/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 ⟶ 2 1 1 , 1 ⟶ 0 0 2 , 2 2 ⟶ } Applying sparse tiling TRFC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (0,0) ↦ 0, (0,1) ↦ 1, (0,2) ↦ 2, (0,4) ↦ 3, (1,0) ↦ 4, (1,1) ↦ 5, (1,2) ↦ 6, (1,4) ↦ 7, (2,0) ↦ 8, (2,1) ↦ 9, (2,2) ↦ 10, (2,4) ↦ 11, (3,0) ↦ 12, (3,1) ↦ 13, (3,2) ↦ 14, (3,4) ↦ 15 }, it remains to prove termination of the 64-rule system { 0 0 ⟶ 0 , 0 1 ⟶ 1 , 0 2 ⟶ 2 , 0 3 ⟶ 3 , 4 0 ⟶ 4 , 4 1 ⟶ 5 , 4 2 ⟶ 6 , 4 3 ⟶ 7 , 8 0 ⟶ 8 , 8 1 ⟶ 9 , 8 2 ⟶ 10 , 8 3 ⟶ 11 , 12 0 ⟶ 12 , 12 1 ⟶ 13 , 12 2 ⟶ 14 , 12 3 ⟶ 15 , 0 1 4 ⟶ 2 9 5 4 , 0 1 5 ⟶ 2 9 5 5 , 0 1 6 ⟶ 2 9 5 6 , 0 1 7 ⟶ 2 9 5 7 , 4 1 4 ⟶ 6 9 5 4 , 4 1 5 ⟶ 6 9 5 5 , 4 1 6 ⟶ 6 9 5 6 , 4 1 7 ⟶ 6 9 5 7 , 8 1 4 ⟶ 10 9 5 4 , 8 1 5 ⟶ 10 9 5 5 , 8 1 6 ⟶ 10 9 5 6 , 8 1 7 ⟶ 10 9 5 7 , 12 1 4 ⟶ 14 9 5 4 , 12 1 5 ⟶ 14 9 5 5 , 12 1 6 ⟶ 14 9 5 6 , 12 1 7 ⟶ 14 9 5 7 , 1 4 ⟶ 0 0 2 8 , 1 5 ⟶ 0 0 2 9 , 1 6 ⟶ 0 0 2 10 , 1 7 ⟶ 0 0 2 11 , 5 4 ⟶ 4 0 2 8 , 5 5 ⟶ 4 0 2 9 , 5 6 ⟶ 4 0 2 10 , 5 7 ⟶ 4 0 2 11 , 9 4 ⟶ 8 0 2 8 , 9 5 ⟶ 8 0 2 9 , 9 6 ⟶ 8 0 2 10 , 9 7 ⟶ 8 0 2 11 , 13 4 ⟶ 12 0 2 8 , 13 5 ⟶ 12 0 2 9 , 13 6 ⟶ 12 0 2 10 , 13 7 ⟶ 12 0 2 11 , 2 10 8 ⟶ 0 , 2 10 9 ⟶ 1 , 2 10 10 ⟶ 2 , 2 10 11 ⟶ 3 , 6 10 8 ⟶ 4 , 6 10 9 ⟶ 5 , 6 10 10 ⟶ 6 , 6 10 11 ⟶ 7 , 10 10 8 ⟶ 8 , 10 10 9 ⟶ 9 , 10 10 10 ⟶ 10 , 10 10 11 ⟶ 11 , 14 10 8 ⟶ 12 , 14 10 9 ⟶ 13 , 14 10 10 ⟶ 14 , 14 10 11 ⟶ 15 } 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 5 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 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, 6 ↦ 5, 8 ↦ 6, 10 ↦ 7, 12 ↦ 8, 9 ↦ 9, 5 ↦ 10, 7 ↦ 11 }, it remains to prove termination of the 28-rule system { 0 0 ⟶ 0 , 0 1 ⟶ 1 , 0 2 ⟶ 2 , 0 3 ⟶ 3 , 4 0 ⟶ 4 , 4 2 ⟶ 5 , 6 0 ⟶ 6 , 6 2 ⟶ 7 , 8 0 ⟶ 8 , 0 1 4 ⟶ 2 9 10 4 , 0 1 10 ⟶ 2 9 10 10 , 0 1 5 ⟶ 2 9 10 5 , 0 1 11 ⟶ 2 9 10 11 , 4 1 4 ⟶ 5 9 10 4 , 4 1 10 ⟶ 5 9 10 10 , 4 1 5 ⟶ 5 9 10 5 , 4 1 11 ⟶ 5 9 10 11 , 6 1 4 ⟶ 7 9 10 4 , 6 1 10 ⟶ 7 9 10 10 , 6 1 5 ⟶ 7 9 10 5 , 6 1 11 ⟶ 7 9 10 11 , 10 4 ⟶ 4 0 2 6 , 10 10 ⟶ 4 0 2 9 , 10 5 ⟶ 4 0 2 7 , 9 4 ⟶ 6 0 2 6 , 9 10 ⟶ 6 0 2 9 , 9 5 ⟶ 6 0 2 7 , 2 7 9 ⟶ 1 