/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 ⟶ 1 , 0 1 ⟶ 1 0 2 0 , 1 1 ⟶ , 2 2 ⟶ } Applying sparse tiling TRFC(2) after reversal [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { 0 ↦ 0, 2 ↦ 1, 1 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15 }, it remains to prove termination of the 64-rule system { 0 0 ⟶ 1 2 , 2 0 ⟶ 3 2 , 4 0 ⟶ 5 2 , 6 0 ⟶ 7 2 , 0 1 ⟶ 1 3 , 2 1 ⟶ 3 3 , 4 1 ⟶ 5 3 , 6 1 ⟶ 7 3 , 0 8 ⟶ 1 9 , 2 8 ⟶ 3 9 , 4 8 ⟶ 5 9 , 6 8 ⟶ 7 9 , 0 10 ⟶ 1 11 , 2 10 ⟶ 3 11 , 4 10 ⟶ 5 11 , 6 10 ⟶ 7 11 , 0 1 2 ⟶ 1 2 8 4 0 , 2 1 2 ⟶ 3 2 8 4 0 , 4 1 2 ⟶ 5 2 8 4 0 , 6 1 2 ⟶ 7 2 8 4 0 , 0 1 3 ⟶ 1 2 8 4 1 , 2 1 3 ⟶ 3 2 8 4 1 , 4 1 3 ⟶ 5 2 8 4 1 , 6 1 3 ⟶ 7 2 8 4 1 , 0 1 9 ⟶ 1 2 8 4 8 , 2 1 9 ⟶ 3 2 8 4 8 , 4 1 9 ⟶ 5 2 8 4 8 , 6 1 9 ⟶ 7 2 8 4 8 , 0 1 11 ⟶ 1 2 8 4 10 , 2 1 11 ⟶ 3 2 8 4 10 , 4 1 11 ⟶ 5 2 8 4 10 , 6 1 11 ⟶ 7 2 8 4 10 , 1 3 2 ⟶ 0 , 3 3 2 ⟶ 2 , 5 3 2 ⟶ 4 , 7 3 2 ⟶ 6 , 1 3 3 ⟶ 1 , 3 3 3 ⟶ 3 , 5 3 3 ⟶ 5 , 7 3 3 ⟶ 7 , 1 3 9 ⟶ 8 , 3 3 9 ⟶ 9 , 5 3 9 ⟶ 12 , 7 3 9 ⟶ 13 , 1 3 11 ⟶ 10 , 3 3 11 ⟶ 11 , 5 3 11 ⟶ 14 , 7 3 11 ⟶ 15 , 8 12 4 ⟶ 0 , 9 12 4 ⟶ 2 , 12 12 4 ⟶ 4 , 13 12 4 ⟶ 6 , 8 12 5 ⟶ 1 , 9 12 5 ⟶ 3 , 12 12 5 ⟶ 5 , 13 12 5 ⟶ 7 , 8 12 12 ⟶ 8 , 9 12 12 ⟶ 9 , 12 12 12 ⟶ 12 , 13 12 12 ⟶ 13 , 8 12 14 ⟶ 10 , 9 12 14 ⟶ 11 , 12 12 14 ⟶ 14 , 13 12 14 ⟶ 15 } 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 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 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 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 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, 8 ↦ 6, 9 ↦ 7, 10 ↦ 8, 11 ↦ 9, 12 ↦ 10 }, it remains to prove termination of the 29-rule system { 0 0 ⟶ 1 2 , 2 0 ⟶ 3 2 , 4 0 ⟶ 5 2 , 0 1 ⟶ 1 3 , 2 1 ⟶ 3 3 , 4 1 ⟶ 5 3 , 0 6 ⟶ 1 7 , 2 6 ⟶ 3 7 , 4 6 ⟶ 5 7 , 0 8 ⟶ 1 9 , 2 8 ⟶ 3 9 , 4 8 ⟶ 5 9 , 0 1 2 ⟶ 1 2 6 4 0 , 2 1 2 ⟶ 3 2 6 4 0 , 4 1 2 ⟶ 5 2 6 4 0 , 0 1 3 ⟶ 1 2 6 4 1 , 2 1 3 ⟶ 3 2 6 4 1 , 4 1 3 ⟶ 5 2 6 4 1 , 0 1 7 ⟶ 1 2 6 4 6 , 2 1 7 ⟶ 3 2 6 4 6 , 4 1 7 ⟶ 5 2 6 4 6 , 0 1 9 ⟶ 1 2 6 4 8 , 2 1 9 ⟶ 3 2 6 4 8 , 4 1 9 ⟶ 5 2 6 4 8 , 5 3 7 ⟶ 10 , 6 10 4 ⟶ 0 , 7 10 4 ⟶ 2 , 6 10 5 ⟶ 1 , 7 10 5 ⟶ 3 } Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols. After renaming modulo the bijection { (0,↑) ↦ 0, (0,↓) ↦ 1, (2,↑) ↦ 2, (4,↑) ↦ 3, (5,↑) ↦ 4, (2,↓) ↦ 5, (1,↓) ↦ 6, (3,↓) ↦ 7, (6,↓) ↦ 8, (7,↑) ↦ 9, (7,↓) ↦ 10, (8,↓) ↦ 11, (9,↓) ↦ 