/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 ⟶ , 0 1 ⟶ 1 0 2 , 1 1 ⟶ , 2 2 ⟶ 1 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 0 ⟶ 2 0 1 , 1 1 ⟶ , 2 2 ⟶ 0 1 } 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 , 1 4 0 ⟶ 2 8 1 4 , 1 4 1 ⟶ 2 8 1 5 , 1 4 2 ⟶ 2 8 1 6 , 1 4 3 ⟶ 2 8 1 7 , 5 4 0 ⟶ 6 8 1 4 , 5 4 1 ⟶ 6 8 1 5 , 5 4 2 ⟶ 6 8 1 6 , 5 4 3 ⟶ 6 8 1 7 , 9 4 0 ⟶ 10 8 1 4 , 9 4 1 ⟶ 10 8 1 5 , 9 4 2 ⟶ 10 8 1 6 , 9 4 3 ⟶ 10 8 1 7 , 13 4 0 ⟶ 14 8 1 4 , 13 4 1 ⟶ 14 8 1 5 , 13 4 2 ⟶ 14 8 1 6 , 13 4 3 ⟶ 14 8 1 7 , 1 5 4 ⟶ 0 , 1 5 5 ⟶ 1 , 1 5 6 ⟶ 2 , 1 5 7 ⟶ 3 , 5 5 4 ⟶ 4 , 5 5 5 ⟶ 5 , 5 5 6 ⟶ 6 , 5 5 7 ⟶ 7 , 9 5 4 ⟶ 8 , 9 5 5 ⟶ 9 , 9 5 6 ⟶ 10 , 9 5 7 ⟶ 11 , 13 5 4 ⟶ 12 , 13 5 5 ⟶ 13 , 13 5 6 ⟶ 14 , 13 5 7 ⟶ 15 , 2 10 8 ⟶ 0 1 4 , 2 10 9 ⟶ 0 1 5 , 2 10 10 ⟶ 0 1 6 , 2 10 11 ⟶ 0 1 7 , 6 10 8 ⟶ 4 1 4 , 6 10 9 ⟶ 4 1 5 , 6 10 10 ⟶ 4 1 6 , 6 10 11 ⟶ 4 1 7 , 10 10 8 ⟶ 8 1 4 , 10 10 9 ⟶ 8 1 5 , 10 10 10 ⟶ 8 1 6 , 10 10 11 ⟶ 8 1 7 , 14 10 8 ⟶ 12 1 4 , 14 10 9 ⟶ 12 1 5 , 14 10 10 ⟶ 12 1 6 , 14 10 11 ⟶ 12 1 7 } 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 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 8 ↦ 0, 1 ↦ 1, 9 ↦ 2, 2 ↦ 3, 10 ↦ 4, 4 ↦ 5, 0 ↦ 6, 5 ↦ 7, 6 ↦ 8, 3 ↦ 9, 7 ↦ 10, 13 ↦ 11 }, it remains to prove termination of the 27-rule system { 0 1 ⟶ 2 , 0 3 ⟶ 4 , 1 5 6 ⟶ 3 0 1 5 , 1 5 1 ⟶ 3 0 1 7 , 1 5 3 ⟶ 3 0 1 8 , 1 5 9 ⟶ 3 0 1 10 , 7 5 6 ⟶ 8 0 1 5 , 7 5 1 ⟶ 8 0 1 7 , 7 5 3 ⟶ 8 0 1 8 , 7 5 9 ⟶ 8 0 1 10 , 2 5 6 ⟶ 4 0 1 5 , 2 5 1 ⟶ 4 0 1 7 , 2 5 3 ⟶ 4 0 1 8 , 2 5 9 ⟶ 4 0 1 10 , 1 7 5 ⟶ 6 , 1 7 7 ⟶ 1 , 1 7 8 ⟶ 3 , 1 7 10 ⟶ 9 , 7 7 5 ⟶ 5 , 7 7 7 ⟶ 7 , 7 7 8 ⟶ 8 , 7 7 10 ⟶ 10 , 2 7 7 ⟶ 2 , 2 7 8 ⟶ 4 , 11 7 7 ⟶ 11 , 3 4 0 ⟶ 6 1 5 , 8 4 0 ⟶ 5 1 5 } Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols. After renaming modulo the bijection { (0,↑) ↦ 0, (1,↓) ↦ 1, (2,↑) ↦ 2, (1,↑) ↦ 3, (5,↓) ↦ 4, (6,↓) ↦ 5, (3,↑) ↦ 6, (0,↓) ↦ 7, (7,↓) ↦ 8, (7,↑) ↦ 9, (3,↓) ↦ 10, (8,↓) ↦ 11, (8,↑) ↦ 12, (9,↓) ↦ 13, (10,↓) ↦ 