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 3: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 1 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 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 0 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 ⎟ ⎝ ⎠ 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 }, it remains to prove termination of the 27-rule system { 0 1 ⟶ 1 , 0 2 ⟶ 2 , 0 3 ⟶ 3 , 4 0 ⟶ 4 , 4 2 ⟶ 5 , 6 0 ⟶ 6 , 6 2 ⟶ 7 , 8 0 ⟶ 8 , 0 1 4 ⟶ 2 9 10 4 , 0 1 10 ⟶ 2 9 10 10 , 0 1 5 ⟶ 2 9 10 5 , 0 1 11 ⟶ 2 9 10 11 , 4 1 4 ⟶ 5 9 10 4 , 4 1 10 ⟶ 5 9 10 10 , 4 1 5 ⟶ 5 9 10 5 , 4 1 11 ⟶ 5 9 10 11 , 6 1 4 ⟶ 7 9 10 4 , 6 1 10 ⟶ 7 9 10 10 , 6 1 5 ⟶ 7 9 10 5 , 6 1 11 ⟶ 7 9 10 11 , 10 4 ⟶ 4 0 2 6 , 10 10 ⟶ 4 0 2 9 , 10 5 ⟶ 4 0 2 7 , 9 4 ⟶ 6 0 2 6 , 9 10 ⟶ 6 0 2 9 , 9 5 ⟶ 6 0 2 7 , 2 7 9 ⟶ 1 } Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols. After renaming modulo the bijection { (4,↑) ↦ 0, (0,↓) ↦ 1, (6,↑) ↦ 2, (8,↑) ↦ 3, (0,↑) ↦ 4, (1,↓) ↦ 5, (4,↓) ↦ 6, (2,↑) ↦ 7, (9,↓) ↦ 8, (10,↓) ↦ 9, (9,↑) ↦ 10, (10,↑) ↦ 11, (5,↓) ↦ 12, (11,↓) ↦ 13, (2,↓) ↦ 14, (6,↓) ↦ 15, (7,↓) ↦ 16, (3,↓) ↦ 17, (8,↓) ↦ 18 }, it remains to prove termination of the 80-rule system { 0 1 ⟶ 0 , 2 1 ⟶ 2 , 3 1 ⟶ 3 , 4 5 6 ⟶ 7 8 9 6 , 4 5 6 ⟶ 10 9 6 , 4 5 6 ⟶ 11 6 , 4 5 9 ⟶ 7 8 9 9 , 4 5 9 ⟶ 10 9 9 , 4 5 9 ⟶ 11 9 , 4 5 12 ⟶ 7 8 9 12 , 4 5 12 ⟶ 10 9 12 , 4 5 12 ⟶ 11 12 , 4 5 13 ⟶ 7 8 9 13 , 4 5 13 ⟶ 10 9 13 , 4 5 13 ⟶ 11 13 , 0 5 6 ⟶ 10 9 6 , 0 5 6 ⟶ 11 6 , 0 5 9 ⟶ 10 9 9 , 0 5 9 ⟶ 11 9 , 0 5 12 ⟶ 10 9 12 , 0 5 12 ⟶ 11 12 , 0 5 13 ⟶ 10 9 13 , 0 5 13 ⟶ 11 13 , 2 5 6 ⟶ 10 9 6 , 2 5 6 ⟶ 11 6 , 2 5 9 ⟶ 10 9 9 , 2 5 9 ⟶ 11 9 , 2 5 12 ⟶ 10 9 12 , 2 5 12 ⟶ 11 12 , 2 5 13 ⟶ 10 9 13 , 2 5 13 ⟶ 11 13 , 11 6 ⟶ 0 1 14 15 , 11 6 ⟶ 4 14 15 , 11 6 ⟶ 7 15 , 11 6 ⟶ 2 , 11 9 ⟶ 0 1 14 8 , 11 9 ⟶ 4 14 8 , 11 9 ⟶ 7 8 , 11 9 ⟶ 10 , 11 12 ⟶ 0 1 14 16 , 11 12 ⟶ 4 14 16 , 11 12 ⟶ 7 16 , 10 6 ⟶ 2 1 14 15 , 10 6 ⟶ 4 14 15 , 10 6 ⟶ 7 15 , 10 6 ⟶ 2 , 10 9 ⟶ 2 1 14 8 , 10 9 ⟶ 4 14 8 , 10 9 ⟶ 7 8 , 10 9 ⟶ 10 , 10 12 ⟶ 2 1 14 16 , 10 12 ⟶ 4 14 16 , 10 12 ⟶ 7 16 , 1 5 →= 5 , 1 14 →= 14 , 1 17 →= 17 , 6 1 →= 6 , 6 14 →= 12 , 15 1 →= 15 , 