/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 { b ↦ 0, a ↦ 1 }, it remains to prove termination of the 2-rule system { 0 0 0 0 ⟶ 1 1 0 0 , 1 0 1 1 ⟶ 0 1 1 0 } Applying sparse tiling TRFC(2) after reversal [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { 0 ↦ 0, 3 ↦ 1, 2 ↦ 2, 1 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6 }, it remains to prove termination of the 15-rule system { 0 0 0 0 0 ⟶ 1 2 3 0 0 , 3 0 0 0 0 ⟶ 2 2 3 0 0 , 4 0 0 0 0 ⟶ 5 2 3 0 0 , 0 0 0 0 1 ⟶ 1 2 3 0 1 , 3 0 0 0 1 ⟶ 2 2 3 0 1 , 4 0 0 0 1 ⟶ 5 2 3 0 1 , 0 0 0 0 6 ⟶ 1 2 3 0 6 , 3 0 0 0 6 ⟶ 2 2 3 0 6 , 4 0 0 0 6 ⟶ 5 2 3 0 6 , 1 3 1 2 3 ⟶ 0 1 2 3 0 , 2 3 1 2 3 ⟶ 3 1 2 3 0 , 5 3 1 2 3 ⟶ 4 1 2 3 0 , 1 3 1 2 2 ⟶ 0 1 2 3 1 , 2 3 1 2 2 ⟶ 3 1 2 3 1 , 5 3 1 2 2 ⟶ 4 1 2 3 1 } 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 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 ⎟ ⎝ ⎠ 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 ⎟ ⎝ ⎠ 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 0 0 0 0 ⟶ 1 2 3 0 0 , 3 0 0 0 0 ⟶ 2 2 3 0 0 , 4 0 0 0 0 ⟶ 5 2 3 0 0 , 0 0 0 0 1 ⟶ 1 2 3 0 1 , 3 0 0 0 1 ⟶ 2 2 3 0 1 , 4 0 0 0 1 ⟶ 5 2 3 0 1 , 0 0 0 0 6 ⟶ 1 2 3 0 6 , 3 0 0 0 6 ⟶ 2 2 3 0 6 , 1 3 1 2 3 ⟶ 0 1 2 3 0 , 2 3 1 2 3 ⟶ 3 1 2 3 0 , 5 3 1 2 3 ⟶ 4 1 2 3 0 , 1 3 1 2 2 ⟶ 0 1 2 3 1 , 2 3 1 2 2 ⟶ 3 1 2 3 1 , 5 3 1 2 2 ⟶ 4 1 2 3 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, (3,↓) ↦ 4, (2,↑) ↦ 5, (3,↑) ↦ 6, (4,↑) ↦ 7, (5,↑) ↦ 8, (1,↓) ↦ 9, (6,↓) ↦ 10, (4,↓) ↦ 11, (5,↓) ↦ 12 }, it remains to prove termination of the 68-rule system { 0 1 1 1 1 ⟶ 2 3 4 1 1 , 0 1 1 1 1 ⟶ 5 4 1 1 , 0 1 1 1 1 ⟶ 6 1 1 , 6 1 1 1 1 ⟶ 5 3 4 1 1 , 6 1 1 1 1 ⟶ 5 4 1 1 , 6 1 1 1 1 ⟶ 6 1 1 , 7 1 1 1 1 ⟶ 8 3 4 1 1 , 7 1 1 1 1 ⟶ 5 4 1 1 , 7 1 1 1 1 ⟶ 6 1 1 , 0 1 1 1 9 ⟶ 2 3 4 1 9 , 0 1 1 1 9 ⟶ 5 4 1 9 , 0 1 1 1 9 ⟶ 6 1 9 , 6 1 1 1 9 ⟶ 5 3 4 1 9 , 6 1 1 1 9 ⟶ 5 4 1 9 , 6 1 1 1 9 ⟶ 6 1 9 , 7 1 1 1 9 ⟶ 8 3 4 1 9 , 7 1 1 1 9 ⟶ 5 4 1 9 , 7 1 1 1 9 ⟶ 6 1 9 , 0 1 1 1 10 ⟶ 2 3 4 1 10 , 0 1 1 1 10 ⟶ 5 4 1 10 , 0 1 1 1 10 ⟶ 6 1 10 , 6 1 1 1 10 ⟶ 5 3 4 1 10 , 6 1 1 1 10 ⟶ 5 4 1 10 , 6 1 1 1 10 ⟶ 6 1 10 , 2 4 9 3 4 ⟶ 0 9 3 4 1 , 2 4 9 3 4 ⟶ 2 