/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 }, it remains to prove termination of the 4-rule system { 0 0 0 1 1 ⟶ 1 1 1 , 1 0 0 0 1 ⟶ 0 0 0 1 0 0 0 , 0 0 0 ⟶ 0 0 , 1 1 ⟶ 0 1 0 } 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 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1 }, it remains to prove termination of the 2-rule system { 0 0 0 1 1 ⟶ 1 1 1 , 1 0 0 0 1 ⟶ 0 0 0 1 0 0 0 } Applying sparse tiling TRFC(2) after reversal [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { 3 ↦ 0, 2 ↦ 1, 1 ↦ 2, 0 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7 }, it remains to prove termination of the 18-rule system { 0 1 1 2 3 3 ⟶ 3 3 3 3 , 1 1 1 2 3 3 ⟶ 2 3 3 3 , 4 1 1 2 3 3 ⟶ 5 3 3 3 , 0 1 1 2 3 0 ⟶ 3 3 3 0 , 1 1 1 2 3 0 ⟶ 2 3 3 0 , 4 1 1 2 3 0 ⟶ 5 3 3 0 , 0 1 1 2 3 6 ⟶ 3 3 3 6 , 1 1 1 2 3 6 ⟶ 2 3 3 6 , 4 1 1 2 3 6 ⟶ 5 3 3 6 , 3 0 1 1 2 3 ⟶ 0 1 1 2 0 1 1 2 , 2 0 1 1 2 3 ⟶ 1 1 1 2 0 1 1 2 , 5 0 1 1 2 3 ⟶ 4 1 1 2 0 1 1 2 , 3 0 1 1 2 0 ⟶ 0 1 1 2 0 1 1 1 , 2 0 1 1 2 0 ⟶ 1 1 1 2 0 1 1 1 , 5 0 1 1 2 0 ⟶ 4 1 1 2 0 1 1 1 , 3 0 1 1 2 6 ⟶ 0 1 1 2 0 1 1 7 , 2 0 1 1 2 6 ⟶ 1 1 1 2 0 1 1 7 , 5 0 1 1 2 6 ⟶ 4 1 1 2 0 1 1 7 } 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6 }, it remains to prove termination of the 15-rule system { 0 1 1 2 3 3 ⟶ 3 3 3 3 , 1 1 1 2 3 3 ⟶ 2 3 3 3 , 4 1 1 2 3 3 ⟶ 5 3 3 3 , 0 1 1 2 3 0 ⟶ 3 3 3 0 , 1 1 1 2 3 0 ⟶ 2 3 3 0 , 4 1 1 2 3 0 ⟶ 5 3 3 0 , 0 1 1 2 3 6 ⟶ 3 3 3 6 , 1 1 1 2 3 6 ⟶ 2 3 3 6 , 4 1 1 2 3 6 ⟶ 5 3 3 6 , 3 0 1 1 2 3 ⟶ 0 1 1 2 0 1 1 2 , 2 0 1 1 2 3 ⟶ 1 1 1 2 0 1 1 2 , 5 0 1 1 2 3 ⟶ 4 1 1 2 0 1 1 2 , 3 0 1 1 2 0 ⟶ 0 1 1 2 0 1 1 1 , 2 0 1 1 2 0 ⟶ 1 1 1 2 0 1 1 1 , 5 0 1 1 2 0 ⟶ 4 1 1 2 0 1 1 1 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 0 ↦ ⎛ ⎞ ⎜ 1 0 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 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 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 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 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 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 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 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 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6 }, it remains to prove termination of the 14-rule system { 0 1 1 2 3 3 ⟶ 3 3 3 3 , 1 1 1 2 3 3 ⟶ 2 3 3 3 , 4 1 1 2 3 3 ⟶ 5 3 3 3 , 0 1 1 2 3 0 ⟶ 3 3 3 0 , 1 1 1 2 3 0 ⟶ 2 3 3 0 , 4 1 1 2 3 0 ⟶ 5 3 3 0 , 0 1 1 2 3 6 ⟶ 3 3 3 6 , 1 1 1 2 3 6 ⟶ 2 3 3 6 , 3 0 1 1 2 3 ⟶ 0 1 1 2 0 1 1 2 , 2 0 1 1 2 3 ⟶ 1 1 1 2 0 1 1 2 , 5 0 1 1 2 3 ⟶ 4 1 1 2 0 1 1 2 , 3 0 1 1 2 0 ⟶ 0 1 1 2 0 1 1 1 , 2 0 1 1 2 0 ⟶ 1 1 1 2 0 1 1 1 , 5 0 1 1 2 0 ⟶ 4 1 1 2 0 1 1 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, (3,↑) ↦ 4, (1,↑) ↦ 5, (2,↑) ↦ 6, (4,↑) ↦ 7, (5,↑) ↦ 8, (0,↓) ↦ 9, (6,↓) ↦ 10, (4,↓) ↦ 11, (5,↓) ↦ 12 }, it remains to prove termination of the 78-rule system { 0 1 1 2 3 3 ⟶ 4 3 3 3 , 0 1 1 2 3 3 ⟶ 4 3 3 , 5 1 1 2 3 3 ⟶ 6 3 3 3 , 5 1 1 2 3 3 ⟶ 4 3 3 , 7 1 1 2 3 3 ⟶ 8 3 3 3 , 7 1 1 2 3 3 ⟶ 4 3 3 , 0 1 1 2 3 9 ⟶ 4 3 3 9 , 0 1 1 2 3 9 ⟶ 4 3 9 , 5 1 1 2 3 9 ⟶ 6 3 3 9 , 5 1 1 2 3 9 ⟶ 4 3 9 , 7 1 1 2 3 9 ⟶ 8 3 3 9 , 7 1 1 2 3 9 ⟶ 4 3 9 , 0 1 1 2 3 10 ⟶ 4 3 3 10 , 0 1 1 2 3 10 ⟶ 4 3 10 , 5 1 1 2 3 10 ⟶ 6 3 3 10 , 5 1 1 2 3 10 ⟶ 4 3 10 , 4 9 1 1 2 3 ⟶ 0 1 1 2 9 1 1 2 , 4 9 1 1 2 3 ⟶ 5 1 2 9 1 1 2 , 4 9 1 1 2 3 ⟶ 5 2 9 1 1 2 , 4 9 1 1 2 3 ⟶ 6 9 1 1 2 , 4 9 1 1 2 3 ⟶ 0 1 1 2 , 4 9 1 1 2 3 ⟶ 5 1 2 , 4 9 1 1 2 3 ⟶ 5 2 , 4 9 1 1 2 3 ⟶ 6 , 6 9 1 1 2 3 ⟶ 5 1 1 2 9 1 1 2 , 6 9 1 1 2 3 ⟶ 5 1 2 9 1 1 2 , 6 9 1 1 2 3 ⟶ 5 2 9 1 1 2 , 6 9 1 1 2 3 ⟶ 6 9 1 1 2 , 6 