/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 }, it remains to prove termination of the 2-rule system { 0 0 0 0 ⟶ 1 0 0 1 , 1 0 1 ⟶ 0 1 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, (0,3) ↦ 4, (1,3) ↦ 5, (2,0) ↦ 6, (2,1) ↦ 7 }, it remains to prove termination of the 18-rule system { 0 0 0 0 0 ⟶ 1 2 0 1 2 , 0 0 0 0 1 ⟶ 1 2 0 1 3 , 0 0 0 0 4 ⟶ 1 2 0 1 5 , 2 0 0 0 0 ⟶ 3 2 0 1 2 , 2 0 0 0 1 ⟶ 3 2 0 1 3 , 2 0 0 0 4 ⟶ 3 2 0 1 5 , 6 0 0 0 0 ⟶ 7 2 0 1 2 , 6 0 0 0 1 ⟶ 7 2 0 1 3 , 6 0 0 0 4 ⟶ 7 2 0 1 5 , 1 2 1 2 ⟶ 0 1 2 0 , 1 2 1 3 ⟶ 0 1 2 1 , 1 2 1 5 ⟶ 0 1 2 4 , 3 2 1 2 ⟶ 2 1 2 0 , 3 2 1 3 ⟶ 2 1 2 1 , 3 2 1 5 ⟶ 2 1 2 4 , 7 2 1 2 ⟶ 6 1 2 0 , 7 2 1 3 ⟶ 6 1 2 1 , 7 2 1 5 ⟶ 6 1 2 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 6: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 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 }, it remains to prove termination of the 17-rule system { 0 0 0 0 0 ⟶ 1 2 0 1 2 , 0 0 0 0 1 ⟶ 1 2 0 1 3 , 0 0 0 0 4 ⟶ 1 2 0 1 5 , 2 0 0 0 0 ⟶ 3 2 0 1 2 , 2 0 0 0 1 ⟶ 3 2 0 1 3 , 2 0 0 0 4 ⟶ 3 2 0 1 5 , 6 0 0 0 0 ⟶ 7 2 0 1 2 , 6 0 0 0 1 ⟶ 7 2 0 1 3 , 1 2 1 2 ⟶ 0 1 2 0 , 1 2 1 3 ⟶ 0 1 2 1 , 1 2 1 5 ⟶ 0 1 2 4 , 3 2 1 2 ⟶ 2 1 2 0 , 3 2 1 3 ⟶ 2 1 2 1 , 3 2 1 5 ⟶ 2 1 2 4 , 7 2 1 2 ⟶ 6 1 2 0 , 7 2 1 3 ⟶ 6 1 2 1 , 7 2 1 5 ⟶ 6 1 2 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 5: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 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 }, it remains to prove termination of the 16-rule system { 0 0 0 0 0 ⟶ 1 2 0 1 2 , 0 0 0 0 1 ⟶ 1 2 0 1 3 , 0 0 0 0 4 ⟶ 1 2 0 1 5 , 2 0 0 0 0 ⟶ 3 2 0 1 2 , 2 0 0 0 1 ⟶ 3 2 0 1 3 , 2 0 0 0 4 ⟶ 3 2 0 1 5 , 6 0 0 0 0 ⟶ 7 2 0 1 2 , 6 0 0 0 1 ⟶ 7 2 0 1 3 , 1 2 1 2 ⟶ 0 1 2 0 , 1 2 1 3 ⟶ 0 1 2 1 , 1 2 1 5 ⟶ 0 1 2 4 , 3 2 1 2 ⟶ 2 1 2 0 , 3 2 1 3 ⟶ 2 1 2 1 , 3 2 1 5 ⟶ 2 1 2 4 , 7 2 1 2 ⟶ 6 1 2 0 , 7 2 1 3 ⟶ 6 1 2 1 } 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, (3,↓) ↦ 6, (3,↑) ↦ 7, (4,↓) ↦ 8, (5,↓) ↦ 9, (6,↑) ↦ 10, (7,↑) ↦ 11, (6,↓) ↦ 12, (7,↓) ↦ 13 }, it remains to prove termination of the 84-rule system { 0 1 1 1 1 ⟶ 2 3 1 4 3 , 0 1 1 1 1 ⟶ 5 1 4 3 , 0 1 1 1 1 ⟶ 0 4 3 , 0 1 1 1 1 ⟶ 2 3 , 0 1 1 1 1 ⟶ 5 , 0 1 1 1 4 ⟶ 2 3 1 4 6 , 0 1 1 1 4 ⟶ 5 1 4 6 , 0 1 1 1 4 ⟶ 0 4 6 , 0 1 1 1 4 ⟶ 2 6 , 0 1 1 1 4 ⟶ 7 , 0 1 1 1 8 ⟶ 2 3 1 4 9 , 0 1 1 1 8 ⟶ 5 1 4 9 , 0 1 1 1 8 ⟶ 0 4 9 , 0 1 1 1 8 ⟶ 2 9 , 5 1 1 1 1 ⟶ 7 3 1 4 3 , 5 1 1 1 1 ⟶ 5 1 4 3 , 5 1 1 1 1 ⟶ 0 4 3 , 5 1 1 1 1 ⟶ 2 3 , 5 1 1 1 1 ⟶ 5 , 5 1 1 1 4 ⟶ 7 3 1 4 6 , 5 1 1 1 4 ⟶ 5 1 4 6 , 5 1 1 1 4 ⟶ 0 4 6 , 5 1 1 1 4 ⟶ 2 6 , 5 1 1 1 4 ⟶ 7 , 5 1 1 1 8 ⟶ 7 3 1 4 9 , 5 1 1 1 8 ⟶ 5 1 4 9 , 5 1 1 1 8 ⟶ 0 4 9 , 5 1 1 1 8 ⟶ 2 9 , 10 1 1 1 1 ⟶ 11 3 1 4 3 , 10 1 1 1 1 ⟶ 5 1 4 3 , 10 1 1 1 1 ⟶ 0 4 3 , 10 1 1 1 1 ⟶ 2 3 , 10 1 1 1 1 ⟶ 5 , 10 1 1 1 4 ⟶ 11 3 1 4 6 , 10 1 1 1 4 ⟶ 5 1 4 6 , 10 1 1 1 4 ⟶ 0 4 6 , 10 1 1 1 4 ⟶ 2 6 , 10 1 1 1 4 ⟶ 7 , 2 3 4 3 ⟶ 0 4 3 1 , 2 3 4 3 ⟶ 2 3 1 , 2 3 4 3 ⟶ 5 1 , 2 3 4 3 ⟶ 0 , 2 3 4 6 ⟶ 0 4 3 4 , 2 3 4 6 ⟶ 2 3 4 , 2 3 4 6 ⟶ 5 4 , 2 3 4 6 ⟶ 2 , 2 3 4 9 ⟶ 0 4 3 8 , 2 3 4 9 ⟶ 2 3 8 , 2 3 4 9 ⟶ 5 8 , 7 3 4 3 ⟶ 5 4 3 1 , 7 3 4 3 ⟶ 2 3 1 , 7 3 4 3 ⟶ 5 1 , 7 3 4 3 ⟶ 0 , 7 3 4 6 ⟶ 5 4 3 4 , 7 3 4 6 ⟶ 2 3 4 , 7 3 4 6 ⟶ 5 4 , 7 3 4 6 ⟶ 2 , 7 3 4 9 ⟶ 5 4 3 8 , 7 3 4 9 ⟶ 2 3 8 , 7 3 4 9 ⟶ 5 8 , 11 3 4 3 ⟶ 10 4 3 1 , 11 3 4 3 ⟶ 2 3 1 , 11 3 4 3 ⟶ 5 1 , 11 3 4 3 ⟶ 0 , 11 3 4 6 ⟶ 10 4 3 4 , 11 3 4 6 ⟶ 2 3 4 , 11 3 4 6 ⟶ 5 4 , 11 3 4 6 ⟶ 2 , 1 1 1 1 1 →= 4 3 1 4 3 , 1 