3 4 1 , 2 4 9 3 4 ⟶ 5 4 1 , 2 4 9 3 4 ⟶ 6 1 , 2 4 9 3 4 ⟶ 0 , 5 4 9 3 4 ⟶ 6 9 3 4 1 , 5 4 9 3 4 ⟶ 2 3 4 1 , 5 4 9 3 4 ⟶ 5 4 1 , 5 4 9 3 4 ⟶ 6 1 , 5 4 9 3 4 ⟶ 0 , 8 4 9 3 4 ⟶ 7 9 3 4 1 , 8 4 9 3 4 ⟶ 2 3 4 1 , 8 4 9 3 4 ⟶ 5 4 1 , 8 4 9 3 4 ⟶ 6 1 , 8 4 9 3 4 ⟶ 0 , 2 4 9 3 3 ⟶ 0 9 3 4 9 , 2 4 9 3 3 ⟶ 2 3 4 9 , 2 4 9 3 3 ⟶ 5 4 9 , 2 4 9 3 3 ⟶ 6 9 , 2 4 9 3 3 ⟶ 2 , 5 4 9 3 3 ⟶ 6 9 3 4 9 , 5 4 9 3 3 ⟶ 2 3 4 9 , 5 4 9 3 3 ⟶ 5 4 9 , 5 4 9 3 3 ⟶ 6 9 , 5 4 9 3 3 ⟶ 2 , 8 4 9 3 3 ⟶ 7 9 3 4 9 , 8 4 9 3 3 ⟶ 2 3 4 9 , 8 4 9 3 3 ⟶ 5 4 9 , 8 4 9 3 3 ⟶ 6 9 , 8 4 9 3 3 ⟶ 2 , 1 1 1 1 1 →= 9 3 4 1 1 , 4 1 1 1 1 →= 3 3 4 1 1 , 11 1 1 1 1 →= 12 3 4 1 1 , 1 1 1 1 9 →= 9 3 4 1 9 , 4 1 1 1 9 →= 3 3 4 1 9 , 11 1 1 1 9 →= 12 3 4 1 9 , 1 1 1 1 10 →= 9 3 4 1 10 , 4 1 1 1 10 →= 3 3 4 1 10 , 9 4 9 3 4 →= 1 9 3 4 1 , 3 4 9 3 4 →= 4 9 3 4 1 , 12 4 9 3 4 →= 11 9 3 4 1 , 9 4 9 3 3 →= 1 9 3 4 9 , 3 4 9 3 3 →= 4 9 3 4 9 , 12 4 9 3 3 →= 11 9 3 4 9 } 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 6 ↦ 5, 5 ↦ 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 1 1 ⟶ 2 3 4 1 1 , 5 1 1 1 1 ⟶ 6 3 4 1 1 , 7 1 1 1 1 ⟶ 8 3 4 1 1 , 0 1 1 1 9 ⟶ 2 3 4 1 9 , 5 1 1 1 9 ⟶ 6 3 4 1 9 , 7 1 1 1 9 ⟶ 8 3 4 1 9 , 0 1 1 1 10 ⟶ 2 3 4 1 10 , 5 1 1 1 10 ⟶ 6 3 4 1 10 , 2 4 9 3 4 ⟶ 0 9 3 4 1 , 6 4 9 3 4 ⟶ 5 9 3 4 1 , 8 4 9 3 4 ⟶ 7 9 3 4 1 , 2 4 9 3 3 ⟶ 0 9 3 4 9 , 6 4 9 3 3 ⟶ 5 9 3 4 9 , 8 4 9 3 3 ⟶ 7 9 3 4 9 , 1 1 1 1 1 →= 9 3 4 1 1 , 4 1 1 1 1 →= 3 3 4 1 1 , 11 1 1 1 1 →= 12 3 4 1 1 , 1 1 1 1 9 →= 9 3 4 1 9 , 4 1 1 1 9 →= 3 3 4 1 9 , 11 1 1 1 9 →= 12 3 4 1 9 , 1 1 1 1 10 →= 9 3 4 1 10 , 4 1 1 1 10 →= 3 3 4 1 10 , 9 4 9 3 4 →= 1 9 3 4 1 , 3 4 9 3 4 →= 4 9 3 4 1 , 12 4 9 3 4 →= 11 9 3 4 1 , 9 4 9 3 3 →= 1 9 3 4 9 , 3 4 9 3 3 →= 4 9 3 4 9 , 12 4 9 3 3 →= 11 9 3 4 9 } 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 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 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 1 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 ⎟ ⎝ ⎠ 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 1 1 ⟶ 2 3 4 1 1 , 5 1 1 1 1 ⟶ 6 3 4 1 1 , 7 1 1 1 1 ⟶ 8 3 4 1 1 , 0 1 1 1 9 ⟶ 2 3 4 1 9 , 5 1 1 1 9 ⟶ 