12, (4,↓) ↦ 13, (6,↑) ↦ 14, (10,↓) ↦ 15, (5,↓) ↦ 16 }, it remains to prove termination of the 87-rule system { 0 1 ⟶ 2 , 2 1 ⟶ 2 , 3 1 ⟶ 4 5 , 3 1 ⟶ 2 , 3 6 ⟶ 4 7 , 0 8 ⟶ 9 , 2 8 ⟶ 9 , 3 8 ⟶ 4 10 , 3 8 ⟶ 9 , 3 11 ⟶ 4 12 , 0 6 5 ⟶ 2 8 13 1 , 0 6 5 ⟶ 14 13 1 , 0 6 5 ⟶ 3 1 , 0 6 5 ⟶ 0 , 2 6 5 ⟶ 2 8 13 1 , 2 6 5 ⟶ 14 13 1 , 2 6 5 ⟶ 3 1 , 2 6 5 ⟶ 0 , 3 6 5 ⟶ 4 5 8 13 1 , 3 6 5 ⟶ 2 8 13 1 , 3 6 5 ⟶ 14 13 1 , 3 6 5 ⟶ 3 1 , 3 6 5 ⟶ 0 , 0 6 7 ⟶ 2 8 13 6 , 0 6 7 ⟶ 14 13 6 , 0 6 7 ⟶ 3 6 , 2 6 7 ⟶ 2 8 13 6 , 2 6 7 ⟶ 14 13 6 , 2 6 7 ⟶ 3 6 , 3 6 7 ⟶ 4 5 8 13 6 , 3 6 7 ⟶ 2 8 13 6 , 3 6 7 ⟶ 14 13 6 , 3 6 7 ⟶ 3 6 , 0 6 10 ⟶ 2 8 13 8 , 0 6 10 ⟶ 14 13 8 , 0 6 10 ⟶ 3 8 , 0 6 10 ⟶ 14 , 2 6 10 ⟶ 2 8 13 8 , 2 6 10 ⟶ 14 13 8 , 2 6 10 ⟶ 3 8 , 2 6 10 ⟶ 14 , 3 6 10 ⟶ 4 5 8 13 8 , 3 6 10 ⟶ 2 8 13 8 , 3 6 10 ⟶ 14 13 8 , 3 6 10 ⟶ 3 8 , 3 6 10 ⟶ 14 , 0 6 12 ⟶ 2 8 13 11 , 0 6 12 ⟶ 14 13 11 , 0 6 12 ⟶ 3 11 , 2 6 12 ⟶ 2 8 13 11 , 2 6 12 ⟶ 14 13 11 , 2 6 12 ⟶ 3 11 , 3 6 12 ⟶ 4 5 8 13 11 , 3 6 12 ⟶ 2 8 13 11 , 3 6 12 ⟶ 14 13 11 , 3 6 12 ⟶ 3 11 , 14 15 13 ⟶ 0 , 9 15 13 ⟶ 2 , 1 1 →= 6 5 , 5 1 →= 7 5 , 13 1 →= 16 5 , 1 6 →= 6 7 , 5 6 →= 7 7 , 13 6 →= 16 7 , 1 8 →= 6 10 , 5 8 →= 7 10 , 13 8 →= 16 10 , 1 11 →= 6 12 , 5 11 →= 7 12 , 13 11 →= 16 12 , 1 6 5 →= 6 5 8 13 1 , 5 6 5 →= 7 5 8 13 1 , 13 6 5 →= 16 5 8 13 1 , 1 6 7 →= 6 5 8 13 6 , 5 6 7 →= 7 5 8 13 6 , 13 6 7 →= 16 5 8 13 6 , 1 6 10 →= 6 5 8 13 8 , 5 6 10 →= 7 5 8 13 8 , 13 6 10 →= 16 5 8 13 8 , 1 6 12 →= 6 5 8 13 11 , 5 6 12 →= 7 5 8 13 11 , 13 6 12 →= 16 5 8 13 11 , 16 7 10 →= 15 , 8 15 13 →= 1 , 10 15 13 →= 5 , 8 15 16 →= 6 , 10 15 16 →= 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 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 1 ↦ 0, 6 ↦ 1, 5 ↦ 2, 7 ↦ 3, 13 ↦ 4, 16 ↦ 5, 8 ↦ 6, 10 ↦ 7, 11 ↦ 8, 12 ↦ 9, 15 ↦ 10 }, it remains to prove termination of the 29-rule system { 0 0 →= 1 2 , 2 0 →= 3 2 , 4 0 →= 5 2 , 0 1 →= 1 3 , 2 1 →= 3 3 , 4 1 →= 5 3 , 0 6 →= 1 7 , 2 6 →= 3 7 , 4 6 →= 5 7 , 0 8 →= 1 9 , 2 8 →= 3 9 , 4 8 →= 5 9 , 0 1 2 →= 1 2 6 4 0 , 2 1 2 →= 3 2 6 4 0 , 4 1 2 →= 5 2 6 4 0 , 0 1 3 →= 1 2 6 4 1 , 2 1 3 →= 3 2 6 4 1 , 4 1 3 →= 5 2 6 4 1 , 0 1 7 →= 1 2 6 4 6 , 2 1 7 →= 3 2 6 4 6 , 4 1 7 →= 5 2 6 4 6 , 0 1 9 →= 1 2 6 4 8 , 2 1 9 →= 3 2 6 4 8 , 4 1 9 →= 5 2 6 4 8 , 5 3 7 →= 10 , 6 10 4 →= 0 , 7 10 4 →= 2 , 6 10 5 →= 1 , 7 10 5 →= 3 } The system is trivially terminating.