9 1 1 2 3 ⟶ 0 1 1 2 , 6 9 1 1 2 3 ⟶ 5 1 2 , 6 9 1 1 2 3 ⟶ 5 2 , 6 9 1 1 2 3 ⟶ 6 , 8 9 1 1 2 3 ⟶ 7 1 1 2 9 1 1 2 , 8 9 1 1 2 3 ⟶ 5 1 2 9 1 1 2 , 8 9 1 1 2 3 ⟶ 5 2 9 1 1 2 , 8 9 1 1 2 3 ⟶ 6 9 1 1 2 , 8 9 1 1 2 3 ⟶ 0 1 1 2 , 8 9 1 1 2 3 ⟶ 5 1 2 , 8 9 1 1 2 3 ⟶ 5 2 , 8 9 1 1 2 3 ⟶ 6 , 4 9 1 1 2 9 ⟶ 0 1 1 2 9 1 1 1 , 4 9 1 1 2 9 ⟶ 5 1 2 9 1 1 1 , 4 9 1 1 2 9 ⟶ 5 2 9 1 1 1 , 4 9 1 1 2 9 ⟶ 6 9 1 1 1 , 4 9 1 1 2 9 ⟶ 0 1 1 1 , 4 9 1 1 2 9 ⟶ 5 1 1 , 4 9 1 1 2 9 ⟶ 5 1 , 4 9 1 1 2 9 ⟶ 5 , 6 9 1 1 2 9 ⟶ 5 1 1 2 9 1 1 1 , 6 9 1 1 2 9 ⟶ 5 1 2 9 1 1 1 , 6 9 1 1 2 9 ⟶ 5 2 9 1 1 1 , 6 9 1 1 2 9 ⟶ 6 9 1 1 1 , 6 9 1 1 2 9 ⟶ 0 1 1 1 , 6 9 1 1 2 9 ⟶ 5 1 1 , 6 9 1 1 2 9 ⟶ 5 1 , 6 9 1 1 2 9 ⟶ 5 , 8 9 1 1 2 9 ⟶ 7 1 1 2 9 1 1 1 , 8 9 1 1 2 9 ⟶ 5 1 2 9 1 1 1 , 8 9 1 1 2 9 ⟶ 5 2 9 1 1 1 , 8 9 1 1 2 9 ⟶ 6 9 1 1 1 , 8 9 1 1 2 9 ⟶ 0 1 1 1 , 8 9 1 1 2 9 ⟶ 5 1 1 , 8 9 1 1 2 9 ⟶ 5 1 , 8 9 1 1 2 9 ⟶ 5 , 9 1 1 2 3 3 →= 3 3 3 3 , 1 1 1 2 3 3 →= 2 3 3 3 , 11 1 1 2 3 3 →= 12 3 3 3 , 9 1 1 2 3 9 →= 3 3 3 9 , 1 1 1 2 3 9 →= 2 3 3 9 , 11 1 1 2 3 9 →= 12 3 3 9 , 9 1 1 2 3 10 →= 3 3 3 10 , 1 1 1 2 3 10 →= 2 3 3 10 , 3 9 1 1 2 3 →= 9 1 1 2 9 1 1 2 , 2 9 1 1 2 3 →= 1 1 1 2 9 1 1 2 , 12 9 1 1 2 3 →= 11 1 1 2 9 1 1 2 , 3 9 1 1 2 9 →= 9 1 1 2 9 1 1 1 , 2 9 1 1 2 9 →= 1 1 1 2 9 1 1 1 , 12 9 1 1 2 9 →= 11 1 1 2 9 1 1 1 } 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 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 }, it remains to prove termination of the 28-rule system { 0 1 1 2 3 3 ⟶ 4 3 3 3 , 5 1 1 2 3 3 ⟶ 6 3 3 3 , 7 1 1 2 3 3 ⟶ 8 3 3 3 , 0 1 1 2 3 9 ⟶ 4 3 3 9 , 5 1 1 2 3 9 ⟶ 6 3 3 9 , 7 1 1 2 3 9 ⟶ 8 3 3 9 , 0 1 1 2 3 10 ⟶ 4 3 3 10 , 5 1 1 2 3 10 ⟶ 6 3 3 10 , 4 9 1 1 2 3 ⟶ 0 1 1 2 9 1 1 2 , 6 9 1 1 2 3 ⟶ 5 1 1 2 9 1 1 2 , 8 9 1 1 2 3 ⟶ 7 1 1 2 9 1 1 2 , 4 9 1 1 2 9 ⟶ 0 1 1 2 9 1 1 1 , 6 9 1 1 2 9 ⟶ 5 1 1 2 9 1 1 1 , 8 9 1 1 2 9 ⟶ 7 1 1 2 9 1 1 1 , 9 1 1 2 3 3 →= 3 3 3 3 , 1 1 1 2 3 3 →= 2 3 3 3 , 11 1 1 2 3 3 →= 12 3 3 3 , 9 1 1 2 3 9 →= 3 3 3 9 , 1 1 1 2 3 9 →= 2 3 3 9 , 11 1 1 2 3 9 →= 12 3 3 9 , 9 1 1 2 3 10 →= 3 3 3 10 , 1 1 1 2 3 10 →= 2 3 3 10 , 3 9 1 1 2 3 →= 9 1 1 2 9 1 1 2 , 2 9 1 1 2 3 →= 1 1 1 2 9 1 1 2 , 12 9 1 1 2 3 →= 11 1 1 2 9 1 1 2 , 3 9 1 1 2 9 →= 9 1 1 2 9 1 1 1 , 2 9 1 1 2 9 →= 1 1 1 2 9 1 1 1 , 12 9 1 1 2 9 →= 11 1 1 2 9 1 1 1 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 0 ↦ ⎛ ⎞ ⎜ 1 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 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 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 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 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 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 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 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 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 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 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 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 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 }, it remains to prove termination of the 27-rule system { 0 1 1 2 3 3 ⟶ 4 3 3 3 , 5 1 1 2 3 3 ⟶ 6 3 3 3 , 7 1 1 2 3 3 ⟶ 8 3 3 3 , 0 1 1 2 3 9 ⟶ 4 3 3 9 , 5 1 1 2 3 9 ⟶ 6 3 3 9 , 7 1 1 2 3 9 ⟶ 8 3 3 9 , 5 1 1 2 3 10 ⟶ 6 3 3 10 , 4 9 1 1 2 3 ⟶ 0 1 1 2 9 1 1 2 , 6 9 1 1 2 3 ⟶ 5 1 1 2 9 1 1 2 , 8 9 1 1 2 3 ⟶ 7 1 1 2 9 1 1 2 , 4 9 1 1 2 9 ⟶ 0 1 1 2 9 1 1 1 , 6 9 1 1 2 9 ⟶ 5 1 1 2 9 1 1 1 , 8 9 1 1 2 9 ⟶ 7 1 1 2 9 1 1 1 , 9 1 1 2 3 3 →= 3 3 3 3 , 1 1 1 2 3 3 →= 2 3 3 3 , 11 1 1 2 3 3 →= 12 3 3 3 , 9 1 1 2 3 9 →= 3 3 3 9 , 1 1 1 2 3 9 →= 2 3 3 9 , 11 1 1 2 3 9 →= 12 3 3 9 , 9 1 1 2 3 10 →= 3 3 3 10 , 1 1 1 2 3 10 →= 2 3 3 10 , 3 9 1 1 2 3 →= 9 1 1 2 9 1 1 2 , 2 9 1 1 2 3 →= 1 1 1 2 9 1 1 2 , 12 9 1 1 2 3 →= 11 1 1 2 9 1 1 2 , 3 9 1 1 2 9 →= 9 1 1 2 9 1 1 1 , 2 9 1 1 2 9 →= 1 1 1 2 9 1 1 1 , 12 9 1 1 2 9 →= 11 1 1 2 9 1 1 1 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 0 ↦ ⎛ ⎞ ⎜ 1 0 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 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 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 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 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 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 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 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 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 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 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, 12 ↦ 11, 10 ↦ 12 }, it remains to prove termination of the 26-rule system { 0 1 1 2 3 3 ⟶ 4 3 3 3 , 5 1 1 2 3 3 ⟶ 6 3 3 3 , 7 1 1 2 3 3 ⟶ 8 3 3 3 , 0 1 1 2 3 9 ⟶ 4 3 3 9 , 5 1 1 2 3 9 ⟶ 6 3 3 9 , 7 1 1 2 3 9 ⟶ 8 3 3 9 , 4 9 1 1 2 3 ⟶ 0 1 1 2 9 1 1 2 , 6 9 1 1 2 3 ⟶ 5 1 1 2 9 1 1 2 , 8 9 1 1 2 3 ⟶ 7 1 1 2 9 1 1 2 , 4 9 1 1 2 9 ⟶ 0 1 1 2 9 1 1 1 , 6 9 1 1 2 9 ⟶ 5 1 1 2 9 1 1 1 , 8 9 1 1 2 9 ⟶ 7 1 1 2 9 1 1 1 , 9 1 1 2 3 3 →= 3 3 3 3 , 1 1 1 2 3 3 →= 2 3 3 3 , 10 1 1 2 3 3 →= 11 3 3 3 , 9 1 1 2 3 9 →= 3 3 3 9 , 1 1 1 2 3 9 →= 2 3 3 9 , 10 1 1 2 3 9 →= 11 3 3 9 , 9 1 1 2 3 12 →= 3 3 3 12 , 1 1 1 2 3 12 →= 2 3 3 12 , 3 9 1 1 2 3 →= 9 1 1 2 9 1 1 2 , 2 9 1 1 2 3 →= 1 1 1 2 9 1 1 2 , 11 9 1 1 2 3 →= 10 1 1 2 9 1 1 2 , 3 9 1 1 2 9 →= 9 1 1 2 9 1 1 1 , 2 9 1 1 2 9 →= 1 1 1 2 9 1 1 1 , 11 9 1 1 2 9 →= 10 1 1 2 9 1 1 1 } Applying sparse tiling TROC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (13,0) ↦ 0, (0,1) ↦ 1, (1,1) ↦ 2, (1,2) ↦ 3, (2,3) ↦ 4, (3,3) ↦ 5, (13,4) ↦ 6, (4,3) ↦ 7, (3,9) ↦ 8, (3,12) ↦ 9, (3,14) ↦ 10, (13,5) ↦ 11, (5,1) ↦ 12, (13,6) ↦ 13, (6,3) ↦ 14, (13,7) ↦ 15, (7,1) ↦ 16, (13,8) ↦ 17, (8,3) ↦ 18, (9,1) ↦ 19, (9,2) ↦ 20, (9,14) ↦ 21, (4,9) ↦ 22, (2,9) ↦ 23, (2,12) ↦ 24, (2,14) ↦ 25, (6,9) ↦ 26, (8,9) ↦ 27, (1,14) ↦ 28, (11,9) ↦ 29, (11,3) ↦ 30, (13,9) ↦ 31, (13,3) ↦ 32, (0,2) ↦ 33, (5,2) ↦ 34, (7,2) ↦ 35, (10,1) ↦ 36, (10,2) ↦ 37, (13,1) ↦ 38, (13,2) ↦ 39, (13,10) ↦ 40, (13,11) ↦ 41, (12,14) ↦ 42 }, it remains to prove termination of the 266-rule system { 0 1 2 3 4 5 5 ⟶ 6 7 5 5 5 , 0 1 2 3 4 5 8 ⟶ 6 7 5 5 8 , 0 1 2 3 4 5 9 ⟶ 6 7 5 5 9 , 0 1 2 3 4 5 10 ⟶ 6 7 5 5 10 , 11 12 2 3 4 5 5 ⟶ 13 14 5 5 5 , 11 12 2 3 4 5 8 ⟶ 13 14 5 5 8 , 11 12 2 3 4 5 9 ⟶ 13 14 5 5 9 , 11 12 2 3 4 5 10 ⟶ 13 14 5 5 10 , 15 16 2 3 4 5 5 ⟶ 17 18 5 5 5 , 15 16 2 3 4 5 8 ⟶ 17 18 5 5 8 , 15 16 2 3 4 5 9 ⟶ 17 18 5 5 9 , 15 16 2 3 4 5 10 ⟶ 17 18 5 5 10 , 0 1 2 3 4 8 19 ⟶ 6 7 5 8 19 , 0 1 2 3 4 8 20 ⟶ 6 7 5 8 20 , 0 1 2 3 4 8 21 ⟶ 6 7 5 8 21 , 11 12 2 3 4 8 19 ⟶ 13 14 5 8 19 , 11 12 2 3 4 8 20 ⟶ 13 14 5 8 20 , 11 12 2 3 4 8 21 ⟶ 13 14 5 8 21 , 15 16 2 3 4 8 19 ⟶ 17 18 5 8 19 , 15 16 2 3 4 8 20 ⟶ 17 18 5 8 20 , 15 16 2 3 4 8 21 ⟶ 17 18 5 8 21 , 6 22 19 2 3 4 5 ⟶ 0 1 2 3 23 19 2 3 4 , 6 22 19 2 3 4 8 ⟶ 0 1 2 3 23 19 2 3 23 , 6 22 19 2 3 4 9 ⟶ 0 1 2 3 23 19 2 3 24 , 6 22 19 2 3 4 10 ⟶ 0 1 2 3 23 19 2 3 25 , 13 26 19 2 3 4 5 ⟶ 11 12 2 3 23 19 2 3 4 , 13 26 19 2 3 4 8 ⟶ 11 12 2 3 23 19 2 3 23 , 13 26 19 2 3 4 9 ⟶ 11 12 2 3 23 19 2 3 24 , 13 26 19 2 3 4 10 ⟶ 11 12 2 3 23 19 2 3 25 , 17 27 19 2 3 4 5 ⟶ 15 16 2 3 23 19 2 3 4 , 17 27 19 2 3 4 8 ⟶ 15 16 2 3 23 19 2 3 23 , 17 27 19 2 3 4 9 ⟶ 15 16 2 3 23 19 