14, (11,↑) ↦ 15, (4,↓) ↦ 16, (2,↓) ↦ 17, (11,↓) ↦ 18 }, it remains to prove termination of the 72-rule system { 0 1 ⟶ 2 , 3 4 5 ⟶ 6 7 1 4 , 3 4 5 ⟶ 0 1 4 , 3 4 5 ⟶ 3 4 , 3 4 1 ⟶ 6 7 1 8 , 3 4 1 ⟶ 0 1 8 , 3 4 1 ⟶ 3 8 , 3 4 1 ⟶ 9 , 3 4 10 ⟶ 6 7 1 11 , 3 4 10 ⟶ 0 1 11 , 3 4 10 ⟶ 3 11 , 3 4 10 ⟶ 12 , 3 4 13 ⟶ 6 7 1 14 , 3 4 13 ⟶ 0 1 14 , 3 4 13 ⟶ 3 14 , 9 4 5 ⟶ 12 7 1 4 , 9 4 5 ⟶ 0 1 4 , 9 4 5 ⟶ 3 4 , 9 4 1 ⟶ 12 7 1 8 , 9 4 1 ⟶ 0 1 8 , 9 4 1 ⟶ 3 8 , 9 4 1 ⟶ 9 , 9 4 10 ⟶ 12 7 1 11 , 9 4 10 ⟶ 0 1 11 , 9 4 10 ⟶ 3 11 , 9 4 10 ⟶ 12 , 9 4 13 ⟶ 12 7 1 14 , 9 4 13 ⟶ 0 1 14 , 9 4 13 ⟶ 3 14 , 2 4 5 ⟶ 0 1 4 , 2 4 5 ⟶ 3 4 , 2 4 1 ⟶ 0 1 8 , 2 4 1 ⟶ 3 8 , 2 4 1 ⟶ 9 , 2 4 10 ⟶ 0 1 11 , 2 4 10 ⟶ 3 11 , 2 4 10 ⟶ 12 , 2 4 13 ⟶ 0 1 14 , 2 4 13 ⟶ 3 14 , 3 8 8 ⟶ 3 , 3 8 11 ⟶ 6 , 2 8 8 ⟶ 2 , 15 8 8 ⟶ 15 , 6 16 7 ⟶ 3 4 , 12 16 7 ⟶ 3 4 , 7 1 →= 17 , 7 10 →= 16 , 1 4 5 →= 10 7 1 4 , 1 4 1 →= 10 7 1 8 , 1 4 10 →= 10 7 1 11 , 1 4 13 →= 10 7 1 14 , 8 4 5 →= 11 7 1 4 , 8 4 1 →= 11 7 1 8 , 8 4 10 →= 11 7 1 11 , 8 4 13 →= 11 7 1 14 , 17 4 5 →= 16 7 1 4 , 17 4 1 →= 16 7 1 8 , 17 4 10 →= 16 7 1 11 , 17 4 13 →= 16 7 1 14 , 1 8 4 →= 5 , 1 8 8 →= 1 , 1 8 11 →= 10 , 1 8 14 →= 13 , 8 8 4 →= 4 , 8 8 8 →= 8 , 8 8 11 →= 11 , 8 8 14 →= 14 , 17 8 8 →= 17 , 17 8 11 →= 16 , 18 8 8 →= 18 , 10 16 7 →= 5 1 4 , 11 16 7 →= 4 1 4 } 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 3 ↦ 0, 8 ↦ 1, 2 ↦ 2, 15 ↦ 3, 7 ↦ 4, 1 ↦ 5, 17 ↦ 6, 10 ↦ 7, 16 ↦ 8, 4 ↦ 9, 5 ↦ 10, 11 ↦ 11, 13 ↦ 12, 14 ↦ 13, 18 ↦ 14 }, it remains to prove termination of the 30-rule system { 0 1 1 ⟶ 0 , 2 1 1 ⟶ 2 , 3 1 1 ⟶ 3 , 4 5 →= 6 , 4 7 →= 8 , 5 9 10 →= 7 4 5 9 , 5 9 5 →= 7 4 5 1 , 5 9 7 →= 7 4 5 11 , 5 9 12 →= 7 4 5 13 , 1 9 10 →= 11 4 5 9 , 1 9 5 →= 11 4 5 1 , 1 9 7 →= 11 4 5 11 , 1 9 12 →= 11 4 5 13 , 6 9 10 →= 8 4 5 9 , 6 9 5 →= 8 4 5 1 , 6 9 7 →= 8 4 5 11 , 6 9 12 →= 8 4 5 13 , 5 1 9 →= 10 , 5 1 1 →= 5 , 5 1 11 →= 7 , 5 1 13 →= 12 , 1 1 9 →= 9 , 1 1 1 →= 1 , 1 1 11 →= 11 , 1 1 13 →= 13 , 6 1 1 →= 6 , 