1 1 1 4 →= 4 3 1 4 6 , 1 1 1 1 8 →= 4 3 1 4 9 , 3 1 1 1 1 →= 6 3 1 4 3 , 3 1 1 1 4 →= 6 3 1 4 6 , 3 1 1 1 8 →= 6 3 1 4 9 , 12 1 1 1 1 →= 13 3 1 4 3 , 12 1 1 1 4 →= 13 3 1 4 6 , 4 3 4 3 →= 1 4 3 1 , 4 3 4 6 →= 1 4 3 4 , 4 3 4 9 →= 1 4 3 8 , 6 3 4 3 →= 3 4 3 1 , 6 3 4 6 →= 3 4 3 4 , 6 3 4 9 →= 3 4 3 8 , 13 3 4 3 →= 12 4 3 1 , 13 3 4 6 →= 12 4 3 4 } 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 6 ↦ 5, 8 ↦ 6, 9 ↦ 7, 5 ↦ 8, 7 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 32-rule system { 0 1 1 1 1 ⟶ 2 3 1 4 3 , 0 1 1 1 4 ⟶ 2 3 1 4 5 , 0 1 1 1 6 ⟶ 2 3 1 4 7 , 8 1 1 1 1 ⟶ 9 3 1 4 3 , 8 1 1 1 4 ⟶ 9 3 1 4 5 , 8 1 1 1 6 ⟶ 9 3 1 4 7 , 10 1 1 1 1 ⟶ 11 3 1 4 3 , 10 1 1 1 4 ⟶ 11 3 1 4 5 , 2 3 4 3 ⟶ 0 4 3 1 , 2 3 4 5 ⟶ 0 4 3 4 , 2 3 4 7 ⟶ 0 4 3 6 , 9 3 4 3 ⟶ 8 4 3 1 , 9 3 4 5 ⟶ 8 4 3 4 , 9 3 4 7 ⟶ 8 4 3 6 , 11 3 4 3 ⟶ 10 4 3 1 , 11 3 4 5 ⟶ 10 4 3 4 , 1 1 1 1 1 →= 4 3 1 4 3 , 1 1 1 1 4 →= 4 3 1 4 5 , 1 1 1 1 6 →= 4 3 1 4 7 , 3 1 1 1 1 →= 5 3 1 4 3 , 3 1 1 1 4 →= 5 3 1 4 5 , 3 1 1 1 6 →= 5 3 1 4 7 , 12 1 1 1 1 →= 13 3 1 4 3 , 12 1 1 1 4 →= 13 3 1 4 5 , 4 3 4 3 →= 1 4 3 1 , 4 3 4 5 →= 1 4 3 4 , 4 3 4 7 →= 1 4 3 6 , 5 3 4 3 →= 3 4 3 1 , 5 3 4 5 →= 3 4 3 4 , 5 3 4 7 →= 3 4 3 6 , 13 3 4 3 →= 12 4 3 1 , 13 3 4 5 →= 12 4 3 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 6: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 8 ↦ 6, 9 ↦ 7, 6 ↦ 8, 7 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 31-rule system { 0 1 1 1 1 ⟶ 2 3 1 4 3 , 0 1 1 1 4 ⟶ 2 3 1 4 5 , 6 1 1 1 1 ⟶ 7 3 1 4 3 , 6 1 1 1 4 ⟶ 7 3 1 4 5 , 6 1 1 1 8 ⟶ 7 3 1 4 9 , 10 1 1 1 1 ⟶ 11 3 1 4 3 , 10 1 1 1 4 ⟶ 11 3 1 4 5 , 2 3 4 3 ⟶ 0 4 3 1 , 2 3 4 5 ⟶ 0 4 3 4 , 2 3 4 9 ⟶ 0 4 3 8 , 7 3 4 3 ⟶ 6 4 3 1 , 7 3 4 5 ⟶ 6 4 3 4 , 7 3 4 9 ⟶ 6 4 3 8 , 11 3 4 3 ⟶ 10 4 3 1 , 11 3 4 5 ⟶ 10 4 3 4 , 1 1 1 1 1 →= 4 3 1 4 3 , 1 1 1 1 4 →= 4 3 1 4 5 , 1 1 1 1 8 →= 4 3 1 4 9 , 3 1 1 1 1 →= 5 3 1 4 3 , 3 1 1 1 4 →= 5 3 1 4 5 , 3 1 1 1 8 →= 5 3 1 4 9 , 12 1 1 1 1 →= 13 3 1 4 3 , 12 1 1 1 4 →= 13 3 1 4 5 , 4 3 4 3 →= 1 4 3 1 , 4 3 4 5 →= 1 4 3 4 , 4 3 4 9 →= 1 4 3 8 , 5 3 4 3 →= 3 4 3 1 , 5 3 4 5 →= 3 4 3 4 , 5 3 4 9 →= 3 4 3 8 , 13 3 4 3 →= 12 4 3 1 , 13 3 4 5 →= 12 4 3 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 6: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 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, 10 ↦ 8, 11 ↦ 9, 9 ↦ 10, 8 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 30-rule system { 0 1 1 1 1 ⟶ 2 3 1 4 3 , 0 1 1 1 4 ⟶ 2 3 1 4 5 , 6 1 1 1 1 ⟶ 7 3 1 4 3 , 6 1 1 1 4 ⟶ 7 3 1 4 5 , 8 1 1 1 1 ⟶ 9 3 1 4 3 , 8 1 1 1 4 ⟶ 9 3 1 4 5 , 2 3 4 3 ⟶ 0 4 3 1 , 2 3 4 5 ⟶ 0 4 3 4 , 2 3 4 10 ⟶ 0 4 3 11 , 7 3 4 3 ⟶ 6 4 3 1 , 7 3 4 5 ⟶ 6 4 3 4 , 7 3 4 10 ⟶ 6 4 3 11 , 9 3 4 3 ⟶ 8 4 3 1 , 9 3 4 5 ⟶ 8 4 3 4 , 1 1 1 1 1 →= 4 3 1 4 3 , 1 1 1 1 4 →= 4 3 1 4 5 , 1 1 1 1 11 →= 4 3 1 4 10 , 3 1 1 1 1 →= 5 3 1 4 3 , 3 1 1 1 4 →= 5 3 1 4 5 , 3 1 1 1 11 →= 5 3 1 4 10 , 12 1 1 1 1 →= 13 3 1 4 3 , 12 1 1 1 4 →= 13 3 1 4 5 , 4 3 4 3 →= 1 4 3 1 , 4 3 4 5 →= 1 4 3 4 , 4 3 4 10 →= 1 4 3 11 , 5 3 4 3 →= 3 4 3 1 , 5 3 4 5 →= 3 4 3 4 , 5 3 4 10 →= 3 4 3 11 , 13 3 4 3 →= 12 4 3 1 , 13 3 4 5 →= 12 4 3 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 5: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 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, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 29-rule system { 0 1 1 1 1 ⟶ 2 3 1 4 3 , 0 1 1 1 4 ⟶ 2 3 1 4 5 , 6 1 1 1 1 ⟶ 7 3 1 