/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, a ↦ 2, B ↦ 3 }, it remains to prove termination of the 4-rule system { 0 1 ⟶ 1 2 3 0 , 3 2 ⟶ 2 1 0 3 , 0 2 ⟶ , 3 1 ⟶ } The system was reversed. After renaming modulo the bijection { 1 ↦ 0, 0 ↦ 1, 3 ↦ 2, 2 ↦ 3 }, it remains to prove termination of the 4-rule system { 0 1 ⟶ 1 2 3 0 , 3 2 ⟶ 2 1 0 3 , 3 1 ⟶ , 0 2 ⟶ } Applying sparse tiling TRFC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (0,0) ↦ 0, (0,1) ↦ 1, (1,0) ↦ 2, (1,2) ↦ 3, (2,3) ↦ 4, (3,0) ↦ 5, (1,1) ↦ 6, (0,2) ↦ 7, (1,3) ↦ 8, (0,3) ↦ 9, (1,5) ↦ 10, (0,5) ↦ 11, (2,0) ↦ 12, (2,1) ↦ 13, (3,1) ↦ 14, (3,2) ↦ 15, (2,2) ↦ 16, (3,3) ↦ 17, (2,5) ↦ 18, (3,5) ↦ 19 }, it remains to prove termination of the 80-rule system { 0 1 2 ⟶ 1 3 4 5 0 , 0 1 6 ⟶ 1 3 4 5 1 , 0 1 3 ⟶ 1 3 4 5 7 , 0 1 8 ⟶ 1 3 4 5 9 , 0 1 10 ⟶ 1 3 4 5 11 , 2 1 2 ⟶ 6 3 4 5 0 , 2 1 6 ⟶ 6 3 4 5 1 , 2 1 3 ⟶ 6 3 4 5 7 , 2 1 8 ⟶ 6 3 4 5 9 , 2 1 10 ⟶ 6 3 4 5 11 , 12 1 2 ⟶ 13 3 4 5 0 , 12 1 6 ⟶ 13 3 4 5 1 , 12 1 3 ⟶ 13 3 4 5 7 , 12 1 8 ⟶ 13 3 4 5 9 , 12 1 10 ⟶ 13 3 4 5 11 , 5 1 2 ⟶ 14 3 4 5 0 , 5 1 6 ⟶ 14 3 4 5 1 , 5 1 3 ⟶ 14 3 4 5 7 , 5 1 8 ⟶ 14 3 4 5 9 , 5 1 10 ⟶ 14 3 4 5 11 , 9 15 12 ⟶ 7 13 2 9 5 , 9 15 13 ⟶ 7 13 2 9 14 , 9 15 16 ⟶ 7 13 2 9 15 , 9 15 4 ⟶ 7 13 2 9 17 , 9 15 18 ⟶ 7 13 2 9 19 , 8 15 12 ⟶ 3 13 2 9 5 , 8 15 13 ⟶ 3 13 2 9 14 , 8 15 16 ⟶ 3 13 2 9 15 , 8 15 4 ⟶ 3 13 2 9 17 , 8 15 18 ⟶ 3 13 2 9 19 , 4 15 12 ⟶ 16 13 2 9 5 , 4 15 13 ⟶ 16 13 2 9 14 , 4 15 16 ⟶ 16 13 2 9 15 , 4 15 4 ⟶ 16 13 2 9 17 , 4 15 18 ⟶ 16 13 2 9 19 , 17 15 12 ⟶ 15 13 2 9 5 , 17 15 13 ⟶ 15 13 2 9 14 , 17 15 16 ⟶ 15 13 2 9 15 , 17 15 4 ⟶ 15 13 2 9 17 , 17 15 18 ⟶ 15 13 2 9 19 , 9 14 2 ⟶ 0 , 9 14 6 ⟶ 1 , 9 14 3 ⟶ 7 , 9 14 8 ⟶ 9 , 9 14 10 ⟶ 11 , 8 14 2 ⟶ 2 , 8 14 6 ⟶ 6 , 8 14 3 ⟶ 3 , 8 14 8 ⟶ 8 , 8 14 10 ⟶ 10 , 4 14 2 ⟶ 12 , 4 14 6 ⟶ 13 , 4 14 3 ⟶ 16 , 4 14 8 ⟶ 4 , 4 14 10 ⟶ 18 , 17 14 2 ⟶ 5 , 17 14 6 ⟶ 14 , 17 14 3 ⟶ 15 , 17 14 8 ⟶ 17 , 17 14 10 ⟶ 19 , 0 7 12 ⟶ 0 , 0 7 13 ⟶ 1 , 0 7 16 ⟶ 7 , 0 7 4 ⟶ 9 , 0 7 18 ⟶ 11 , 2 7 12 ⟶ 2 , 2 7 13 ⟶ 6 , 2 7 16 ⟶ 3 , 2 7 4 ⟶ 8 , 2 7 18 ⟶ 10 , 12 7 12 ⟶ 12 , 12 7 13 ⟶ 13 , 12 7 16 ⟶ 16 , 12 7 4 ⟶ 4 , 12 7 18 ⟶ 18 , 5 7 12 ⟶ 5 , 5 7 13 ⟶ 14 , 5 7 16 ⟶ 15 , 5 7 4 ⟶ 17 , 5 7 18 ⟶ 19 } 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 2 ⎟ ⎜ 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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, 14 ↦ 8, 9 ↦ 9, 15 ↦ 10, 13 ↦ 11, 16 ↦ 12, 17 ↦ 13 }, it remains to prove termination of the 28-rule system { 0 1 2 ⟶ 1 3 4 5 0 , 0 1 6 ⟶ 1 3 4 5 1 , 0 1 3 ⟶ 1 3 4 5 7 , 2 1 2 ⟶ 6 3 4 5 0 , 2 1 6 ⟶ 6 3 4 5 1 , 2 1 3 ⟶ 6 3 4 5 7 , 5 1 2 ⟶ 8 3 4 5 0 , 5 1 6 ⟶ 8 3 4 5 1 , 5 1 3 ⟶ 8 3 4 5 7 , 9 10 11 ⟶ 7 11 2 9 8 , 9 10 12 ⟶ 7 11 2 9 10 , 9 10 4 ⟶ 7 11 2 9 13 , 4 10 11 ⟶ 12 11 2 9 8 , 4 10 12 ⟶ 12 11 2 9 10 , 4 10 4 ⟶ 12 11 2 9 13 , 13 10 11 ⟶ 10 11 2 9 8 , 13 10 12 ⟶ 10 11 2 9 10 , 13 10 4 ⟶ 10 11 2 9 13 , 9 8 2 ⟶ 0 , 9 8 6 ⟶ 1 , 9 8 3 ⟶ 7 , 4 8 3 ⟶ 12 , 13 8 3 ⟶ 10 , 0 7 11 ⟶ 1 , 2 7 11 ⟶ 6 , 5 7 11 ⟶ 8 , 5 7 12 ⟶ 10 , 5 7 4 ⟶ 13 } Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols. After renaming modulo the bijection { (0,↑) ↦ 0, (1,↓) ↦ 1, (2,↓) ↦ 2, (4,↑) ↦ 3, (5,↓) ↦ 4, (0,↓) ↦ 5, (5,↑) ↦ 6, (6,↓) ↦ 7, (3,↓) ↦ 8, (7,↓) ↦ 9, (2,↑) ↦ 10, (9,↑) ↦ 11, (10,↓) ↦ 12, (11,↓) ↦ 13, (9,↓) ↦ 14, (8,↓) ↦ 15, (12,↓) ↦ 16, (4,↓) ↦ 17, (13,↓) ↦ 18, (13,↑) ↦ 19 }, it remains to prove termination of the 72-rule system { 0 1 2 ⟶ 3 4 5 , 0 1 2 ⟶ 6 5 , 0 1 2 ⟶ 0 , 0 1 7 ⟶ 3 4 1 , 0 1 7 ⟶ 6 1 , 0 1 8 ⟶ 3 4 9 , 0 1 8 ⟶ 6 9 , 10 1 2 ⟶ 3 4 5 , 10 1 2 ⟶ 6 5 , 10 1 2 ⟶ 0 , 10 1 7 ⟶ 3 4 1 , 10 1 7 ⟶ 6 1 , 10 1 8 ⟶ 3 4 9 , 10 1 8 ⟶ 6 9 , 6 1 2 ⟶ 3 4 5 , 6 1 2 ⟶ 6 5 , 6 1 2 ⟶ 0 , 6 1 7 ⟶ 3 4 1 , 6 1 7 ⟶ 6 1 , 6 1 8 ⟶ 3 4 9 , 6 1 8 ⟶ 6 9 , 11 12 13 ⟶ 10 14 15 , 11 12 13 ⟶ 11 15 , 11 12 16 ⟶ 10 14 12 , 11 12 16 ⟶ 11 12 , 11 12 17 ⟶ 10 14 18 , 11 12 17 ⟶ 11 18 , 11 12 17 ⟶ 19 , 3 12 13 ⟶ 10 14 15 , 3 12 13 ⟶ 11 15 , 3 12 16 ⟶ 10 14 12 , 3 12 16 ⟶ 11 12 , 3 12 17 ⟶ 10 14 18 , 3 12 17 ⟶ 11 18 , 3 12 17 ⟶ 19 , 19 12 13 ⟶ 10 14 15 , 19 12 13 ⟶ 11 15 , 19 12 16 ⟶ 10 14 12 , 19 12 16 ⟶ 11 12 , 19 12 17 ⟶ 10 14 18 , 19 12 17 ⟶ 11 18 , 19 12 17 ⟶ 19 , 11 15 2 ⟶ 0 , 6 9 17 ⟶ 19 , 5 1 2 →= 1 8 17 4 5 , 5 1 7 →= 1 8 17 4 1 , 5 1 8 →= 1 8 17 4 9 , 2 1 2 →= 7 8 17 4 5 , 2 1 7 →= 7 8 17 4 1 , 2 1 8 →= 7 8 17 4 9 , 4 1 2 →= 15 8 17 4 5 , 4 1 7 →= 15 8 17 4 1 , 4 1 8 →= 15 8 17 4 9 , 14 12 13 →= 9 13 2 14 15 , 14 12 16 →= 9 13 2 14 12 , 14 12 17 →= 9 13 2 14 18 , 17 12 13 →= 16 13 2 14 15 , 17 12 16 →= 16 13 2 14 12 , 17 12 17 →= 16 13 2 14 18 , 18 12 13 →= 12 13 2 14 15 , 18 12 16 →= 12 13 2 14 12 , 18 12 17 →= 12 13 2 14 18 , 14 15 2 →= 5 , 14 15 7 →= 1 , 14 15 8 →= 9 , 17 15 8 →= 16 , 18 15 8 →= 12 , 5 9 13 →= 1 , 2 9 13 →= 7 , 4 9 13 →= 15 , 4 9 16 →= 12 , 4 9 17 →= 18 } 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 5 ↦ 0, 1 ↦ 1, 2 ↦ 2, 8 ↦ 3, 17 ↦ 4, 4 ↦ 5, 7 ↦ 6, 9 ↦ 7, 15 ↦ 8, 14 ↦ 9, 12 ↦ 10, 13 ↦ 11, 16 ↦ 12, 18 ↦ 13 }, it remains to prove termination of the 28-rule system { 0 1 2 →= 1 3 4 5 0 , 0 1 6 →= 1 3 4 5 1 , 0 1 3 →= 1 3 4 5 7 , 2 1 2 →= 6 3 4 5 0 , 2 1 6 →= 6 3 4 5 1 , 2 1 3 →= 6 3 4 5 7 , 5 1 2 →= 8 3 4 5 0 , 5 1 6 →= 8 3 4 5 1 , 5 1 3 →= 8 3 4 5 7 , 9 10 11 →= 7 11 2 9 8 , 9 10 12 →= 7 11 2 9 10 , 9 10 4 →= 7 11 2 9 13 , 4 10 11 →= 12 11 2 9 8 , 4 10 12 →= 12 11 2 9 10 , 4 10 4 →= 12 11 2 9 13 , 13 10 11 →= 10 11 2 9 8 , 13 10 12 →= 10 11 2 9 10 , 13 10 4 →= 10 11 2 9 13 , 9 8 2 →= 0 , 9 8 6 →= 1 , 9 8 3 →= 7 , 4 8 3 →= 12 , 13 8 3 →= 10 , 0 7 11 →= 1 , 2 7 11 →= 6 , 5 7 11 →= 8 , 5 7 12 →= 10 , 5 7 4 →= 13 } The system is trivially terminating.