/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 ⟶ 1 , 0 1 ⟶ 1 0 2 , 1 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 4-rule system { 0 ⟶ 1 , 1 0 ⟶ 2 0 1 , 1 1 ⟶ , 2 2 ⟶ 0 } Applying sparse tiling TRFC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (0,0) ↦ 0, (0,1) ↦ 1, (1,0) ↦ 2, (1,1) ↦ 3, (0,2) ↦ 4, (1,2) ↦ 5, (0,4) ↦ 6, (1,4) ↦ 7, (2,0) ↦ 8, (2,1) ↦ 9, (3,0) ↦ 10, (3,1) ↦ 11, (2,2) ↦ 12, (3,2) ↦ 13, (2,4) ↦ 14, (3,4) ↦ 15 }, it remains to prove termination of the 64-rule system { 0 0 ⟶ 1 2 , 0 1 ⟶ 1 3 , 0 4 ⟶ 1 5 , 0 6 ⟶ 1 7 , 2 0 ⟶ 3 2 , 2 1 ⟶ 3 3 , 2 4 ⟶ 3 5 , 2 6 ⟶ 3 7 , 8 0 ⟶ 9 2 , 8 1 ⟶ 9 3 , 8 4 ⟶ 9 5 , 8 6 ⟶ 9 7 , 10 0 ⟶ 11 2 , 10 1 ⟶ 11 3 , 10 4 ⟶ 11 5 , 10 6 ⟶ 11 7 , 1 2 0 ⟶ 4 8 1 2 , 1 2 1 ⟶ 4 8 1 3 , 1 2 4 ⟶ 4 8 1 5 , 1 2 6 ⟶ 4 8 1 7 , 3 2 0 ⟶ 5 8 1 2 , 3 2 1 ⟶ 5 8 1 3 , 3 2 4 ⟶ 5 8 1 5 , 3 2 6 ⟶ 5 8 1 7 , 9 2 0 ⟶ 12 8 1 2 , 9 2 1 ⟶ 12 8 1 3 , 9 2 4 ⟶ 12 8 1 5 , 9 2 6 ⟶ 12 8 1 7 , 11 2 0 ⟶ 13 8 1 2 , 11 2 1 ⟶ 13 8 1 3 , 11 2 4 ⟶ 13 8 1 5 , 11 2 6 ⟶ 13 8 1 7 , 1 3 2 ⟶ 0 , 1 3 3 ⟶ 1 , 1 3 5 ⟶ 4 , 1 3 7 ⟶ 6 , 3 3 2 ⟶ 2 , 3 3 3 ⟶ 3 , 3 3 5 ⟶ 5 , 3 3 7 ⟶ 7 , 9 3 2 ⟶ 8 , 9 3 3 ⟶ 9 , 9 3 5 ⟶ 12 , 9 3 7 ⟶ 14 , 11 3 2 ⟶ 10 , 11 3 3 ⟶ 11 , 11 3 5 ⟶ 13 , 11 3 7 ⟶ 15 , 4 12 8 ⟶ 0 0 , 4 12 9 ⟶ 0 1 , 4 12 12 ⟶ 0 4 , 4 12 14 ⟶ 0 6 , 5 12 8 ⟶ 2 0 , 5 12 9 ⟶ 2 1 , 5 12 12 ⟶ 2 4 , 5 12 14 ⟶ 2 6 , 12 12 8 ⟶ 8 0 , 12 12 9 ⟶ 8 1 , 12 12 12 ⟶ 8 4 , 12 12 14 ⟶ 8 6 , 13 12 8 ⟶ 10 0 , 13 12 9 ⟶ 10 1 , 13 12 12 ⟶ 10 4 , 13 12 14 ⟶ 10 6 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 8 ↦ 0, 0 ↦ 1, 9 ↦ 2, 2 ↦ 3, 1 ↦ 4, 3 ↦ 5, 4 ↦ 6, 5 ↦ 7, 6 ↦ 8, 7 ↦ 9, 12 ↦ 10, 11 ↦ 11 }, it remains to prove termination of the 29-rule system { 0 1 ⟶ 2 3 , 0 4 ⟶ 2 5 , 0 6 ⟶ 2 7 , 0 8 ⟶ 2 9 , 4 3 1 ⟶ 6 0 4 3 , 4 3 4 ⟶ 6 0 4 5 , 4 3 6 ⟶ 6 0 4 7 , 4 3 8 ⟶ 6 0 4 9 , 5 3 1 ⟶ 7 0 4 3 , 5 3 4 ⟶ 7 0 4 5 , 5 3 6 ⟶ 7 0 4 7 , 5 3 8 ⟶ 7 0 4 9 , 2 3 1 ⟶ 10 0 4 3 , 2 3 4 ⟶ 10 0 4 5 , 2 3 6 ⟶ 10 0 4 7 , 2 3 8 ⟶ 10 0 4 9 , 4 5 3 ⟶ 1 , 4 5 5 ⟶ 4 , 4 5 7 ⟶ 6 , 4 5 9 ⟶ 8 , 5 5 3 ⟶ 3 , 5 5 5 ⟶ 5 , 5 5 7 ⟶ 7 , 5 5 9 ⟶ 9 , 2 5 5 ⟶ 2 , 2 5 7 ⟶ 10 , 11 5 5 ⟶ 11 , 6 10 0 ⟶ 1 1 , 7 10 0 ⟶ 3 1 } Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols. After renaming modulo the bijection { (0,↑) ↦ 0, (1,↓) ↦ 1, (2,↑) ↦ 2, (3,↓) ↦ 3, (4,↓) ↦ 4, (5,↓) ↦ 5, (5,↑) ↦ 6, (6,↓) ↦ 7, (7,↓) ↦ 8, (7,↑) ↦ 9, (8,↓) ↦ 