4 3 , 6 1 1 1 4 ⟶ 7 3 1 4 5 , 8 1 1 1 1 ⟶ 9 3 1 4 3 , 8 1 1 1 4 ⟶ 9 3 1 4 5 , 2 3 4 3 ⟶ 0 4 3 1 , 2 3 4 5 ⟶ 0 4 3 4 , 7 3 4 3 ⟶ 6 4 3 1 , 7 3 4 5 ⟶ 6 4 3 4 , 7 3 4 10 ⟶ 6 4 3 11 , 9 3 4 3 ⟶ 8 4 3 1 , 9 3 4 5 ⟶ 8 4 3 4 , 1 1 1 1 1 →= 4 3 1 4 3 , 1 1 1 1 4 →= 4 3 1 4 5 , 1 1 1 1 11 →= 4 3 1 4 10 , 3 1 1 1 1 →= 5 3 1 4 3 , 3 1 1 1 4 →= 5 3 1 4 5 , 3 1 1 1 11 →= 5 3 1 4 10 , 12 1 1 1 1 →= 13 3 1 4 3 , 12 1 1 1 4 →= 13 3 1 4 5 , 4 3 4 3 →= 1 4 3 1 , 4 3 4 5 →= 1 4 3 4 , 4 3 4 10 →= 1 4 3 11 , 5 3 4 3 →= 3 4 3 1 , 5 3 4 5 →= 3 4 3 4 , 5 3 4 10 →= 3 4 3 11 , 13 3 4 3 →= 12 4 3 1 , 13 3 4 5 →= 12 4 3 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 5: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 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, 11 ↦ 10, 10 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 28-rule system { 0 1 1 1 1 ⟶ 2 3 1 4 3 , 0 1 1 1 4 ⟶ 2 3 1 4 5 , 6 1 1 1 1 ⟶ 7 3 1 4 3 , 6 1 1 1 4 ⟶ 7 3 1 4 5 , 8 1 1 1 1 ⟶ 9 3 1 4 3 , 8 1 1 1 4 ⟶ 9 3 1 4 5 , 2 3 4 3 ⟶ 0 4 3 1 , 2 3 4 5 ⟶ 0 4 3 4 , 7 3 4 3 ⟶ 6 4 3 1 , 7 3 4 5 ⟶ 6 4 3 4 , 9 3 4 3 ⟶ 8 4 3 1 , 9 3 4 5 ⟶ 8 4 3 4 , 1 1 1 1 1 →= 4 3 1 4 3 , 1 1 1 1 4 →= 4 3 1 4 5 , 1 1 1 1 10 →= 4 3 1 4 11 , 3 1 1 1 1 →= 5 3 1 4 3 , 3 1 1 1 4 →= 5 3 1 4 5 , 3 1 1 1 10 →= 5 3 1 4 11 , 12 1 1 1 1 →= 13 3 1 4 3 , 12 1 1 1 4 →= 13 3 1 4 5 , 4 3 4 3 →= 1 4 3 1 , 4 3 4 5 →= 1 4 3 4 , 4 3 4 11 →= 1 4 3 10 , 5 3 4 3 →= 3 4 3 1 , 5 3 4 5 →= 3 4 3 4 , 5 3 4 11 →= 3 4 3 10 , 13 3 4 3 →= 12 4 3 1 , 13 3 4 5 →= 12 4 3 4 } Applying sparse tiling TROC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (14,0) ↦ 0, (0,1) ↦ 1, (1,1) ↦ 2, (14,2) ↦ 3, (2,3) ↦ 4, (3,1) ↦ 5, (1,4) ↦ 6, (4,3) ↦ 7, (3,4) ↦ 8, (1,10) ↦ 9, (3,10) ↦ 10, (1,15) ↦ 11, (3,15) ↦ 12, (4,5) ↦ 13, (5,3) ↦ 14, (5,5) ↦ 15, (4,11) ↦ 16, (5,11) ↦ 17, (4,15) ↦ 18, (5,15) ↦ 19, (14,6) ↦ 20, (6,1) ↦ 21, (14,7) ↦ 22, (7,3) ↦ 23, (14,8) ↦ 24, (8,1) ↦ 25, (14,9) ↦ 26, (9,3) ↦ 27, (0,4) ↦ 28, (6,4) ↦ 29, (8,4) ↦ 30, (12,1) ↦ 31, (12,4) ↦ 32, (14,1) ↦ 33, (14,4) ↦ 34, (10,15) ↦ 35, (11,15) ↦ 36, (2,5) ↦ 37, (7,5) ↦ 38, (9,5) ↦ 39, (13,3) ↦ 40, (13,5) ↦ 41, (14,3) ↦ 42, (14,5) ↦ 43, (14,12) ↦ 44, (14,13) ↦ 45 }, it remains to prove termination of the 316-rule system { 0 1 2 2 2 2 ⟶ 3 4 5 6 7 5 , 0 1 2 2 2 6 ⟶ 3 4 5 6 7 8 , 0 1 2 2 2 9 ⟶ 3 4 5 6 7 10 , 0 1 2 2 2 11 ⟶ 3 4 5 6 7 12 , 0 1 2 2 6 7 ⟶ 3 4 5 6 13 14 , 0 1 2 2 6 13 ⟶ 3 4 5 6 13 15 , 0 1 2 2 6 16 ⟶ 3 4 5 6 13 17 , 0 1 2 2 6 18 ⟶ 3 4 5 6 13 19 , 20 21 2 2 2 2 ⟶ 22 23 5 6 7 5 , 20 21 2 2 2 6 ⟶ 22 23 5 6 7 8 , 20 21 2 2 2 9 ⟶ 22 23 5 6 7 10 , 20 21 2 2 2 11 ⟶ 22 23 5 6 7 12 , 20 21 2 2 6 7 ⟶ 22 23 5 6 13 14 , 20 21 2 2 6 13 ⟶ 22 23 5 6 13 15 , 20 21 2 2 6 16 ⟶ 22 23 5 6 13 17 , 20 21 2 2 6 18 ⟶ 22 23 5 6 13 19 , 24 25 2 2 2 2 ⟶ 26 27 5 6 7 5 , 24 25 2 2 2 6 ⟶ 26 27 5 6 7 8 , 24 25 2 2 2 9 ⟶ 26 27 5 6 7 10 , 24 25 2 2 2 11 ⟶ 26 27 5 6 7 12 , 24 25 2 2 6 7 ⟶ 26 27 5 6 13 14 , 24 25 2 2 6 13 ⟶ 26 27 5 6 13 15 , 24 25 2 2 6 16 ⟶ 26 27 5 6 13 17 , 24 25 2 2 6 18 ⟶ 26 27 5 6 13 19 , 3 4 8 7 5 ⟶ 0 28 7 5 2 , 3 4 8 7 8 ⟶ 0 28 7 5 6 , 3 4 8 7 10 ⟶ 0 28 7 5 9 , 3 4 8 7 12 ⟶ 0 28 7 5 11 , 3 4 8 13 14 ⟶ 0 28 7 8 7 , 3 4 8 13 15 ⟶ 0 28 7 8 13 , 3 4 8 13 17 ⟶ 0 28 7 8 16 , 3 4 8 13 19 ⟶ 0 28 7 8 18 , 22 23 8 7 5 ⟶ 20 29 7 5 2 , 22 23 8 7 8 ⟶ 20 29 7 5 6 , 22 23 8 7 10 ⟶ 20 29 7 5 9 , 22 23 8 7 12 ⟶ 20 29 7 5 11 , 22 23 8 13 14 ⟶ 20 29 7 8 7 , 22 23 8 13 15 ⟶ 20 29 7 8 13 , 22 23 8 13 17 ⟶ 20 29 