/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 5-rule system { 0 ⟶ , 0 ⟶ 1 , 0 1 ⟶ 1 0 2 , 1 ⟶ , 2 2 ⟶ 0 } The system was reversed. After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2 }, it remains to prove termination of the 5-rule system { 0 ⟶ , 0 ⟶ 1 , 1 0 ⟶ 2 0 1 , 1 ⟶ , 2 2 ⟶ 0 } Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols. After renaming modulo the bijection { (0,↑) ↦ 0, (1,↑) ↦ 1, (0,↓) ↦ 2, (2,↑) ↦ 3, (1,↓) ↦ 4, (2,↓) ↦ 5 }, it remains to prove termination of the 10-rule system { 0 ⟶ 1 , 1 2 ⟶ 3 2 4 , 1 2 ⟶ 0 4 , 1 2 ⟶ 1 , 3 5 ⟶ 0 , 2 →= , 2 →= 4 , 4 2 →= 5 2 4 , 4 →= , 5 5 →= 2 } Applying sparse tiling TROC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (6,0) ↦ 0, (0,2) ↦ 1, (6,1) ↦ 2, (1,2) ↦ 3, (0,4) ↦ 4, (1,4) ↦ 5, (0,5) ↦ 6, (1,5) ↦ 7, (0,7) ↦ 8, (1,7) ↦ 9, (2,2) ↦ 10, (6,3) ↦ 11, (3,2) ↦ 12, (2,4) ↦ 13, (4,2) ↦ 14, (4,4) ↦ 15, (2,5) ↦ 16, (4,5) ↦ 17, (2,7) ↦ 18, (4,7) ↦ 19, (3,5) ↦ 20, (5,2) ↦ 21, (5,4) ↦ 22, (5,5) ↦ 23, (5,7) ↦ 24, (3,4) ↦ 25, (3,7) ↦ 26, (6,2) ↦ 27, (6,4) ↦ 28, (6,5) ↦ 29, (6,7) ↦ 30 }, it remains to prove termination of the 160-rule system { 0 1 ⟶ 2 3 , 0 4 ⟶ 2 5 , 0 6 ⟶ 2 7 , 0 8 ⟶ 2 9 , 2 3 10 ⟶ 11 12 13 14 , 2 3 13 ⟶ 11 12 13 15 , 2 3 16 ⟶ 11 12 13 17 , 2 3 18 ⟶ 11 12 13 19 , 2 3 10 ⟶ 0 4 14 , 2 3 13 ⟶ 0 4 15 , 2 3 16 ⟶ 0 4 17 , 2 3 18 ⟶ 0 4 19 , 2 3 10 ⟶ 2 3 , 2 3 13 ⟶ 2 5 , 2 3 16 ⟶ 2 7 , 2 3 18 ⟶ 2 9 , 11 20 21 ⟶ 0 1 , 11 20 22 ⟶ 0 4 , 11 20 23 ⟶ 0 6 , 11 20 24 ⟶ 0 8 , 1 10 →= 1 , 1 13 →= 4 , 1 16 →= 6 , 1 18 →= 8 , 3 10 →= 3 , 3 13 →= 5 , 3 16 →= 7 , 3 18 →= 9 , 10 10 →= 10 , 10 13 →= 13 , 10 16 →= 16 , 10 18 →= 18 , 12 10 →= 12 , 12 13 →= 25 , 12 16 →= 20 , 12 18 →= 26 , 14 10 →= 14 , 14 13 →= 15 , 14 16 →= 17 , 14 18 →= 19 , 21 10 →= 21 , 21 13 →= 22 , 21 16 →= 23 , 21 18 →= 24 , 27 10 →= 27 , 27 13 →= 28 , 27 16 →= 29 , 27 18 →= 30 , 1 10 →= 4 14 , 1 13 →= 4 15 , 1 16 →= 4 17 , 1 18 →= 4 19 , 3 10 →= 5 14 , 3 13 →= 5 15 , 3 16 →= 5 17 , 3 18 →= 5 19 , 10 10 →= 13 14 , 10 13 →= 13 15 , 10 16 →= 13 17 , 10 18 →= 13 19 , 12 10 →= 25 14 , 12 13 →= 25 15 , 12 16 →= 25 17 , 12 18 →= 25 19 , 14 10 →= 15 14 , 14 13 →= 15 15 , 14 16 →= 15 17 , 14 18 →= 15 19 , 21 10 →= 22 14 , 21 13 →= 22 15 , 21 16 →= 22 17 , 21 18 →= 22 19 , 27 10 →= 28 14 , 27 13 →= 28 15 , 27 16 →= 28 17 , 27 18 →= 28 19 , 4 14 10 →= 6 21 13 14 , 4 14 13 →= 6 21 13 15 , 4 14 16 →= 6 21 13 17 , 4 14 18 →= 6 21 13 19 , 5 14 10 →= 7 21 13 14 , 5 14 13 →= 7 21 13 15 , 5 14 16 →= 7 21 13 17 , 5 14 18 →= 7 21 13 19 , 13 14 10 →= 16 21 13 14 , 13 14 13 →= 16 21 13 15 , 13 14 16 →= 16 21 13 17 , 13 14 18 →= 16 21 13 19 , 25 14 10 →= 20 21 13 14 , 25 14 13 →= 20 21 13 15 , 25 14 16 →= 20 21 13 17 , 25 14 18 →= 20 21 13 19 , 