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