7 8 16 , 22 23 8 13 19 ⟶ 20 29 7 8 18 , 26 27 8 7 5 ⟶ 24 30 7 5 2 , 26 27 8 7 8 ⟶ 24 30 7 5 6 , 26 27 8 7 10 ⟶ 24 30 7 5 9 , 26 27 8 7 12 ⟶ 24 30 7 5 11 , 26 27 8 13 14 ⟶ 24 30 7 8 7 , 26 27 8 13 15 ⟶ 24 30 7 8 13 , 26 27 8 13 17 ⟶ 24 30 7 8 16 , 26 27 8 13 19 ⟶ 24 30 7 8 18 , 1 2 2 2 2 2 →= 28 7 5 6 7 5 , 1 2 2 2 2 6 →= 28 7 5 6 7 8 , 1 2 2 2 2 9 →= 28 7 5 6 7 10 , 1 2 2 2 2 11 →= 28 7 5 6 7 12 , 2 2 2 2 2 2 →= 6 7 5 6 7 5 , 2 2 2 2 2 6 →= 6 7 5 6 7 8 , 2 2 2 2 2 9 →= 6 7 5 6 7 10 , 2 2 2 2 2 11 →= 6 7 5 6 7 12 , 5 2 2 2 2 2 →= 8 7 5 6 7 5 , 5 2 2 2 2 6 →= 8 7 5 6 7 8 , 5 2 2 2 2 9 →= 8 7 5 6 7 10 , 5 2 2 2 2 11 →= 8 7 5 6 7 12 , 21 2 2 2 2 2 →= 29 7 5 6 7 5 , 21 2 2 2 2 6 →= 29 7 5 6 7 8 , 21 2 2 2 2 9 →= 29 7 5 6 7 10 , 21 2 2 2 2 11 →= 29 7 5 6 7 12 , 25 2 2 2 2 2 →= 30 7 5 6 7 5 , 25 2 2 2 2 6 →= 30 7 5 6 7 8 , 25 2 2 2 2 9 →= 30 7 5 6 7 10 , 25 2 2 2 2 11 →= 30 7 5 6 7 12 , 31 2 2 2 2 2 →= 32 7 5 6 7 5 , 31 2 2 2 2 6 →= 32 7 5 6 7 8 , 31 2 2 2 2 9 →= 32 7 5 6 7 10 , 31 2 2 2 2 11 →= 32 7 5 6 7 12 , 33 2 2 2 2 2 →= 34 7 5 6 7 5 , 33 2 2 2 2 6 →= 34 7 5 6 7 8 , 33 2 2 2 2 9 →= 34 7 5 6 7 10 , 33 2 2 2 2 11 →= 34 7 5 6 7 12 , 1 2 2 2 6 7 →= 28 7 5 6 13 14 , 1 2 2 2 6 13 →= 28 7 5 6 13 15 , 1 2 2 2 6 16 →= 28 7 5 6 13 17 , 1 2 2 2 6 18 →= 28 7 5 6 13 19 , 2 2 2 2 6 7 →= 6 7 5 6 13 14 , 2 2 2 2 6 13 →= 6 7 5 6 13 15 , 2 2 2 2 6 16 →= 6 7 5 6 13 17 , 2 2 2 2 6 18 →= 6 7 5 6 13 19 , 5 2 2 2 6 7 →= 8 7 5 6 13 14 , 5 2 2 2 6 13 →= 8 7 5 6 13 15 , 5 2 2 2 6 16 →= 8 7 5 6 13 17 , 5 2 2 2 6 18 →= 8 7 5 6 13 19 , 21 2 2 2 6 7 →= 29 7 5 6 13 14 , 21 2 2 2 6 13 →= 29 7 5 6 13 15 , 21 2 2 2 6 16 →= 29 7 5 6 13 17 , 21 2 2 2 6 18 →= 29 7 5 6 13 19 , 25 2 2 2 6 7 →= 30 7 5 6 13 14 , 25 2 2 2 6 13 →= 30 7 5 6 13 15 , 25 2 2 2 6 16 →= 30 7 5 6 13 17 , 25 2 2 2 6 18 →= 30 7 5 6 13 19 , 31 2 2 2 6 7 →= 32 7 5 6 13 14 , 31 2 2 2 6 13 →= 32 7 5 6 13 15 , 31 2 2 2 6 16 →= 32 7 5 6 13 17 , 31 2 2 2 6 18 →= 32 7 5 6 13 19 , 33 2 2 2 6 7 →= 34 7 5 6 13 14 , 33 2 2 2 6 13 →= 34 7 5 6 13 15 , 33 2 2 2 6 16 →= 34 7 5 6 13 17 , 33 2 2 2 6 18 →= 34 7 5 6 13 19 , 1 2 2 2 9 35 →= 28 7 5 6 16 36 , 2 2 2 2 9 35 →= 6 7 5 6 16 36 , 5 2 2 2 9 35 →= 8 7 5 6 16 36 , 21 2 2 2 9 35 →= 29 7 5 6 16 36 , 25 2 2 2 9 35 →= 30 7 5 6 16 36 , 31 2 2 2 9 35 →= 32 7 5 6 16 36 , 33 2 2 2 9 35 →= 34 7 5 6 16 36 , 4 5 2 2 2 2 →= 37 14 5 6 7 5 , 4 5 2 2 2 6 →= 37 14 5 6 7 8 , 4 5 2 2 2 9 →= 37 14 5 6 7 10 , 4 5 2 2 2 11 →= 37 14 5 6 7 12 , 7 5 2 2 2 2 →= 13 14 5 6 7 5 , 7 5 2 2 2 6 →= 13 14 5 6 7 8 , 7 5 2 2 2 9 →= 13 14 5 6 7 10 , 7 5 2 2 2 11 →= 13 14 5 6 7 12 , 14 5 2 2 2 2 →= 15 14 5 6 7 5 , 14 5 2 2 2 6 →= 15 14 5 6 7 8 , 14 5 2 2 2 9 →= 15 14 5 6 7 10 , 14 5 2 2 2 11 →= 15 14 5 6 7 12 , 23 5 2 2 2 2 →= 38 14 5 6 7 5 , 23 5 2 2 2 6 →= 38 14 5 6 7 8 , 23 5 2 2 2 9 →= 38 14 5 6 7 10 , 23 5 2 2 2 11 →= 38 14 5 6 7 12 , 27 5 2 2 2 2 →= 39 14 5 6 7 5 , 27 5 2 2 2 6 →= 39 14 5 6 7 8 , 27 5 2 2 2 9 →= 39 14 5 6 7 10 , 27 5 2 2 2 11 →= 39 14 5 6 7 12 , 40 5 2 2 2 2 →= 41 14 5 6 7 5 , 40 5 2 2 2 6 →= 41 14 5 6 7 8 , 40 5 2 2 2 9 →= 41 14 5 6 7 10 , 40 5 2 2 2 11 →= 41 14 5 6 7 12 , 42 5 2 2 2 2 →= 43 14 5 6 7 5 , 42 5 2 2 2 6 →= 43 14 5 6 7 8 , 42 5 2 2 2 9 →= 43 14 5 6 7 10 , 42 5 2 2 2 11 →= 43 14 5 6 7 12 , 4 5 2 2 6 7 →= 37 14 5 6 13 14 , 4 5 2 2 6 13 →= 37 14 5 6 13 15 , 4 5 2 2 6 16 →= 37 14 5 6 13 17 , 4 5 2 2 6 18 →= 37 14 5 6 13 19 , 7 5 2 2 6 7 →= 13 14 5 6 13 14 , 7 5 2 2 6 13 →= 