15 14 →= 16 , 18 1 →= 18 , 1 5 6 →= 14 8 9 6 , 1 5 9 →= 14 8 9 9 , 1 5 12 →= 14 8 9 12 , 1 5 13 →= 14 8 9 13 , 6 5 6 →= 12 8 9 6 , 6 5 9 →= 12 8 9 9 , 6 5 12 →= 12 8 9 12 , 6 5 13 →= 12 8 9 13 , 15 5 6 →= 16 8 9 6 , 15 5 9 →= 16 8 9 9 , 15 5 12 →= 16 8 9 12 , 15 5 13 →= 16 8 9 13 , 9 6 →= 6 1 14 15 , 9 9 →= 6 1 14 8 , 9 12 →= 6 1 14 16 , 8 6 →= 15 1 14 15 , 8 9 →= 15 1 14 8 , 8 12 →= 15 1 14 16 , 14 16 8 →= 5 } 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 4 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 5 ↦ 4, 14 ↦ 5, 17 ↦ 6, 6 ↦ 7, 12 ↦ 8, 15 ↦ 9, 16 ↦ 10, 18 ↦ 11, 8 ↦ 12, 9 ↦ 13, 13 ↦ 14 }, it remains to prove termination of the 30-rule system { 0 1 ⟶ 0 , 2 1 ⟶ 2 , 3 1 ⟶ 3 , 1 4 →= 4 , 1 5 →= 5 , 1 6 →= 6 , 7 1 →= 7 , 7 5 →= 8 , 9 1 →= 9 , 9 5 →= 10 , 11 1 →= 11 , 1 4 7 →= 5 12 13 7 , 1 4 13 →= 5 12 13 13 , 1 4 8 →= 5 12 13 8 , 1 4 14 →= 5 12 13 14 , 7 4 7 →= 8 12 13 7 , 7 4 13 →= 8 12 13 13 , 7 4 8 →= 8 12 13 8 , 7 4 14 →= 8 12 13 14 , 9 4 7 →= 10 12 13 7 , 9 4 13 →= 10 12 13 13 , 9 4 8 →= 10 12 13 8 , 9 4 14 →= 10 12 13 14 , 13 7 →= 7 1 5 9 , 13 13 →= 7 1 5 12 , 13 8 →= 7 1 5 10 , 12 7 →= 9 1 5 9 , 12 13 →= 9 1 5 12 , 12 8 →= 9 1 5 10 , 5 10 12 →= 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 3: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 1 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 1 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 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 0 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 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 2 ↦ 0, 1 ↦ 1, 3 ↦ 2, 4 ↦ 3, 5 ↦ 4, 6 ↦ 5, 7 ↦ 6, 8 ↦ 7, 9 ↦ 8, 10 ↦ 9, 11 ↦ 10, 12 ↦ 11, 13 ↦ 12, 14 ↦ 13 }, it remains to prove termination of the 29-rule system { 0 1 ⟶ 0 , 2 1 ⟶ 2 , 1 3 →= 3 , 1 4 →= 4 , 1 5 →= 5 , 6 1 →= 6 , 6 4 →= 7 , 8 1 →= 8 , 8 4 →= 9 , 10 1 →= 10 , 1 3 6 →= 4 11 12 6 , 1 3 12 →= 4 11 12 12 , 1 3 7 →= 4 11 12 7 , 1 3 13 →= 4 11 12 13 , 6 3 6 →= 7 11 12 6 , 6 3 12 →= 7 11 12 12 , 6 3 7 →= 7 11 12 7 , 6 3 13 →= 7 11 12 13 , 8 3 6 →= 9 11 12 6 , 8 3 12 →= 9 11 12 12 , 8 3 7 →= 9 11 12 7 , 8 3 13 →= 9 11 12 13 , 12 