10, (9,↓) ↦ 11, (4,↑) ↦ 12, (6,↑) ↦ 13, (0,↓) ↦ 14, (11,↑) ↦ 15, (2,↓) ↦ 16, (10,↓) ↦ 17, (11,↓) ↦ 18 }, it remains to prove termination of the 77-rule system { 0 1 ⟶ 2 3 , 0 4 ⟶ 2 5 , 0 4 ⟶ 6 , 0 7 ⟶ 2 8 , 0 7 ⟶ 9 , 0 10 ⟶ 2 11 , 12 3 1 ⟶ 13 14 4 3 , 12 3 1 ⟶ 0 4 3 , 12 3 1 ⟶ 12 3 , 12 3 4 ⟶ 13 14 4 5 , 12 3 4 ⟶ 0 4 5 , 12 3 4 ⟶ 12 5 , 12 3 4 ⟶ 6 , 12 3 7 ⟶ 13 14 4 8 , 12 3 7 ⟶ 0 4 8 , 12 3 7 ⟶ 12 8 , 12 3 7 ⟶ 9 , 12 3 10 ⟶ 13 14 4 11 , 12 3 10 ⟶ 0 4 11 , 12 3 10 ⟶ 12 11 , 6 3 1 ⟶ 9 14 4 3 , 6 3 1 ⟶ 0 4 3 , 6 3 1 ⟶ 12 3 , 6 3 4 ⟶ 9 14 4 5 , 6 3 4 ⟶ 0 4 5 , 6 3 4 ⟶ 12 5 , 6 3 4 ⟶ 6 , 6 3 7 ⟶ 9 14 4 8 , 6 3 7 ⟶ 0 4 8 , 6 3 7 ⟶ 12 8 , 6 3 7 ⟶ 9 , 6 3 10 ⟶ 9 14 4 11 , 6 3 10 ⟶ 0 4 11 , 6 3 10 ⟶ 12 11 , 2 3 1 ⟶ 0 4 3 , 2 3 1 ⟶ 12 3 , 2 3 4 ⟶ 0 4 5 , 2 3 4 ⟶ 12 5 , 2 3 4 ⟶ 6 , 2 3 7 ⟶ 0 4 8 , 2 3 7 ⟶ 12 8 , 2 3 7 ⟶ 9 , 2 3 10 ⟶ 0 4 11 , 2 3 10 ⟶ 12 11 , 12 5 5 ⟶ 12 , 12 5 8 ⟶ 13 , 2 5 5 ⟶ 2 , 15 5 5 ⟶ 15 , 14 1 →= 16 3 , 14 4 →= 16 5 , 14 7 →= 16 8 , 14 10 →= 16 11 , 4 3 1 →= 7 14 4 3 , 4 3 4 →= 7 14 4 5 , 4 3 7 →= 7 14 4 8 , 4 3 10 →= 7 14 4 11 , 5 3 1 →= 8 14 4 3 , 5 3 4 →= 8 14 4 5 , 5 3 7 →= 8 14 4 8 , 5 3 10 →= 8 14 4 11 , 16 3 1 →= 17 14 4 3 , 16 3 4 →= 17 14 4 5 , 16 3 7 →= 17 14 4 8 , 16 3 10 →= 17 14 4 11 , 4 5 3 →= 1 , 4 5 5 →= 4 , 4 5 8 →= 7 , 4 5 11 →= 10 , 5 5 3 →= 3 , 5 5 5 →= 5 , 5 5 8 →= 8 , 5 5 11 →= 11 , 16 5 5 →= 16 , 16 5 8 →= 17 , 18 5 5 →= 18 , 7 17 14 →= 1 1 , 8 17 14 →= 3 1 } 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 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 12 ↦ 0, 5 ↦ 1, 2 ↦ 2, 15 ↦ 3, 14 ↦ 4, 1 ↦ 5, 16 ↦ 6, 3 ↦ 7, 4 ↦ 8, 7 ↦ 9, 8 ↦ 10, 10 ↦ 11, 11 ↦ 12, 17 ↦ 13, 18 ↦ 14 }, it remains to prove termination of the 32-rule system { 0 1 1 ⟶ 0 , 2 1 1 ⟶ 2 , 3 1 1 ⟶ 3 , 4 5 →= 6 7 , 4 8 →= 6 1 , 4 9 →= 6 10 , 4 11 →= 6 12 , 8 7 5 →= 9 4 8 7 , 8 7 8 →= 9 4 8 1 , 8 7 9 →= 9 4 8 10 , 8 7 11 →= 9 4 8 12 , 1 7 5 →= 10 4 8 7 , 1 7 8 →= 10 4 8 1 , 1 7 9 →= 10 4 8 10 , 1 7 11 →= 10 4 8 12 , 6 7 5 →= 13 4 8 7 , 6 7 8 →= 13 4 8 1 , 6 7 9 →= 13 4 8 10 , 6 7 11 →= 13 4 8 12 , 8 1 7 →= 5 , 8 1 1 →= 8 , 8 1 10 →= 9 , 8 1 12 →= 11 , 1 1 7 →= 7 , 1 1 1 →= 1 , 1 1 10 →= 10 , 1 1 12 →= 12 , 6 1 1 →= 6 , 6 1 10 →= 13 , 14 1 1 →= 14 , 9 13 4 →= 5 5 , 10 13 4 →= 7 5 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 4: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 