13 14 5 6 13 15 , 7 5 2 2 6 16 →= 13 14 5 6 13 17 , 7 5 2 2 6 18 →= 13 14 5 6 13 19 , 14 5 2 2 6 7 →= 15 14 5 6 13 14 , 14 5 2 2 6 13 →= 15 14 5 6 13 15 , 14 5 2 2 6 16 →= 15 14 5 6 13 17 , 14 5 2 2 6 18 →= 15 14 5 6 13 19 , 23 5 2 2 6 7 →= 38 14 5 6 13 14 , 23 5 2 2 6 13 →= 38 14 5 6 13 15 , 23 5 2 2 6 16 →= 38 14 5 6 13 17 , 23 5 2 2 6 18 →= 38 14 5 6 13 19 , 27 5 2 2 6 7 →= 39 14 5 6 13 14 , 27 5 2 2 6 13 →= 39 14 5 6 13 15 , 27 5 2 2 6 16 →= 39 14 5 6 13 17 , 27 5 2 2 6 18 →= 39 14 5 6 13 19 , 40 5 2 2 6 7 →= 41 14 5 6 13 14 , 40 5 2 2 6 13 →= 41 14 5 6 13 15 , 40 5 2 2 6 16 →= 41 14 5 6 13 17 , 40 5 2 2 6 18 →= 41 14 5 6 13 19 , 42 5 2 2 6 7 →= 43 14 5 6 13 14 , 42 5 2 2 6 13 →= 43 14 5 6 13 15 , 42 5 2 2 6 16 →= 43 14 5 6 13 17 , 42 5 2 2 6 18 →= 43 14 5 6 13 19 , 4 5 2 2 9 35 →= 37 14 5 6 16 36 , 7 5 2 2 9 35 →= 13 14 5 6 16 36 , 14 5 2 2 9 35 →= 15 14 5 6 16 36 , 23 5 2 2 9 35 →= 38 14 5 6 16 36 , 27 5 2 2 9 35 →= 39 14 5 6 16 36 , 40 5 2 2 9 35 →= 41 14 5 6 16 36 , 42 5 2 2 9 35 →= 43 14 5 6 16 36 , 44 31 2 2 2 2 →= 45 40 5 6 7 5 , 44 31 2 2 2 6 →= 45 40 5 6 7 8 , 44 31 2 2 2 9 →= 45 40 5 6 7 10 , 44 31 2 2 2 11 →= 45 40 5 6 7 12 , 44 31 2 2 6 7 →= 45 40 5 6 13 14 , 44 31 2 2 6 13 →= 45 40 5 6 13 15 , 44 31 2 2 6 16 →= 45 40 5 6 13 17 , 44 31 2 2 6 18 →= 45 40 5 6 13 19 , 28 7 8 7 5 →= 1 6 7 5 2 , 28 7 8 7 8 →= 1 6 7 5 6 , 28 7 8 7 10 →= 1 6 7 5 9 , 28 7 8 7 12 →= 1 6 7 5 11 , 6 7 8 7 5 →= 2 6 7 5 2 , 6 7 8 7 8 →= 2 6 7 5 6 , 6 7 8 7 10 →= 2 6 7 5 9 , 6 7 8 7 12 →= 2 6 7 5 11 , 8 7 8 7 5 →= 5 6 7 5 2 , 8 7 8 7 8 →= 5 6 7 5 6 , 8 7 8 7 10 →= 5 6 7 5 9 , 8 7 8 7 12 →= 5 6 7 5 11 , 29 7 8 7 5 →= 21 6 7 5 2 , 29 7 8 7 8 →= 21 6 7 5 6 , 29 7 8 7 10 →= 21 6 7 5 9 , 29 7 8 7 12 →= 21 6 7 5 11 , 30 7 8 7 5 →= 25 6 7 5 2 , 30 7 8 7 8 →= 25 6 7 5 6 , 30 7 8 7 10 →= 25 6 7 5 9 , 30 7 8 7 12 →= 25 6 7 5 11 , 32 7 8 7 5 →= 31 6 7 5 2 , 32 7 8 7 8 →= 31 6 7 5 6 , 32 7 8 7 10 →= 31 6 7 5 9 , 32 7 8 7 12 →= 31 6 7 5 11 , 34 7 8 7 5 →= 33 6 7 5 2 , 34 7 8 7 8 →= 33 6 7 5 6 , 34 7 8 7 10 →= 33 6 7 5 9 , 34 7 8 7 12 →= 33 6 7 5 11 , 28 7 8 13 14 →= 1 6 7 8 7 , 28 7 8 13 15 →= 1 6 7 8 13 , 28 7 8 13 17 →= 1 6 7 8 16 , 28 7 8 13 19 →= 1 6 7 8 18 , 6 7 8 13 14 →= 2 6 7 8 7 , 6 7 8 13 15 →= 2 6 7 8 13 , 6 7 8 13 17 →= 2 6 7 8 16 , 6 7 8 13 19 →= 2 6 7 8 18 , 8 7 8 13 14 →= 5 6 7 8 7 , 8 7 8 13 15 →= 5 6 7 8 13 , 8 7 8 13 17 →= 5 6 7 8 16 , 8 7 8 13 19 →= 5 6 7 8 18 , 29 7 8 13 14 →= 21 6 7 8 7 , 29 7 8 13 15 →= 21 6 7 8 13 , 29 7 8 13 17 →= 21 6 7 8 16 , 29 7 8 13 19 →= 21 6 7 8 18 , 30 7 8 13 14 →= 25 6 7 8 7 , 30 7 8 13 15 →= 25 6 7 8 13 , 30 7 8 13 17 →= 25 6 7 8 16 , 30 7 8 13 19 →= 25 6 7 8 18 , 32 7 8 13 14 →= 31 6 7 8 7 , 32 7 8 13 15 →= 31 6 7 8 13 , 32 7 8 13 17 →= 31 6 7 8 16 , 32 7 8 13 19 →= 31 6 7 8 18 , 34 7 8 13 14 →= 33 6 7 8 7 , 34 7 8 13 15 →= 33 6 7 8 13 , 34 7 8 13 17 →= 33 6 7 8 16 , 34 7 8 13 19 →= 33 6 7 8 18 , 28 7 8 16 36 →= 1 6 7 10 35 , 6 7 8 16 36 →= 2 6 7 10 35 , 8 7 8 16 36 →= 5 6 7 10 35 , 29 7 8 16 36 →= 21 6 7 10 35 , 30 7 8 16 36 →= 25 6 7 10 35 , 32 7 8 16 36 →= 31 6 7 10 35 , 34 7 8 16 36 →= 33 6 7 10 35 , 37 14 8 7 5 →= 4 8 7 5 2 , 37 14 8 7 8 →= 4 8 7 5 6 , 37 14 8 7 10 →= 4 8 7 5 9 , 37 14 8 7 12 →= 4 8 7 5 11 , 13 14 8 7 5 →= 7 8 7 5 2 , 13 14 8 7 8 →= 7 8 7 5 6 , 13 14 8 7 10 →= 7 8 7 5 9 , 13 14 8 7 12 →= 7 8 7 5 11 , 15 14 8 7 5 →= 14 8 7 5 2 , 15 14 8 7 8 →= 14 8 7 5 6 , 15 14 8 7 10 →= 14 8 7 5 9 , 15 14 8 