6 3 4 1 9 , 7 1 1 1 9 ⟶ 8 3 4 1 9 , 5 1 1 1 10 ⟶ 6 3 4 1 10 , 2 4 9 3 4 ⟶ 0 9 3 4 1 , 6 4 9 3 4 ⟶ 5 9 3 4 1 , 8 4 9 3 4 ⟶ 7 9 3 4 1 , 2 4 9 3 3 ⟶ 0 9 3 4 9 , 6 4 9 3 3 ⟶ 5 9 3 4 9 , 8 4 9 3 3 ⟶ 7 9 3 4 9 , 1 1 1 1 1 →= 9 3 4 1 1 , 4 1 1 1 1 →= 3 3 4 1 1 , 11 1 1 1 1 →= 12 3 4 1 1 , 1 1 1 1 9 →= 9 3 4 1 9 , 4 1 1 1 9 →= 3 3 4 1 9 , 11 1 1 1 9 →= 12 3 4 1 9 , 1 1 1 1 10 →= 9 3 4 1 10 , 4 1 1 1 10 →= 3 3 4 1 10 , 9 4 9 3 4 →= 1 9 3 4 1 , 3 4 9 3 4 →= 4 9 3 4 1 , 12 4 9 3 4 →= 11 9 3 4 1 , 9 4 9 3 3 →= 1 9 3 4 9 , 3 4 9 3 3 →= 4 9 3 4 9 , 12 4 9 3 3 →= 11 9 3 4 9 } 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 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 ⎟ ⎝ ⎠ 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 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 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 1 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 ⎟ ⎝ ⎠ 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 1 1 ⟶ 2 3 4 1 1 , 5 1 1 1 1 ⟶ 6 3 4 1 1 , 7 1 1 1 1 ⟶ 8 3 4 1 1 , 0 1 1 1 9 ⟶ 2 3 4 1 9 , 5 1 1 1 9 ⟶ 6 3 4 1 9 , 7 1 1 1 9 ⟶ 8 3 4 1 9 , 2 4 9 3 4 ⟶ 0 9 3 4 1 , 6 4 9 3 4 ⟶ 5 9 3 4 1 , 8 4 9 3 4 ⟶ 7 9 3 4 1 , 2 4 9 3 3 ⟶ 0 9 3 4 9 , 6 4 9 3 3 ⟶ 5 9 3 4 9 , 8 4 9 3 3 ⟶ 7 9 3 4 9 , 1 1 1 1 1 →= 9 3 4 1 1 , 4 1 1 1 1 →= 3 3 4 1 1 , 10 1 1 1 1 →= 11 3 4 1 1 , 1 1 1 1 9 →= 9 3 4 1 9 , 4 1 1 1 9 →= 3 3 4 1 9 , 10 1 1 1 9 →= 11 3 4 1 9 , 1 1 1 1 12 →= 9 3 4 1 12 , 4 1 1 1 12 →= 3 3 4 1 12 , 9 4 9 3 4 →= 1 9 3 4 1 , 3 4 9 3 4 →= 4 9 3 4 1 , 11 4 9 3 4 →= 10 9 3 4 1 , 9 4 9 3 3 →= 1 9 3 4 9 , 3 4 9 3 3 →= 4 9 3 4 9 , 11 4 9 3 3 →= 10 9 3 4 9 } Applying sparse tiling TROC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (13,0) ↦ 0, (0,1) ↦ 1, (1,1) ↦ 2, (13,2) ↦ 3, (2,3) ↦ 4, (3,4) ↦ 5, (4,1) ↦ 6, (1,9) ↦ 7, (1,12) ↦ 8, (1,14) ↦ 9, (13,5) ↦ 10, (5,1) ↦ 11, (13,6) ↦ 12, (6,3) ↦ 13, (13,7) ↦ 14, (7,1) ↦ 15, (13,8) ↦ 16, (8,3) ↦ 17, (9,3) ↦ 18, (9,4) ↦ 19, (9,14) ↦ 20, (2,4) ↦ 21, (4,9) ↦ 22, (0,9) ↦ 23, (6,4) ↦ 24, (5,9) ↦ 25, (8,4) ↦ 26, (7,9) ↦ 27, (3,3) ↦ 28, (10,1) ↦ 29, (10,9) ↦ 30, (13,1) ↦ 31, (13,9) ↦ 32, (11,4) ↦ 33, (11,3) ↦ 34, (13,4) ↦ 