 5 8 →= 5 8 19 9 , 7 5 10 →= 5 8 19 11 , 19 5 2 →= 18 8 19 5 , 19 5 6 →= 18 8 19 7 , 19 5 8 →= 18 8 19 9 , 19 5 10 →= 18 8 19 11 , 23 5 2 →= 22 8 19 5 , 23 5 6 →= 22 8 19 7 , 23 5 8 →= 22 8 19 9 , 23 5 10 →= 22 8 19 11 , 13 18 →= 12 5 , 13 19 →= 12 7 , 13 20 →= 12 9 , 13 21 →= 12 11 , 8 18 →= 6 5 , 8 19 →= 6 7 , 8 20 →= 6 9 , 8 21 →= 6 11 , 16 18 →= 4 5 , 16 19 →= 4 7 , 16 20 →= 4 9 , 16 21 →= 4 11 , 9 18 →= 7 5 , 9 19 →= 7 7 , 9 20 →= 7 9 , 9 21 →= 7 11 , 20 18 →= 19 5 , 20 19 →= 19 7 , 20 20 →= 19 9 , 20 21 →= 19 11 , 24 18 →= 23 5 , 24 19 →= 23 7 , 24 20 →= 23 9 , 24 21 →= 23 11 , 13 20 18 →= 13 18 2 , 13 20 19 →= 13 18 6 , 13 20 20 →= 13 18 8 , 13 20 21 →= 13 18 10 , 8 20 18 →= 8 18 2 , 8 20 19 →= 8 18 6 , 8 20 20 →= 8 18 8 , 8 20 21 →= 8 18 10 , 16 20 18 →= 16 18 2 , 16 20 19 →= 16 18 6 , 16 20 20 →= 16 18 8 , 16 20 21 →= 16 18 10 , 9 20 18 →= 9 18 2 , 9 20 19 →= 9 18 6 , 9 20 20 →= 9 18 8 , 9 20 21 →= 9 18 10 , 20 20 18 →= 20 18 2 , 20 20 19 →= 20 18 6 , 20 20 20 →= 20 18 8 , 20 20 21 →= 20 18 10 , 24 20 18 →= 24 18 2 , 24 20 19 →= 24 18 6 , 24 20 20 →= 24 18 8 , 24 20 21 →= 24 18 10 } 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 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 23 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 24 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 25 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 3 ↦ 0, 16 ↦ 1, 18 ↦ 2, 15 ↦ 3, 2 ↦ 4, 19 ↦ 5, 6 ↦ 6, 8 ↦ 7, 20 ↦ 8, 5 ↦ 9, 7 ↦ 10, 9 ↦ 11, 13 ↦ 12, 24 ↦ 13 }, it remains to prove termination of the 29-rule system { 0 1 2 ⟶ 0 3 4 , 0 1 5 ⟶ 0 3 6 , 3 7 →= 1 , 2 7 →= 8 , 6 9 4 →= 4 7 5 9 , 6 9 6 →= 4 7 5 10 , 6 9 7 →= 4 7 5 11 , 10 9 4 →= 9 7 5 9 , 10 9 6 →= 9 7 5 10 , 10 9 7 →= 9 7 5 11 , 5 9 4 →= 2 7 5 9 , 5 9 6 →= 2 7 5 10 , 5 9 7 →= 2 7 5 11 , 7 2 →= 6 9 , 7 5 →= 6 10 , 11 2 →= 10 9 , 11 5 →= 10 10 , 12 8 2 →= 12 2 4 , 12 8 5 →= 12 2 6 , 7 8 2 →= 7 2 4 , 7 8 5 →= 7 2 6 , 1 8 2 →= 1 2 4 , 1 8 5 →= 1 2 6 , 11 8 2 →= 11 2 4 , 11 8 5 →= 11 2 6 , 8 8 2 →= 8 2 4 , 8 8 5 →= 8 2 6 , 13 8 2 →= 13 2 4 , 13 8 5 →= 13 2 6 } 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 1 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 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 , 0 1 5 ⟶ 0 3 6 , 2 7 →= 8 , 6 9 4 →= 4 7 5 9 , 6 9 6 →= 4 7 5 10 , 6 9 7 →= 4 7 5 11 , 10 9 4 →= 9 7 5 9 , 10 9 6 →= 9 7 5 10 , 10 9 7 →= 9 7 5 11 , 5 9 4 →= 2 7 5 9 , 5 9 6 →= 2 7 5 10 , 5 9 7 →= 2 7 5 11 , 7 2 →= 6 9 , 7 5 →= 6 10 , 11 2 →= 10 9 , 11 5 →= 10 10 , 12 8 2 →= 12 2 4 , 12 8 5 →= 12 2 6 , 7 8 2 →= 7 2 4 , 7 8 5 →= 7 2 6 , 1 8 2 →= 1 2 4 , 1 8 5 →= 1 2 6 , 11 8 2 →= 11 2 4 , 11 8 5 →= 11 2 6 , 8 8 2 →= 8 2 4 , 8 8 5 →= 8 2 6 , 13 8 2 →= 13 2 4 , 13 8 5 →= 13 2 6 } 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 { 2 ↦ 0, 7 ↦ 1, 8 ↦ 2, 6 ↦ 3, 9 ↦ 4, 4 ↦ 5, 5 ↦ 6, 10 ↦ 7, 11 ↦ 8, 12 ↦ 9, 1 ↦ 10, 13 ↦ 11 }, it remains to prove termination of the 26-rule system { 0 1 →= 2 , 3 4 5 →= 5 1 6 4 , 3 4 3 →= 5 1 6 7 , 3 4 1 →= 5 1 6 8 , 7 4 5 →= 4 1 6 4 , 7 4 3 →= 4 1 6 7 , 7 4 1 →= 4 1 6 8 , 6 4 5 →= 0 1 6 4 , 6 4 3 →= 0 1 6 7 , 6 4 1 →= 0 1 6 8 , 1 0 →= 3 4 , 1 6 →= 3 7 , 8 0 →= 7 4 , 8 6 →= 7 7 , 9 2 0 →= 9 0 5 , 9 2 6 →= 9 0 3 , 1 2 0 →= 1 0 5 , 1 2 6 →= 1 0 3 , 10 2 0 →= 10 0 5 , 10 2 6 →= 10 0 3 , 8 2 0 →= 8 0 5 , 8 2 6 →= 8 0 3 , 2 2 0 →= 2 0 5 , 2 2 6 →= 2 0 3 , 11 2 0 →= 11 0 5 , 11 2 6 →= 11 0 3 } The system is trivially terminating.