7 12 →= 14 8 7 5 11 , 38 14 8 7 5 →= 23 8 7 5 2 , 38 14 8 7 8 →= 23 8 7 5 6 , 38 14 8 7 10 →= 23 8 7 5 9 , 38 14 8 7 12 →= 23 8 7 5 11 , 39 14 8 7 5 →= 27 8 7 5 2 , 39 14 8 7 8 →= 27 8 7 5 6 , 39 14 8 7 10 →= 27 8 7 5 9 , 39 14 8 7 12 →= 27 8 7 5 11 , 41 14 8 7 5 →= 40 8 7 5 2 , 41 14 8 7 8 →= 40 8 7 5 6 , 41 14 8 7 10 →= 40 8 7 5 9 , 41 14 8 7 12 →= 40 8 7 5 11 , 43 14 8 7 5 →= 42 8 7 5 2 , 43 14 8 7 8 →= 42 8 7 5 6 , 43 14 8 7 10 →= 42 8 7 5 9 , 43 14 8 7 12 →= 42 8 7 5 11 , 37 14 8 13 14 →= 4 8 7 8 7 , 37 14 8 13 15 →= 4 8 7 8 13 , 37 14 8 13 17 →= 4 8 7 8 16 , 37 14 8 13 19 →= 4 8 7 8 18 , 13 14 8 13 14 →= 7 8 7 8 7 , 13 14 8 13 15 →= 7 8 7 8 13 , 13 14 8 13 17 →= 7 8 7 8 16 , 13 14 8 13 19 →= 7 8 7 8 18 , 15 14 8 13 14 →= 14 8 7 8 7 , 15 14 8 13 15 →= 14 8 7 8 13 , 15 14 8 13 17 →= 14 8 7 8 16 , 15 14 8 13 19 →= 14 8 7 8 18 , 38 14 8 13 14 →= 23 8 7 8 7 , 38 14 8 13 15 →= 23 8 7 8 13 , 38 14 8 13 17 →= 23 8 7 8 16 , 38 14 8 13 19 →= 23 8 7 8 18 , 39 14 8 13 14 →= 27 8 7 8 7 , 39 14 8 13 15 →= 27 8 7 8 13 , 39 14 8 13 17 →= 27 8 7 8 16 , 39 14 8 13 19 →= 27 8 7 8 18 , 41 14 8 13 14 →= 40 8 7 8 7 , 41 14 8 13 15 →= 40 8 7 8 13 , 41 14 8 13 17 →= 40 8 7 8 16 , 41 14 8 13 19 →= 40 8 7 8 18 , 43 14 8 13 14 →= 42 8 7 8 7 , 43 14 8 13 15 →= 42 8 7 8 13 , 43 14 8 13 17 →= 42 8 7 8 16 , 43 14 8 13 19 →= 42 8 7 8 18 , 37 14 8 16 36 →= 4 8 7 10 35 , 13 14 8 16 36 →= 7 8 7 10 35 , 15 14 8 16 36 →= 14 8 7 10 35 , 38 14 8 16 36 →= 23 8 7 10 35 , 39 14 8 16 36 →= 27 8 7 10 35 , 41 14 8 16 36 →= 40 8 7 10 35 , 43 14 8 16 36 →= 42 8 7 10 35 , 45 40 8 7 5 →= 44 32 7 5 2 , 45 40 8 7 8 →= 44 32 7 5 6 , 45 40 8 7 10 →= 44 32 7 5 9 , 45 40 8 7 12 →= 44 32 7 5 11 , 45 40 8 13 14 →= 44 32 7 8 7 , 45 40 8 13 15 →= 44 32 7 8 13 , 45 40 8 13 17 →= 44 32 7 8 16 , 45 40 8 13 19 →= 44 32 7 8 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 23 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 24 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 25 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 26 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 27 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 28 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 29 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 30 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 31 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 32 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 33 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 34 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 35 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 36 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 37 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 38 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 39 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 40 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 41 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 42 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 43 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 44 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 45 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 3 ↦ 0, 4 ↦ 1, 8 ↦ 2, 7 ↦ 3, 5 ↦ 4, 0 ↦ 5, 28 ↦ 6, 2 ↦ 7, 13 ↦ 8, 14 ↦ 9, 15 ↦ 10, 17 ↦ 11, 16 ↦ 12, 19 ↦ 13, 18 ↦ 14, 22 ↦ 15, 23 ↦ 16, 20 ↦ 17, 29 ↦ 18, 26 ↦ 19, 27 ↦ 20, 24 ↦ 21, 30 ↦ 22, 6 ↦ 23, 37 ↦ 24, 38 ↦ 25, 39 ↦ 26, 40 ↦ 27, 