6 1 11 →= 8 , 14 1 1 →= 14 , 7 8 4 →= 10 5 9 , 11 8 4 →= 9 5 9 } 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 ⎟ ⎝ ⎠ 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 { 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 1 ⟶ 0 , 2 1 1 ⟶ 2 , 3 4 →= 5 , 3 6 →= 7 , 4 8 9 →= 6 3 4 8 , 4 8 4 →= 6 3 4 1 , 4 8 6 →= 6 3 4 10 , 4 8 11 →= 6 3 4 12 , 1 8 9 →= 10 3 4 8 , 1 8 4 →= 10 3 4 1 , 1 8 6 →= 10 3 4 10 , 1 8 11 →= 10 3 4 12 , 5 8 9 →= 7 3 4 8 , 5 8 4 →= 7 3 4 1 , 5 8 6 →= 7 3 4 10 , 5 8 11 →= 7 3 4 12 , 4 1 8 →= 9 , 4 1 1 →= 4 , 4 1 10 →= 6 , 4 1 12 →= 11 , 1 1 8 →= 8 , 1 1 1 →= 1 , 1 1 10 →= 10 , 1 1 12 →= 12 , 5 1 1 →= 5 , 5 1 10 →= 7 , 13 1 1 →= 13 , 6 7 3 →= 9 4 8 , 10 7 3 →= 8 4 8 } 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 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 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 1 ⟶ 0 , 2 3 →= 4 , 2 5 →= 6 , 3 7 8 →= 5 2 3 7 , 3 7 3 →= 5 2 3 1 , 3 7 5 →= 5 2 3 9 , 3 7 10 →= 5 2 3 11 , 1 7 8 →= 9 2 3 7 , 1 7 3 →= 9 2 3 1 , 1 7 5 →= 9 2 3 9 , 1 7 10 →= 9 2 3 11 , 4 7 8 →= 6 2 3 7 , 4 7 3 →= 6 2 3 1 , 4 7 5 →= 6 2 3 9 , 4 7 10 →= 6 2 3 11 , 3 1 7 →= 8 , 3 1 1 →= 3 , 3 1 9 →= 5 , 3 1 11 →= 10 , 1 1 7 →= 7 , 1 1 1 →= 1 , 1 1 9 →= 9 , 1 1 11 →= 11 , 4 1 1 →= 4 , 4 1 9 →= 6 , 12 1 1 →= 12 , 5 6 2 →= 8 3 7 , 9 6 2 →= 7 3 7 } 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, 7 ↦ 5, 8 ↦ 6, 1 ↦ 7, 9 ↦ 8, 10 ↦ 9, 11 ↦ 10, 12 ↦ 11 }, it remains to prove termination of the 27-rule system { 0 1 →= 2 , 0 3 →= 4 , 1 5 6 →= 3 0 1 5 , 1 5 1 →= 3 0 1 7 , 1 5 3 →= 3 0 1 8 , 1 5 9 →= 3 0 1 10 , 7 5 6 →= 8 0 1 5 , 7 5 1 →= 8 0 1 7 , 7 5 3 →= 8 0 1 8 , 7 5 9 →= 8 0 1 10 , 2 5 6 →= 4 0 1 5 , 2 5 1 →= 4 0 1 7 , 2 5 3 →= 4 0 1 8 , 2 5 9 →= 4 0 1 10 , 1 7 5 →= 6 , 1 7 7 →= 1 , 1 7 8 →= 3 , 1 7 10 →= 9 , 7 7 5 →= 5 , 7 7 7 →= 7 , 7 7 8 →= 8 , 7 7 10 →= 10 , 2 7 7 →= 2 , 2 7 8 →= 4 , 11 7 7 →= 11 , 3 4 0 →= 6 1 5 , 8 4 0 →= 5 1 5 } The system is trivially terminating.