2 3 24 , 17 27 19 2 3 4 10 ⟶ 15 16 2 3 23 19 2 3 25 , 6 22 19 2 3 23 19 ⟶ 0 1 2 3 23 19 2 2 2 , 6 22 19 2 3 23 20 ⟶ 0 1 2 3 23 19 2 2 3 , 6 22 19 2 3 23 21 ⟶ 0 1 2 3 23 19 2 2 28 , 13 26 19 2 3 23 19 ⟶ 11 12 2 3 23 19 2 2 2 , 13 26 19 2 3 23 20 ⟶ 11 12 2 3 23 19 2 2 3 , 13 26 19 2 3 23 21 ⟶ 11 12 2 3 23 19 2 2 28 , 17 27 19 2 3 23 19 ⟶ 15 16 2 3 23 19 2 2 2 , 17 27 19 2 3 23 20 ⟶ 15 16 2 3 23 19 2 2 3 , 17 27 19 2 3 23 21 ⟶ 15 16 2 3 23 19 2 2 28 , 23 19 2 3 4 5 5 →= 4 5 5 5 5 , 23 19 2 3 4 5 8 →= 4 5 5 5 8 , 23 19 2 3 4 5 9 →= 4 5 5 5 9 , 23 19 2 3 4 5 10 →= 4 5 5 5 10 , 8 19 2 3 4 5 5 →= 5 5 5 5 5 , 8 19 2 3 4 5 8 →= 5 5 5 5 8 , 8 19 2 3 4 5 9 →= 5 5 5 5 9 , 8 19 2 3 4 5 10 →= 5 5 5 5 10 , 22 19 2 3 4 5 5 →= 7 5 5 5 5 , 22 19 2 3 4 5 8 →= 7 5 5 5 8 , 22 19 2 3 4 5 9 →= 7 5 5 5 9 , 22 19 2 3 4 5 10 →= 7 5 5 5 10 , 26 19 2 3 4 5 5 →= 14 5 5 5 5 , 26 19 2 3 4 5 8 →= 14 5 5 5 8 , 26 19 2 3 4 5 9 →= 14 5 5 5 9 , 26 19 2 3 4 5 10 →= 14 5 5 5 10 , 27 19 2 3 4 5 5 →= 18 5 5 5 5 , 27 19 2 3 4 5 8 →= 18 5 5 5 8 , 27 19 2 3 4 5 9 →= 18 5 5 5 9 , 27 19 2 3 4 5 10 →= 18 5 5 5 10 , 29 19 2 3 4 5 5 →= 30 5 5 5 5 , 29 19 2 3 4 5 8 →= 30 5 5 5 8 , 29 19 2 3 4 5 9 →= 30 5 5 5 9 , 29 19 2 3 4 5 10 →= 30 5 5 5 10 , 31 19 2 3 4 5 5 →= 32 5 5 5 5 , 31 19 2 3 4 5 8 →= 32 5 5 5 8 , 31 19 2 3 4 5 9 →= 32 5 5 5 9 , 31 19 2 3 4 5 10 →= 32 5 5 5 10 , 1 2 2 3 4 5 5 →= 33 4 5 5 5 , 1 2 2 3 4 5 8 →= 33 4 5 5 8 , 1 2 2 3 4 5 9 →= 33 4 5 5 9 , 1 2 2 3 4 5 10 →= 33 4 5 5 10 , 2 2 2 3 4 5 5 →= 3 4 5 5 5 , 2 2 2 3 4 5 8 →= 3 4 5 5 8 , 2 2 2 3 4 5 9 →= 3 4 5 5 9 , 2 2 2 3 4 5 10 →= 3 4 5 5 10 , 12 2 2 3 4 5 5 →= 34 4 5 5 5 , 12 2 2 3 4 5 8 →= 34 4 5 5 8 , 12 2 2 3 4 5 9 →= 34 4 5 5 9 , 12 2 2 3 4 5 10 →= 34 4 5 5 10 , 16 2 2 3 4 5 5 →= 35 4 5 5 5 , 16 2 2 3 4 5 8 →= 35 4 5 5 8 , 16 2 2 3 4 5 9 →= 35 4 5 5 9 , 16 2 2 3 4 5 10 →= 35 4 5 5 10 , 19 2 2 3 4 5 5 →= 20 4 5 5 5 , 19 2 2 3 4 5 8 →= 20 4 5 5 8 , 19 2 2 3 4 5 9 →= 20 4 5 5 9 , 19 2 2 3 4 5 10 →= 20 4 5 5 10 , 36 2 2 3 4 5 5 →= 37 4 5 5 5 , 36 2 2 3 4 5 8 →= 37 4 5 5 8 , 36 2 2 3 4 5 9 →= 37 4 5 5 9 , 36 2 2 3 4 5 10 →= 37 4 5 5 10 , 38 2 2 3 4 5 5 →= 39 4 5 5 5 , 38 2 2 3 4 5 8 →= 39 4 5 5 8 , 38 2 2 3 4 5 9 →= 39 4 5 5 9 , 38 2 2 3 4 5 10 →= 39 4 5 5 10 , 40 36 2 3 4 5 5 →= 41 30 5 5 5 , 40 36 2 3 4 5 8 →= 41 30 5 5 8 , 40 36 2 3 4 5 9 →= 41 30 5 5 9 , 40 36 2 3 4 5 10 →= 41 30 5 5 10 , 23 19 2 3 4 8 19 →= 4 5 5 8 19 , 23 19 2 3 4 8 20 →= 4 5 5 8 20 , 23 19 2 3 4 8 21 →= 4 5 5 8 21 , 8 19 2 3 4 8 19 →= 5 5 5 8 19 , 8 19 2 3 4 8 20 →= 5 5 5 8 20 , 8 19 2 3 4 8 21 →= 5 5 5 8 21 , 22 19 2 3 4 8 19 →= 7 5 5 8 19 , 22 19 2 3 4 8 20 →= 7 5 5 8 20 , 22 19 2 3 4 8 21 →= 7 5 5 8 21 , 26 19 2 3 4 8 19 →= 14 5 5 8 19 , 26 19 2 3 4 8 20 →= 14 5 5 8 20 , 26 19 2 3 4 8 21 →= 14 5 5 8 21 , 27 19 2 3 4 8 19 →= 18 5 5 8 19 , 27 19 2 3 4 8 20 →= 18 5 5 8 20 , 27 19 2 3 4 8 21 →= 18 5 5 8 21 , 29 19 2 3 4 8 19 →= 30 5 5 8 19 , 29 19 2 3 4 8 20 →= 30 5 5 8 20 , 29 19 2 3 4 8 21 →= 30 5 5 8 21 , 31 19 2 3 4 8 19 →= 32 5 5 8 19 , 31 19 2 3 4 8 20 →= 32 5 5 8 20 , 31 19 2 3 4 8 21 →= 32 5 5 8 21 , 1 2 2 3 4 8 19 →= 33 4 5 8 19 , 1 2 2 3 4 8 20 →= 33 4 5 8 20 , 1 2 2 3 4 8 21 →= 33 4 5 8 21 , 2 2 2 3 4 8 19 →= 3 4 5 8 19 , 2 2 2 3 4 8 20 →= 3 4 5 8 20 , 2 2 2 3 4 8 21 →= 3 4 5 8 21 , 12 2 2 3 4 8 19 →= 34 4 5 8 19 , 12 2 2 3 4 8 20 →= 34 4 5 8 20 , 12 2 2 3 4 8 21 →= 34 4 5 8 21 , 16 2 2 3 4 8 19 →= 35 4 5 8 19 , 16 2 2 3 4 8 20 →= 35 4 5 8 20 , 16 2 2 3 4 8 21 →= 35 4 5 8 21 , 19 2 2 3 4 8 19 →= 20 4 5 8 19 , 19 2 2 3 4 8 20 →= 20 4 5 8 20 , 19 2 2 3 4 8 21 →= 20 4 5 8 21 , 36 2 2 3 4 8 19 →= 37 4 5 8 