41 ↦ 28, 42 ↦ 29, 43 ↦ 30, 36 ↦ 31, 10 ↦ 32, 35 ↦ 33, 45 ↦ 34, 44 ↦ 35, 32 ↦ 36 }, it remains to prove termination of the 108-rule system { 0 1 2 3 4 ⟶ 5 6 3 4 7 , 0 1 2 8 9 ⟶ 5 6 3 2 3 , 0 1 2 8 10 ⟶ 5 6 3 2 8 , 0 1 2 8 11 ⟶ 5 6 3 2 12 , 0 1 2 8 13 ⟶ 5 6 3 2 14 , 15 16 2 3 4 ⟶ 17 18 3 4 7 , 15 16 2 8 9 ⟶ 17 18 3 2 3 , 15 16 2 8 10 ⟶ 17 18 3 2 8 , 15 16 2 8 11 ⟶ 17 18 3 2 12 , 15 16 2 8 13 ⟶ 17 18 3 2 14 , 19 20 2 3 4 ⟶ 21 22 3 4 7 , 19 20 2 8 9 ⟶ 21 22 3 2 3 , 19 20 2 8 10 ⟶ 21 22 3 2 8 , 19 20 2 8 11 ⟶ 21 22 3 2 12 , 19 20 2 8 13 ⟶ 21 22 3 2 14 , 4 7 7 7 7 23 →= 2 3 4 23 3 2 , 4 7 7 7 23 3 →= 2 3 4 23 8 9 , 4 7 7 7 23 8 →= 2 3 4 23 8 10 , 4 7 7 7 23 12 →= 2 3 4 23 8 11 , 4 7 7 7 23 14 →= 2 3 4 23 8 13 , 1 4 7 7 7 23 →= 24 9 4 23 3 2 , 3 4 7 7 7 23 →= 8 9 4 23 3 2 , 9 4 7 7 7 23 →= 10 9 4 23 3 2 , 16 4 7 7 7 23 →= 25 9 4 23 3 2 , 20 4 7 7 7 23 →= 26 9 4 23 3 2 , 27 4 7 7 7 23 →= 28 9 4 23 3 2 , 29 4 7 7 7 23 →= 30 9 4 23 3 2 , 1 4 7 7 23 3 →= 24 9 4 23 8 9 , 1 4 7 7 23 8 →= 24 9 4 23 8 10 , 1 4 7 7 23 12 →= 24 9 4 23 8 11 , 1 4 7 7 23 14 →= 24 9 4 23 8 13 , 3 4 7 7 23 3 →= 8 9 4 23 8 9 , 3 4 7 7 23 8 →= 8 9 4 23 8 10 , 3 4 7 7 23 12 →= 8 9 4 23 8 11 , 3 4 7 7 23 14 →= 8 9 4 23 8 13 , 9 4 7 7 23 3 →= 10 9 4 23 8 9 , 9 4 7 7 23 8 →= 10 9 4 23 8 10 , 9 4 7 7 23 12 →= 10 9 4 23 8 11 , 9 4 7 7 23 14 →= 10 9 4 23 8 13 , 16 4 7 7 23 3 →= 25 9 4 23 8 9 , 16 4 7 7 23 8 →= 25 9 4 23 8 10 , 16 4 7 7 23 12 →= 25 9 4 23 8 11 , 16 4 7 7 23 14 →= 25 9 4 23 8 13 , 20 4 7 7 23 3 →= 26 9 4 23 8 9 , 20 4 7 7 23 8 →= 26 9 4 23 8 10 , 20 4 7 7 23 12 →= 26 9 4 23 8 11 , 20 4 7 7 23 14 →= 26 9 4 23 8 13 , 27 4 7 7 23 3 →= 28 9 4 23 8 9 , 27 4 7 7 23 8 →= 28 9 4 23 8 10 , 27 4 7 7 23 12 →= 28 9 4 23 8 11 , 27 4 7 7 23 14 →= 28 9 4 23 8 13 , 29 4 7 7 23 3 →= 30 9 4 23 8 9 , 29 4 7 7 23 8 →= 30 9 4 23 8 10 , 29 4 7 7 23 12 →= 30 9 4 23 8 11 , 29 4 7 7 23 14 →= 30 9 4 23 8 13 , 23 3 2 3 4 →= 7 23 3 4 7 , 23 3 2 8 9 →= 7 23 3 2 3 , 23 3 2 8 10 →= 7 23 3 2 8 , 23 3 2 8 11 →= 7 23 3 2 12 , 23 3 2 8 13 →= 7 23 3 2 14 , 23 3 2 12 31 →= 7 23 3 32 33 , 24 9 2 3 4 →= 1 2 3 4 7 , 8 9 2 3 4 →= 3 2 3 4 7 , 10 9 2 3 4 →= 9 2 3 4 7 , 25 9 2 3 4 →= 16 2 3 4 7 , 26 9 2 3 4 →= 20 2 3 4 7 , 28 9 2 3 4 →= 27 2 3 4 7 , 30 9 2 3 4 →= 29 2 3 4 7 , 24 9 2 8 9 →= 1 2 3 2 3 , 24 9 2 8 10 →= 1 2 3 2 8 , 24 9 2 8 11 →= 1 2 3 2 12 , 24 9 2 8 13 →= 1 2 3 2 14 , 8 9 2 8 9 →= 3 2 3 2 3 , 8 9 2 8 10 →= 3 2 3 2 8 , 8 9 2 8 11 →= 3 2 3 2 12 , 8 9 2 8 13 →= 3 2 3 2 14 , 10 9 2 8 9 →= 9 2 3 2 3 , 10 9 2 8 10 →= 9 2 3 2 8 , 10 9 2 8 11 →= 9 2 3 2 12 , 10 9 2 8 13 →= 9 2 3 2 14 , 25 9 2 8 9 →= 16 2 3 2 3 , 25 9 2 8 10 →= 16 2 3 2 8 , 25 9 2 8 11 →= 16 2 3 2 12 , 25 9 2 8 13 →= 16 2 3 2 14 , 26 9 2 8 9 →= 20 2 3 2 3 , 26 9 2 8 10 →= 20 2 3 2 8 , 26 9 2 8 11 →= 20 2 3 2 12 , 26 9 2 8 13 →= 20 2 3 2 14 , 28 9 2 8 9 →= 27 2 3 2 3 , 28 9 2 8 10 →= 27 2 3 2 8 , 28 9 2 8 11 →= 27 2 3 2 12 , 28 9 2 8 13 →= 27 2 3 2 14 , 30 9 2 8 9 →= 29 2 3 2 3 , 30 9 2 8 10 →= 29 2 3 2 8 , 30 9 2 8 11 →= 29 2 3 2 12 , 30 9 2 8 13 →= 29 2 3 2 14 , 24 9 2 12 31 →= 1 2 3 32 33 , 8 9 2 12 31 →= 3 2 3 32 33 , 10 9 2 12 31 →= 9 2 3 32 33 , 25 9 2 12 31 →= 16 2 3 32 33 , 26 9 2 12 31 →= 20 2 3 32 33 , 28 9 2 12 31 →= 27 2 3 32 33 , 30 9 2 12 31 →= 29 2 3 32 33 , 34 27 2 3 4 →= 35 36 3 4 7 , 34 27 2 8 9 →= 35 36 3 2 3 , 34 27 2 8 10 →= 35 36 3 2 8 , 34 27 2 8 11 →= 35 36 3 2 12 , 34 27 2 8 13 →= 35 36 3 2 14 } 