6 →= 6 1 4 8 , 12 12 →= 6 1 4 11 , 12 7 →= 6 1 4 9 , 11 6 →= 8 1 4 8 , 11 12 →= 8 1 4 11 , 11 7 →= 8 1 4 9 , 4 9 11 →= 3 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 3: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 1 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 1 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 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 0 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 { 2 ↦ 0, 1 ↦ 1, 3 ↦ 2, 4 ↦ 3, 5 ↦ 4, 6 ↦ 5, 7 ↦ 6, 8 ↦ 7, 9 ↦ 8, 10 ↦ 9, 11 ↦ 10, 12 ↦ 11, 13 ↦ 12 }, it remains to prove termination of the 28-rule system { 0 1 ⟶ 0 , 1 2 →= 2 , 1 3 →= 3 , 1 4 →= 4 , 5 1 →= 5 , 5 3 →= 6 , 7 1 →= 7 , 7 3 →= 8 , 9 1 →= 9 , 1 2 5 →= 3 10 11 5 , 1 2 11 →= 3 10 11 11 , 1 2 6 →= 3 10 11 6 , 1 2 12 →= 3 10 11 12 , 5 2 5 →= 6 10 11 5 , 5 2 11 →= 6 10 11 11 , 5 2 6 →= 6 10 11 6 , 5 2 12 →= 6 10 11 12 , 7 2 5 →= 8 10 11 5 , 7 2 11 →= 8 10 11 11 , 7 2 6 →= 8 10 11 6 , 7 2 12 →= 8 10 11 12 , 11 5 →= 5 1 3 7 , 11 11 →= 5 1 3 10 , 11 6 →= 5 1 3 8 , 10 5 →= 7 1 3 7 , 10 11 →= 7 1 3 10 , 10 6 →= 7 1 3 8 , 3 8 10 →= 2 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 3: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 1 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 1 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 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 0 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 ⎟ ⎝ ⎠ After renaming modulo the bijection { 1 ↦ 0, 2 ↦ 1, 3 ↦ 2, 4 ↦ 3, 5 ↦ 4, 6 ↦ 5, 7 ↦ 6, 8 ↦ 7, 9 ↦ 8, 10 ↦ 9, 11 ↦ 10, 12 ↦ 11 }, it remains to prove termination of the 27-rule system { 0 1 →= 1 , 0 2 →= 2 , 0 3 →= 3 , 4 0 →= 4 , 4 2 →= 5 , 6 0 →= 6 , 6 2 →= 7 , 8 0 →= 8 , 0 1 4 →= 2 9 10 4 , 0 1 10 →= 2 9 10 10 , 0 1 5 →= 2 9 10 5 , 0 1 11 →= 2 9 10 11 , 4 1 4 →= 5 9 10 4 , 4 1 10 →= 5 9 10 10 , 4 1 5 →= 5 9 10 5 , 4 1 11 →= 5 9 10 11 , 6 1 4 →= 7 9 10 4 , 6 1 10 →= 7 9 10 10 , 6 1 5 →= 7 9 10 5 , 6 1 11 →= 7 9 10 11 , 10 4 →= 4 0 2 6 , 10 10 →= 4 0 2 9 , 10 5 →= 4 0 2 7 , 9 4 →= 6 0 2 6 , 9 10 →= 6 0 2 9 , 9 5 →= 6 0 2 7 , 2 7 9 →= 1 } The system is trivially terminating.