19 , 36 2 2 3 4 8 20 →= 37 4 5 8 20 , 36 2 2 3 4 8 21 →= 37 4 5 8 21 , 38 2 2 3 4 8 19 →= 39 4 5 8 19 , 38 2 2 3 4 8 20 →= 39 4 5 8 20 , 38 2 2 3 4 8 21 →= 39 4 5 8 21 , 40 36 2 3 4 8 19 →= 41 30 5 8 19 , 40 36 2 3 4 8 20 →= 41 30 5 8 20 , 40 36 2 3 4 8 21 →= 41 30 5 8 21 , 23 19 2 3 4 9 42 →= 4 5 5 9 42 , 8 19 2 3 4 9 42 →= 5 5 5 9 42 , 22 19 2 3 4 9 42 →= 7 5 5 9 42 , 26 19 2 3 4 9 42 →= 14 5 5 9 42 , 27 19 2 3 4 9 42 →= 18 5 5 9 42 , 29 19 2 3 4 9 42 →= 30 5 5 9 42 , 31 19 2 3 4 9 42 →= 32 5 5 9 42 , 1 2 2 3 4 9 42 →= 33 4 5 9 42 , 2 2 2 3 4 9 42 →= 3 4 5 9 42 , 12 2 2 3 4 9 42 →= 34 4 5 9 42 , 16 2 2 3 4 9 42 →= 35 4 5 9 42 , 19 2 2 3 4 9 42 →= 20 4 5 9 42 , 36 2 2 3 4 9 42 →= 37 4 5 9 42 , 38 2 2 3 4 9 42 →= 39 4 5 9 42 , 4 8 19 2 3 4 5 →= 23 19 2 3 23 19 2 3 4 , 4 8 19 2 3 4 8 →= 23 19 2 3 23 19 2 3 23 , 4 8 19 2 3 4 9 →= 23 19 2 3 23 19 2 3 24 , 4 8 19 2 3 4 10 →= 23 19 2 3 23 19 2 3 25 , 5 8 19 2 3 4 5 →= 8 19 2 3 23 19 2 3 4 , 5 8 19 2 3 4 8 →= 8 19 2 3 23 19 2 3 23 , 5 8 19 2 3 4 9 →= 8 19 2 3 23 19 2 3 24 , 5 8 19 2 3 4 10 →= 8 19 2 3 23 19 2 3 25 , 7 8 19 2 3 4 5 →= 22 19 2 3 23 19 2 3 4 , 7 8 19 2 3 4 8 →= 22 19 2 3 23 19 2 3 23 , 7 8 19 2 3 4 9 →= 22 19 2 3 23 19 2 3 24 , 7 8 19 2 3 4 10 →= 22 19 2 3 23 19 2 3 25 , 14 8 19 2 3 4 5 →= 26 19 2 3 23 19 2 3 4 , 14 8 19 2 3 4 8 →= 26 19 2 3 23 19 2 3 23 , 14 8 19 2 3 4 9 →= 26 19 2 3 23 19 2 3 24 , 14 8 19 2 3 4 10 →= 26 19 2 3 23 19 2 3 25 , 18 8 19 2 3 4 5 →= 27 19 2 3 23 19 2 3 4 , 18 8 19 2 3 4 8 →= 27 19 2 3 23 19 2 3 23 , 18 8 19 2 3 4 9 →= 27 19 2 3 23 19 2 3 24 , 18 8 19 2 3 4 10 →= 27 19 2 3 23 19 2 3 25 , 30 8 19 2 3 4 5 →= 29 19 2 3 23 19 2 3 4 , 30 8 19 2 3 4 8 →= 29 19 2 3 23 19 2 3 23 , 30 8 19 2 3 4 9 →= 29 19 2 3 23 19 2 3 24 , 30 8 19 2 3 4 10 →= 29 19 2 3 23 19 2 3 25 , 32 8 19 2 3 4 5 →= 31 19 2 3 23 19 2 3 4 , 32 8 19 2 3 4 8 →= 31 19 2 3 23 19 2 3 23 , 32 8 19 2 3 4 9 →= 31 19 2 3 23 19 2 3 24 , 32 8 19 2 3 4 10 →= 31 19 2 3 23 19 2 3 25 , 33 23 19 2 3 4 5 →= 1 2 2 3 23 19 2 3 4 , 33 23 19 2 3 4 8 →= 1 2 2 3 23 19 2 3 23 , 33 23 19 2 3 4 9 →= 1 2 2 3 23 19 2 3 24 , 33 23 19 2 3 4 10 →= 1 2 2 3 23 19 2 3 25 , 3 23 19 2 3 4 5 →= 2 2 2 3 23 19 2 3 4 , 3 23 19 2 3 4 8 →= 2 2 2 3 23 19 2 3 23 , 3 23 19 2 3 4 9 →= 2 2 2 3 23 19 2 3 24 , 3 23 19 2 3 4 10 →= 2 2 2 3 23 19 2 3 25 , 34 23 19 2 3 4 5 →= 12 2 2 3 23 19 2 3 4 , 34 23 19 2 3 4 8 →= 12 2 2 3 23 19 2 3 23 , 34 23 19 2 3 4 9 →= 12 2 2 3 23 19 2 3 24 , 34 23 19 2 3 4 10 →= 12 2 2 3 23 19 2 3 25 , 35 23 19 2 3 4 5 →= 16 2 2 3 23 19 2 3 4 , 35 23 19 2 3 4 8 →= 16 2 2 3 23 19 2 3 23 , 35 23 19 2 3 4 9 →= 16 2 2 3 23 19 2 3 24 , 35 23 19 2 3 4 10 →= 16 2 2 3 23 19 2 3 25 , 20 23 19 2 3 4 5 →= 19 2 2 3 23 19 2 3 4 , 20 23 19 2 3 4 8 →= 19 2 2 3 23 19 2 3 23 , 20 23 19 2 3 4 9 →= 19 2 2 3 23 19 2 3 24 , 20 23 19 2 3 4 10 →= 19 2 2 3 23 19 2 3 25 , 37 23 19 2 3 4 5 →= 36 2 2 3 23 19 2 3 4 , 37 23 19 2 3 4 8 →= 36 2 2 3 23 19 2 3 23 , 37 23 19 2 3 4 9 →= 36 2 2 3 23 19 2 3 24 , 37 23 19 2 3 4 10 →= 36 2 2 3 23 19 2 3 25 , 39 23 19 2 3 4 5 →= 38 2 2 3 23 19 2 3 4 , 39 23 19 2 3 4 8 →= 38 2 2 3 23 19 2 3 23 , 39 23 19 2 3 4 9 →= 38 2 2 3 23 19 2 3 24 , 39 23 19 2 3 4 10 →= 38 2 2 3 23 19 2 3 25 , 41 29 19 2 3 4 5 →= 40 36 2 3 23 19 2 3 4 , 41 29 19 2 3 4 8 →= 40 36 2 3 23 19 2 3 23 , 41 29 19 2 3 4 9 →= 40 36 2 3 23 19 2 3 24 , 41 29 19 2 3 4 10 →= 40 36 2 3 23 19 2 3 25 , 4 8 19 2 3 23 19 →= 23 19 2 3 23 19 2 2 2 , 4 8 19 2 3 23 20 →= 23 19 2 3 23 19 2 2 3 , 4 8 19 2 3 23 21 →= 23 19 2 3 23 19 2 2 28 , 5 8 19 2 3 23 19 →= 8 19 2 3 23 19 2 2 2 , 5 8 19 2 3 23 20 →= 8 19 2 3 23 19 2 2 3 , 5 8 19 2 3 23 21 →= 8 19 2 3 23 19 2 2 28 , 7 8 19 2 3 23 19 →= 22 19 2 3 23 19 2 2 2 , 7 8 19 2 3 23 20 →= 22 19 2 3 23 19 2 2 3 , 7 8 19 2 3 23 21 →= 22 19 2 3 23 19 2 2 28 , 14 8 19 2 3 23 19 →= 26 19 2 3 23 19 2 2 2 , 14 8 19 2 3 23 20 →= 26 19 2 3 23 19 2 2 3 , 14 8 19 2 3 23 21 →= 26 19 2 3 23 19 2 2 28 , 18 8 19 2 3 23 19 →= 27 19 2 3 23 19 2 2 2 , 18 8 19 2 3 23 20 →= 27 19 2 3 23 19 2 2 3 , 18 8 19 2 3 23 21 →= 27 19 2 3 23 19 2 2 28 , 30 8 19 2 3 23 19 →= 29 19 2 3 23 19 2 2 2 , 30 8 19 2 3 23 20 →= 29 19 2 3 23 19 2 2 3 , 30 8 19 2 3 23 21 →= 29 19 2 3 23 19 2 2 28 , 32 8 19 2 3 23 19 →= 31 19 2 3 23 19 2 2 2 , 32 8 19 2 3 23 20 →= 31 19 2 3 23 19 2 2 3 , 32 8 19 2 3 23 21 →= 31 19 2 3 23 19 2 2 28 , 33 23 19 2 3 23 19 →= 1 2 2 3 23 19 2 2 2 , 33 23 19 2 3 23 20 →= 1 2 2 3 23 19 2 2 3 , 33 23 19 2 3 23 21 →= 1 2 2 3 23 19 2 2 28 , 3 23 19 2 3 23 19 →= 2 2 2 3 23 19 2 2 2 , 3 23 19 2 3 23 20 →= 2 2 2 3 23 19 2 2 3 , 3 23 19 2 3 23 21 →= 2 2 2 3 23 19 2 2 28 , 34 23 19 2 3 23 19 →= 12 2 2 3 23 19 2 2 2 , 34 23 19 2 3 23 20 →= 12 2 2 3 23 19 2 2 3 , 34 23 19 2 3 23 21 →= 12 2 2 3 23 19 2 2 28 , 35 23 19 2 3 23 19 →= 16 2 2 3 23 19 2 2 2 , 35 23 19 2 3 23 20 →= 16 2 2 3 23 19 2 2 3 , 35 23 19 2 3 23 21 →= 16 2 2 3 23 19 2 2 28 , 20 23 19 2 3 23 19 →= 19 2 2 3 23 19 2 2 2 , 20 23 19 2 3 23 20 →= 19 2 2 3 23 19 2 2 3 , 20 23 19 2 3 23 21 →= 19 2 2 3 23 19 2 2 28 , 37 23 19 2 3 23 19 →= 36 2 2 3 23 19 2 2 2 , 37 23 19 2 3 23 20 →= 36 2 2 3 23 19 2 2 3 , 37 23 19 2 3 23 21 →= 36 2 2 3 23 19 2 2 28 , 39 23 19 2 3 23 19 →= 38 2 2 3 23 19 2 2 2 , 39 23 19 2 3 23 20 →= 38 2 2 3 23 19 2 2 3 , 39 23 19 2 3 23 21 →= 38 2 2 3 23 19 2 2 28 , 41 29 19 2 3 23 19 →= 40 36 2 3 23 19 2 2 2 , 41 29 19 2 3 23 20 →= 40 36 2 3 23 19 2 2 3 , 41 29 19 2 3 23 21 →= 40 36 2 3 23 19 2 2 28 } 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 23 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 24 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 25 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 26 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 27 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 28 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 29 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 30 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 31 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 32 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 33 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 34 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 35 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 36 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 37 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 38 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 39 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 40 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 41 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 42 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 6 ↦ 0, 22 ↦ 1, 19 ↦ 2, 2 ↦ 3, 3 ↦ 4, 4 ↦ 5, 5 ↦ 6, 0 ↦ 7, 1 ↦ 8, 23 ↦ 9, 13 ↦ 10, 26 ↦ 11, 11 ↦ 12, 12 ↦ 13, 17 ↦ 14, 27 ↦ 15, 15 ↦ 16, 16 ↦ 17, 20 ↦ 18, 8 ↦ 19, 9 ↦ 20, 10 ↦ 21, 33 ↦ 22, 34 ↦ 23, 35 ↦ 24, 36 ↦ 25, 37 ↦ 26, 38 ↦ 27, 39 ↦ 28, 21 ↦ 29, 42 ↦ 30, 41 ↦ 31, 29 ↦ 32, 40 ↦ 33 }, it remains to prove termination of the 100-rule system { 0 1 2 3 4 5 6 ⟶ 7 8 3 4 9 2 3 4 5 , 10 11 2 3 4 5 6 ⟶ 12 13 3 4 9 2 3 4 5 , 14 15 2 3 4 5 6 ⟶ 16 17 3 4 9 2 3 4 5 , 0 1 2 3 4 9 2 ⟶ 7 8 3 4 9 2 3 3 3 , 0 1 2 3 4 9 18 ⟶ 7 8 3 4 9 2 3 3 4 , 10 11 2 3 4 9 2 ⟶ 12 13 3 4 9 2 3 3 