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 23 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 24 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 25 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 26 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 27 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 28 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 29 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 30 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 31 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 32 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 33 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 34 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 35 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 36 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 4 ↦ 0, 7 ↦ 1, 23 ↦ 2, 2 ↦ 3, 3 ↦ 4, 8 ↦ 5, 9 ↦ 6, 10 ↦ 7, 12 ↦ 8, 11 ↦ 9, 14 ↦ 10, 13 ↦ 11, 1 ↦ 12, 24 ↦ 13, 16 ↦ 14, 25 ↦ 15, 20 ↦ 16, 26 ↦ 17, 27 ↦ 18, 28 ↦ 19, 29 ↦ 20, 30 ↦ 21 }, it remains to prove termination of the 80-rule system { 0 1 1 1 1 2 →= 3 4 0 2 4 3 , 0 1 1 1 2 4 →= 3 4 0 2 5 6 , 0 1 1 1 2 5 →= 3 4 0 2 5 7 , 0 1 1 1 2 8 →= 3 4 0 2 5 9 , 0 1 1 1 2 10 →= 3 4 0 2 5 11 , 12 0 1 1 1 2 →= 13 6 0 2 4 3 , 4 0 1 1 1 2 →= 5 6 0 2 4 3 , 6 0 1 1 1 2 →= 7 6 0 2 4 3 , 14 0 1 1 1 2 →= 15 6 0 2 4 3 , 16 0 1 1 1 2 →= 17 6 0 2 4 3 , 18 0 1 1 1 2 →= 19 6 0 2 4 3 , 20 0 1 1 1 2 →= 21 6 0 2 4 3 , 12 0 1 1 2 4 →= 13 6 0 2 5 6 , 12 0 1 1 2 5 →= 13 6 0 2 5 7 , 12 0 1 1 2 8 →= 13 6 0 2 5 9 , 12 0 1 1 2 10 →= 13 6 0 2 5 11 , 4 0 1 1 2 4 →= 5 6 0 2 5 6 , 4 0 1 1 2 5 →= 5 6 0 2 5 7 , 4 0 1 1 2 8 →= 5 6 0 2 5 9 , 4 0 1 1 2 10 →= 5 6 0 2 5 11 , 6 0 1 1 2 4 →= 7 6 0 2 5 6 , 6 0 1 1 2 5 →= 7 6 0 2 5 7 , 6 0 1 1 2 8 →= 7 6 0 2 5 9 , 6 0 1 1 2 10 →= 7 6 0 2 5 11 , 14 0 1 1 2 4 →= 15 6 0 2 5 6 , 14 0 1 1 2 5 →= 15 6 0 2 5 7 , 14 0 1 1 2 8 →= 15 6 0 2 5 9 , 14 0 1 1 2 10 →= 15 6 0 2 5 11 , 16 0 1 1 2 4 →= 17 6 0 2 5 6 , 16 0 1 1 2 5 →= 17 6 0 2 5 7 , 16 0 1 1 2 8 →= 17 6 0 2 5 9 , 16 0 1 1 2 10 →= 17 6 0 2 5 11 , 18 0 1 1 2 4 →= 19 6 0 2 5 6 , 18 0 1 1 2 5 →= 19 6 0 2 5 7 , 18 0 1 1 2 8 →= 19 6 0 2 5 9 , 18 0 1 1 2 10 →= 19 6 0 2 5 11 , 20 0 1 1 2 4 →= 21 6 0 2 5 6 , 20 0 1 1 2 5 →= 21 6 0 2 5 7 , 20 0 1 1 2 8 →= 21 6 0 2 5 9 , 20 0 1 1 2 10 →= 21 6 0 2 5 11 , 2 4 3 4 0 →= 1 2 4 0 1 , 2 4 3 5 6 →= 1 2 4 3 4 , 2 4 3 5 7 →= 1 2 4 3 5 , 2 4 3 5 9 →= 1 2 4 3 8 , 2 4 3 5 11 →= 1 2 4 3 10 , 13 6 3 4 0 →= 12 3 4 0 1 , 5 6 3 4 0 →= 4 3 4 0 1 , 7 6 3 4 0 →= 6 3 4 0 1 , 15 6 3 4 0 →= 14 3 4 0 1 , 17 6 3 4 0 →= 16 3 4 0 1 , 19 6 3 4 0 →= 18 3 4 0 1 , 21 6 3 4 0 →= 20 3 4 0 1 , 13 6 3 5 6 →= 12 3 4 3 4 , 13 6 3 5 7 →= 12 3 4 3 5 , 13 6 3 5 9 →= 12 3 4 3 8 , 13 6 3 5 11 →= 12 3 4 3 10 , 5 6 3 5 6 →= 4 3 4 3 4 , 5 6 3 5 7 →= 4 3 4 3 5 , 5 6 3 5 9 →= 4 3 4 3 8 , 5 6 3 5 11 →= 4 3 4 3 10 , 7 6 3 5 6 →= 6 3 4 3 4 , 7 6 3 5 7 →= 6 3 4 3 5 , 7 6 3 5 9 →= 6 3 4 3 8 , 7 6 3 5 11 →= 6 3 4 3 10 , 15 6 3 5 6 →= 14 3 4 3 4 , 15 6 3 5 7 →= 14 3 4 3 5 , 15 6 3 5 9 →= 14 3 4 3 8 , 15 6 3 5 11 →= 14 3 4 3 10 , 17 6 3 5 6 →= 16 3 4 3 4 , 17 6 3 5 7 →= 16 3 4 3 5 , 17 6 3 5 9 →= 16 3 4 3 8 , 17 6 3 5 11 →= 16 3 4 3 10 , 19 6 3 5 6 →= 18 3 4 3 4 , 19 6 3 5 7 →= 18 3 4 3 5 , 19 6 3 5 9 →= 18 3 4 3 8 , 19 6 3 5 11 →= 18 3 4 3 10 , 21 6 3 5 6 →= 20 3 4 3 4 , 21 6 3 5 7 →= 20 3 4 3 5 , 21 6 3 5 9 →= 20 3 4 3 8 , 21 6 3 5 11 →= 20 3 4 3 10 } The system is trivially terminating.