1 ⎟ ⎜ 0 1 1 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 1 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 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 31-rule system { 0 1 1 ⟶ 0 , 2 1 1 ⟶ 2 , 3 4 →= 5 6 , 3 7 →= 5 1 , 3 8 →= 5 9 , 3 10 →= 5 11 , 7 6 4 →= 8 3 7 6 , 7 6 7 →= 8 3 7 1 , 7 6 8 →= 8 3 7 9 , 7 6 10 →= 8 3 7 11 , 1 6 4 →= 9 3 7 6 , 1 6 7 →= 9 3 7 1 , 1 6 8 →= 9 3 7 9 , 1 6 10 →= 9 3 7 11 , 5 6 4 →= 12 3 7 6 , 5 6 7 →= 12 3 7 1 , 5 6 8 →= 12 3 7 9 , 5 6 10 →= 12 3 7 11 , 7 1 6 →= 4 , 7 1 1 →= 7 , 7 1 9 →= 8 , 7 1 11 →= 10 , 1 1 6 →= 6 , 1 1 1 →= 1 , 1 1 9 →= 9 , 1 1 11 →= 11 , 5 1 1 →= 5 , 5 1 9 →= 12 , 13 1 1 →= 13 , 8 12 3 →= 4 4 , 9 12 3 →= 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 4: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 1 ⎟ ⎜ 0 1 1 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 1 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 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 30-rule system { 0 1 1 ⟶ 0 , 2 3 →= 4 5 , 2 6 →= 4 1 , 2 7 →= 4 8 , 2 9 →= 4 10 , 6 5 3 →= 7 2 6 5 , 6 5 6 →= 7 2 6 1 , 6 5 7 →= 7 2 6 8 , 6 5 9 →= 7 2 6 10 , 1 5 3 →= 8 2 6 5 , 1 5 6 →= 8 2 6 1 , 1 5 7 →= 8 2 6 8 , 1 5 9 →= 8 2 6 10 , 4 5 3 →= 11 2 6 5 , 4 5 6 →= 11 2 6 1 , 4 5 7 →= 11 2 6 8 , 4 5 9 →= 11 2 6 10 , 6 1 5 →= 3 , 6 1 1 →= 6 , 6 1 8 →= 7 , 6 1 10 →= 9 , 1 1 5 →= 5 , 1 1 1 →= 1 , 1 1 8 →= 8 , 1 1 10 →= 10 , 4 1 1 →= 4 , 4 1 8 →= 11 , 12 1 1 →= 12 , 7 11 2 →= 3 3 , 8 11 2 →= 5 3 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 4: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 1 ⎟ ⎜ 0 1 1 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 2 ↦ 0, 3 ↦ 1, 4 ↦ 2, 5 ↦ 3, 6 ↦ 4, 1 ↦ 5, 7 ↦ 6, 8 ↦ 7, 9 ↦ 8, 10 ↦ 9, 11 ↦ 10, 12 ↦ 11 }, it remains to prove termination of the 29-rule system { 0 1 →= 2 3 , 0 4 →= 2 5 , 0 6 →= 2 7 , 0 8 →= 2 9 , 4 3 1 →= 6 0 4 3 , 4 3 4 →= 6 0 4 5 , 4 3 6 →= 6 0 4 7 , 4 3 8 →= 6 0 4 9 , 5 3 1 →= 7 0 4 3 , 5 3 4 →= 7 0 4 5 , 5 3 6 →= 7 0 4 7 , 5 3 8 →= 7 0 4 9 , 2 3 1 →= 10 0 4 3 , 2 3 4 →= 10 0 4 5 , 2 3 6 →= 10 0 4 7 , 2 3 8 →= 10 0 4 9 , 4 5 3 →= 1 , 4 5 5 →= 4 , 4 5 7 →= 6 , 4 5 9 →= 8 , 5 5 3 →= 3 , 5 5 5 →= 5 , 5 5 7 →= 7 , 5 5 9 →= 9 , 2 5 5 →= 2 , 2 5 7 →= 10 , 11 5 5 →= 11 , 6 10 0 →= 1 1 , 7 10 0 →= 3 1 } The system is trivially terminating.