3 , 10 11 2 3 4 9 18 ⟶ 12 13 3 4 9 2 3 3 4 , 14 15 2 3 4 9 2 ⟶ 16 17 3 4 9 2 3 3 3 , 14 15 2 3 4 9 18 ⟶ 16 17 3 4 9 2 3 3 4 , 9 2 3 4 5 6 6 →= 5 6 6 6 6 , 9 2 3 4 5 6 19 →= 5 6 6 6 19 , 9 2 3 4 5 6 20 →= 5 6 6 6 20 , 9 2 3 4 5 6 21 →= 5 6 6 6 21 , 8 3 3 4 5 6 6 →= 22 5 6 6 6 , 8 3 3 4 5 6 19 →= 22 5 6 6 19 , 8 3 3 4 5 6 20 →= 22 5 6 6 20 , 8 3 3 4 5 6 21 →= 22 5 6 6 21 , 3 3 3 4 5 6 6 →= 4 5 6 6 6 , 3 3 3 4 5 6 19 →= 4 5 6 6 19 , 3 3 3 4 5 6 20 →= 4 5 6 6 20 , 3 3 3 4 5 6 21 →= 4 5 6 6 21 , 13 3 3 4 5 6 6 →= 23 5 6 6 6 , 13 3 3 4 5 6 19 →= 23 5 6 6 19 , 13 3 3 4 5 6 20 →= 23 5 6 6 20 , 13 3 3 4 5 6 21 →= 23 5 6 6 21 , 17 3 3 4 5 6 6 →= 24 5 6 6 6 , 17 3 3 4 5 6 19 →= 24 5 6 6 19 , 17 3 3 4 5 6 20 →= 24 5 6 6 20 , 17 3 3 4 5 6 21 →= 24 5 6 6 21 , 2 3 3 4 5 6 6 →= 18 5 6 6 6 , 2 3 3 4 5 6 19 →= 18 5 6 6 19 , 2 3 3 4 5 6 20 →= 18 5 6 6 20 , 2 3 3 4 5 6 21 →= 18 5 6 6 21 , 25 3 3 4 5 6 6 →= 26 5 6 6 6 , 25 3 3 4 5 6 19 →= 26 5 6 6 19 , 25 3 3 4 5 6 20 →= 26 5 6 6 20 , 25 3 3 4 5 6 21 →= 26 5 6 6 21 , 27 3 3 4 5 6 6 →= 28 5 6 6 6 , 27 3 3 4 5 6 19 →= 28 5 6 6 19 , 27 3 3 4 5 6 20 →= 28 5 6 6 20 , 27 3 3 4 5 6 21 →= 28 5 6 6 21 , 9 2 3 4 5 19 2 →= 5 6 6 19 2 , 9 2 3 4 5 19 18 →= 5 6 6 19 18 , 9 2 3 4 5 19 29 →= 5 6 6 19 29 , 8 3 3 4 5 19 2 →= 22 5 6 19 2 , 8 3 3 4 5 19 18 →= 22 5 6 19 18 , 8 3 3 4 5 19 29 →= 22 5 6 19 29 , 3 3 3 4 5 19 2 →= 4 5 6 19 2 , 3 3 3 4 5 19 18 →= 4 5 6 19 18 , 3 3 3 4 5 19 29 →= 4 5 6 19 29 , 13 3 3 4 5 19 2 →= 23 5 6 19 2 , 13 3 3 4 5 19 18 →= 23 5 6 19 18 , 13 3 3 4 5 19 29 →= 23 5 6 19 29 , 17 3 3 4 5 19 2 →= 24 5 6 19 2 , 17 3 3 4 5 19 18 →= 24 5 6 19 18 , 17 3 3 4 5 19 29 →= 24 5 6 19 29 , 2 3 3 4 5 19 2 →= 18 5 6 19 2 , 2 3 3 4 5 19 18 →= 18 5 6 19 18 , 2 3 3 4 5 19 29 →= 18 5 6 19 29 , 25 3 3 4 5 19 2 →= 26 5 6 19 2 , 25 3 3 4 5 19 18 →= 26 5 6 19 18 , 25 3 3 4 5 19 29 →= 26 5 6 19 29 , 27 3 3 4 5 19 2 →= 28 5 6 19 2 , 27 3 3 4 5 19 18 →= 28 5 6 19 18 , 27 3 3 4 5 19 29 →= 28 5 6 19 29 , 9 2 3 4 5 20 30 →= 5 6 6 20 30 , 8 3 3 4 5 20 30 →= 22 5 6 20 30 , 3 3 3 4 5 20 30 →= 4 5 6 20 30 , 13 3 3 4 5 20 30 →= 23 5 6 20 30 , 17 3 3 4 5 20 30 →= 24 5 6 20 30 , 2 3 3 4 5 20 30 →= 18 5 6 20 30 , 25 3 3 4 5 20 30 →= 26 5 6 20 30 , 27 3 3 4 5 20 30 →= 28 5 6 20 30 , 6 19 2 3 4 5 6 →= 19 2 3 4 9 2 3 4 5 , 22 9 2 3 4 5 6 →= 8 3 3 4 9 2 3 4 5 , 4 9 2 3 4 5 6 →= 3 3 3 4 9 2 3 4 5 , 23 9 2 3 4 5 6 →= 13 3 3 4 9 2 3 4 5 , 24 9 2 3 4 5 6 →= 17 3 3 4 9 2 3 4 5 , 18 9 2 3 4 5 6 →= 2 3 3 4 9 2 3 4 5 , 26 9 2 3 4 5 6 →= 25 3 3 4 9 2 3 4 5 , 28 9 2 3 4 5 6 →= 27 3 3 4 9 2 3 4 5 , 31 32 2 3 4 5 6 →= 33 25 3 4 9 2 3 4 5 , 6 19 2 3 4 9 2 →= 19 2 3 4 9 2 3 3 3 , 6 19 2 3 4 9 18 →= 19 2 3 4 9 2 3 3 4 , 22 9 2 3 4 9 2 →= 8 3 3 4 9 2 3 3 3 , 22 9 2 3 4 9 18 →= 8 3 3 4 9 2 3 3 4 , 4 9 2 3 4 9 2 →= 3 3 3 4 9 2 3 3 3 , 4 9 2 3 4 9 18 →= 3 3 3 4 9 2 3 3 4 , 23 9 2 3 4 9 2 →= 13 3 3 4 9 2 3 3 3 , 23 9 2 3 4 9 18 →= 13 3 3 4 9 2 3 3 4 , 24 9 2 3 4 9 2 →= 17 3 3 4 9 2 3 3 3 , 24 9 2 3 4 9 18 →= 17 3 3 4 9 2 3 3 4 , 18 9 2 3 4 9 2 →= 2 3 3 4 9 2 3 3 3 , 18 9 2 3 4 9 18 →= 2 3 3 4 9 2 3 3 4 , 26 9 2 3 4 9 2 →= 25 3 3 4 9 2 3 3 3 , 26 9 2 3 4 9 18 →= 25 3 3 4 9 2 3 3 4 , 28 9 2 3 4 9 2 →= 27 3 3 4 9 2 3 3 3 , 28 9 2 3 4 9 18 →= 27 3 3 4 9 2 3 3 4 , 31 32 2 3 4 9 2 →= 33 25 3 4 9 2 3 3 3 , 31 32 2 3 4 9 18 →= 33 25 3 4 9 2 3 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 23 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 24 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 25 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 26 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 27 