15 14 10 →= 17 21 13 14 , 15 14 13 →= 17 21 13 15 , 15 14 16 →= 17 21 13 17 , 15 14 18 →= 17 21 13 19 , 22 14 10 →= 23 21 13 14 , 22 14 13 →= 23 21 13 15 , 22 14 16 →= 23 21 13 17 , 22 14 18 →= 23 21 13 19 , 28 14 10 →= 29 21 13 14 , 28 14 13 →= 29 21 13 15 , 28 14 16 →= 29 21 13 17 , 28 14 18 →= 29 21 13 19 , 4 14 →= 1 , 4 15 →= 4 , 4 17 →= 6 , 4 19 →= 8 , 5 14 →= 3 , 5 15 →= 5 , 5 17 →= 7 , 5 19 →= 9 , 13 14 →= 10 , 13 15 →= 13 , 13 17 →= 16 , 13 19 →= 18 , 25 14 →= 12 , 25 15 →= 25 , 25 17 →= 20 , 25 19 →= 26 , 15 14 →= 14 , 15 15 →= 15 , 15 17 →= 17 , 15 19 →= 19 , 22 14 →= 21 , 22 15 →= 22 , 22 17 →= 23 , 22 19 →= 24 , 28 14 →= 27 , 28 15 →= 28 , 28 17 →= 29 , 28 19 →= 30 , 6 23 21 →= 1 10 , 6 23 22 →= 1 13 , 6 23 23 →= 1 16 , 6 23 24 →= 1 18 , 7 23 21 →= 3 10 , 7 23 22 →= 3 13 , 7 23 23 →= 3 16 , 7 23 24 →= 3 18 , 16 23 21 →= 10 10 , 16 23 22 →= 10 13 , 16 23 23 →= 10 16 , 16 23 24 →= 10 18 , 20 23 21 →= 12 10 , 20 23 22 →= 12 13 , 20 23 23 →= 12 16 , 20 23 24 →= 12 18 , 17 23 21 →= 14 10 , 17 23 22 →= 14 13 , 17 23 23 →= 14 16 , 17 23 24 →= 14 18 , 23 23 21 →= 21 10 , 23 23 22 →= 21 13 , 23 23 23 →= 21 16 , 23 23 24 →= 21 18 , 29 23 21 →= 27 10 , 29 23 22 →= 27 13 , 29 23 23 →= 27 16 , 29 23 24 →= 27 18 } 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 1 ⎟ ⎜ 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 5 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 4 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 1 4 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 23 ↦ ⎛ ⎞ ⎜ 1 5 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 24 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 25 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 26 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 27 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 28 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 29 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 30 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 21 ↦ 0, 13 ↦ 1, 22 ↦ 2, 16 ↦ 3, 23 ↦ 4, 10 ↦ 5, 14 ↦ 6, 15 ↦ 7, 17 ↦ 8, 18 ↦ 9, 19 ↦ 10, 4 ↦ 11, 5 ↦ 12, 25 ↦ 13, 28 ↦ 14 }, it remains to prove termination of the 34-rule system { 0 1 →= 2 , 0 3 →= 4 , 0 5 →= 2 6 , 0 1 →= 2 7 , 0 3 →= 2 8 , 0 9 →= 2 10 , 1 6 5 →= 3 0 1 6 , 1 6 1 →= 3 0 1 7 , 1 6 3 →= 3 0 1 8 , 1 6 9 →= 3 0 1 10 , 7 6 5 →= 8 0 1 6 , 7 6 1 →= 8 0 1 7 , 7 6 3 →= 8 0 1 8 , 7 6 9 →= 8 0 1 10 , 2 6 5 →= 4 0 1 6 , 2 6 1 →= 4 0 1 7 , 2 6 3 →= 4 0 1 8 , 2 6 9 →= 4 0 1 10 , 11 7 →= 11 , 12 7 →= 12 , 1 6 →= 5 , 1 7 →= 1 , 1 8 →= 3 , 1 10 →= 9 , 13 7 →= 13 , 7 6 →= 6 , 7 7 →= 7 , 7 8 →= 8 , 7 10 →= 10 , 2 7 →= 2 , 2 8 →= 4 , 14 7 →= 14 , 3 4 0 →= 5 5 , 8 4 0 →= 6 5 } The system is trivially terminating.