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 28 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 29 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 30 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 31 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 32 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 33 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 9 ↦ 0, 2 ↦ 1, 3 ↦ 2, 4 ↦ 3, 5 ↦ 4, 6 ↦ 5, 19 ↦ 6, 20 ↦ 7, 21 ↦ 8, 8 ↦ 9, 22 ↦ 10, 13 ↦ 11, 23 ↦ 12, 17 ↦ 13, 24 ↦ 14, 18 ↦ 15, 25 ↦ 16, 26 ↦ 17, 27 ↦ 18, 28 ↦ 19, 29 ↦ 20, 30 ↦ 21 }, it remains to prove termination of the 88-rule system { 0 1 2 3 4 5 5 →= 4 5 5 5 5 , 0 1 2 3 4 5 6 →= 4 5 5 5 6 , 0 1 2 3 4 5 7 →= 4 5 5 5 7 , 0 1 2 3 4 5 8 →= 4 5 5 5 8 , 9 2 2 3 4 5 5 →= 10 4 5 5 5 , 9 2 2 3 4 5 6 →= 10 4 5 5 6 , 9 2 2 3 4 5 7 →= 10 4 5 5 7 , 9 2 2 3 4 5 8 →= 10 4 5 5 8 , 2 2 2 3 4 5 5 →= 3 4 5 5 5 , 2 2 2 3 4 5 6 →= 3 4 5 5 6 , 2 2 2 3 4 5 7 →= 3 4 5 5 7 , 2 2 2 3 4 5 8 →= 3 4 5 5 8 , 11 2 2 3 4 5 5 →= 12 4 5 5 5 , 11 2 2 3 4 5 6 →= 12 4 5 5 6 , 11 2 2 3 4 5 7 →= 12 4 5 5 7 , 11 2 2 3 4 5 8 →= 12 4 5 5 8 , 13 2 2 3 4 5 5 →= 14 4 5 5 5 , 13 2 2 3 4 5 6 →= 14 4 5 5 6 , 13 2 2 3 4 5 7 →= 14 4 5 5 7 , 13 2 2 3 4 5 8 →= 14 4 5 5 8 , 1 2 2 3 4 5 5 →= 15 4 5 5 5 , 1 2 2 3 4 5 6 →= 15 4 5 5 6 , 1 2 2 3 4 5 7 →= 15 4 5 5 7 , 1 2 2 3 4 5 8 →= 15 4 5 5 8 , 16 2 2 3 4 5 5 →= 17 4 5 5 5 , 16 2 2 3 4 5 6 →= 17 4 5 5 6 , 16 2 2 3 4 5 7 →= 17 4 5 5 7 , 16 2 2 3 4 5 8 →= 17 4 5 5 8 , 18 2 2 3 4 5 5 →= 19 4 5 5 5 , 18 2 2 3 4 5 6 →= 19 4 5 5 6 , 18 2 2 3 4 5 7 →= 19 4 5 5 7 , 18 2 2 3 4 5 8 →= 19 4 5 5 8 , 0 1 2 3 4 6 1 →= 4 5 5 6 1 , 0 1 2 3 4 6 15 →= 4 5 5 6 15 , 0 1 2 3 4 6 20 →= 4 5 5 6 20 , 9 2 2 3 4 6 1 →= 10 4 5 6 1 , 9 2 2 3 4 6 15 →= 10 4 5 6 15 , 9 2 2 3 4 6 20 →= 10 4 5 6 20 , 2 2 2 3 4 6 1 →= 3 4 5 6 1 , 2 2 2 3 4 6 15 →= 3 4 5 6 15 , 2 2 2 3 4 6 20 →= 3 4 5 6 20 , 11 2 2 3 4 6 1 →= 12 4 5 6 1 , 11 2 2 3 4 6 15 →= 12 4 5 6 15 , 11 2 2 3 4 6 20 →= 12 4 5 6 20 , 13 2 2 3 4 6 1 →= 14 4 5 6 1 , 13 2 2 3 4 6 15 →= 14 4 5 6 15 , 13 2 2 3 4 6 20 →= 14 4 5 6 20 , 1 2 2 3 4 6 1 →= 15 4 5 6 1 , 1 2 2 3 4 6 15 →= 15 4 5 6 15 , 1 2 2 3 4 6 20 →= 15 4 5 6 20 , 16 2 2 3 4 6 1 →= 17 4 5 6 1 , 16 2 2 3 4 6 15 →= 17 4 5 6 15 , 16 2 2 3 4 6 20 →= 17 4 5 6 20 , 18 2 2 3 4 6 1 →= 19 4 5 6 1 , 18 2 2 3 4 6 15 →= 19 4 5 6 15 , 18 2 2 3 4 6 20 →= 19 4 5 6 20 , 0 1 2 3 4 7 21 →= 4 5 5 7 21 , 9 2 2 3 4 7 21 →= 10 4 5 7 21 , 2 2 2 3 4 7 21 →= 3 4 5 7 21 , 11 2 2 3 4 7 21 →= 12 4 5 7 21 , 13 2 2 3 4 7 21 →= 14 4 5 7 21 , 1 2 2 3 4 7 21 →= 15 4 5 7 21 , 16 2 2 3 4 7 21 →= 17 4 5 7 21 , 18 2 2 3 4 7 21 →= 19 4 5 7 21 , 5 6 1 2 3 4 5 →= 6 1 2 3 0 1 2 3 4 , 10 0 1 2 3 4 5 →= 9 2 2 3 0 1 2 3 4 , 3 0 1 2 3 4 5 →= 2 2 2 3 0 1 2 3 4 , 12 0 1 2 3 4 5 →= 11 2 2 3 0 1 2 3 4 , 14 0 1 2 3 4 5 →= 13 2 2 3 0 1 2 3 4 , 15 0 1 2 3 4 5 →= 1 2 2 3 0 1 2 3 4 , 17 0 1 2 3 4 5 →= 16 2 2 3 0 1 2 3 4 , 19 0 1 2 3 4 5 →= 18 2 2 3 0 1 2 3 4 , 5 6 1 2 3 0 1 →= 6 1 2 3 0 1 2 2 2 , 5 6 1 2 3 0 15 →= 6 1 2 3 0 1 2 2 3 , 10 0 1 2 3 0 1 →= 9 2 2 3 0 1 2 2 2 , 10 0 1 2 3 0 15 →= 9 2 2 3 0 1 2 2 3 , 3 0 1 2 3 0 1 →= 2 2 2 3 0 1 2 2 2 , 3 0 1 2 3 0 15 →= 2 2 2 3 0 1 2 2 3 , 12 0 1 2 3 0 1 →= 11 2 2 3 0 1 2 2 2 , 12 0 1 2 3 0 15 →= 11 2 2 3 0 1 2 2 3 , 14 0 1 2 3 0 1 →= 13 2 2 3 0 1 2 2 2 , 14 0 1 2 3 0 15 →= 13 2 2 3 0 1 2 2 3 , 15 0 1 2 3 0 1 →= 1 2 2 3 0 1 2 2 2 , 15 0 1 2 3 0 15 →= 1 2 2 3 0 1 2 2 3 , 17 0 1 2 3 0 1 →= 16 2 2 3 0 1 2 2 2 , 17 0 1 2 3 0 15 →= 16 2 2 3 0 1 2 2 3 , 19 0 1 2 3 0 1 →= 18 2 2 3 0 1 2 2 2 , 19 0 1 2 3 0 15 →= 18 2 2 3 0 1 2 2 3 } The system is trivially terminating.