35, (13,3) ↦ 36, (13,10) ↦ 37, (13,11) ↦ 38, (12,14) ↦ 39 }, it remains to prove termination of the 212-rule system { 0 1 2 2 2 2 ⟶ 3 4 5 6 2 2 , 0 1 2 2 2 7 ⟶ 3 4 5 6 2 7 , 0 1 2 2 2 8 ⟶ 3 4 5 6 2 8 , 0 1 2 2 2 9 ⟶ 3 4 5 6 2 9 , 10 11 2 2 2 2 ⟶ 12 13 5 6 2 2 , 10 11 2 2 2 7 ⟶ 12 13 5 6 2 7 , 10 11 2 2 2 8 ⟶ 12 13 5 6 2 8 , 10 11 2 2 2 9 ⟶ 12 13 5 6 2 9 , 14 15 2 2 2 2 ⟶ 16 17 5 6 2 2 , 14 15 2 2 2 7 ⟶ 16 17 5 6 2 7 , 14 15 2 2 2 8 ⟶ 16 17 5 6 2 8 , 14 15 2 2 2 9 ⟶ 16 17 5 6 2 9 , 0 1 2 2 7 18 ⟶ 3 4 5 6 7 18 , 0 1 2 2 7 19 ⟶ 3 4 5 6 7 19 , 0 1 2 2 7 20 ⟶ 3 4 5 6 7 20 , 10 11 2 2 7 18 ⟶ 12 13 5 6 7 18 , 10 11 2 2 7 19 ⟶ 12 13 5 6 7 19 , 10 11 2 2 7 20 ⟶ 12 13 5 6 7 20 , 14 15 2 2 7 18 ⟶ 16 17 5 6 7 18 , 14 15 2 2 7 19 ⟶ 16 17 5 6 7 19 , 14 15 2 2 7 20 ⟶ 16 17 5 6 7 20 , 3 21 22 18 5 6 ⟶ 0 23 18 5 6 2 , 3 21 22 18 5 22 ⟶ 0 23 18 5 6 7 , 12 24 22 18 5 6 ⟶ 10 25 18 5 6 2 , 12 24 22 18 5 22 ⟶ 10 25 18 5 6 7 , 16 26 22 18 5 6 ⟶ 14 27 18 5 6 2 , 16 26 22 18 5 22 ⟶ 14 27 18 5 6 7 , 3 21 22 18 28 28 ⟶ 0 23 18 5 22 18 , 3 21 22 18 28 5 ⟶ 0 23 18 5 22 19 , 12 24 22 18 28 28 ⟶ 10 25 18 5 22 18 , 12 24 22 18 28 5 ⟶ 10 25 18 5 22 19 , 16 26 22 18 28 28 ⟶ 14 27 18 5 22 18 , 16 26 22 18 28 5 ⟶ 14 27 18 5 22 19 , 1 2 2 2 2 2 →= 23 18 5 6 2 2 , 1 2 2 2 2 7 →= 23 18 5 6 2 7 , 1 2 2 2 2 8 →= 23 18 5 6 2 8 , 1 2 2 2 2 9 →= 23 18 5 6 2 9 , 2 2 2 2 2 2 →= 7 18 5 6 2 2 , 2 2 2 2 2 7 →= 7 18 5 6 2 7 , 2 2 2 2 2 8 →= 7 18 5 6 2 8 , 2 2 2 2 2 9 →= 7 18 5 6 2 9 , 6 2 2 2 2 2 →= 22 18 5 6 2 2 , 6 2 2 2 2 7 →= 22 18 5 6 2 7 , 6 2 2 2 2 8 →= 22 18 5 6 2 8 , 6 2 2 2 2 9 →= 22 18 5 6 2 9 , 11 2 2 2 2 2 →= 25 18 5 6 2 2 , 11 2 2 2 2 7 →= 25 18 5 6 2 7 , 11 2 2 2 2 8 →= 25 18 5 6 2 8 , 11 2 2 2 2 9 →= 25 18 5 6 2 9 , 15 2 2 2 2 2 →= 27 18 5 6 2 2 , 15 2 2 2 2 7 →= 27 18 5 6 2 7 , 15 2 2 2 2 8 →= 27 18 5 6 2 8 , 15 2 2 2 2 9 →= 27 18 5 6 2 9 , 29 2 2 2 2 2 →= 30 18 5 6 2 2 , 29 2 2 2 2 7 →= 30 18 5 6 2 7 , 29 2 2 2 2 8 →= 30 18 5 6 2 8 , 29 2 2 2 2 9 →= 30 18 5 6 2 9 , 31 2 2 2 2 2 →= 32 18 5 6 2 2 , 31 2 2 2 2 7 →= 32 18 5 6 2 7 , 31 2 2 2 2 8 →= 32 18 5 6 2 8 , 31 2 2 2 2 9 →= 32 18 5 6 2 9 , 21 6 2 2 2 2 →= 4 28 5 6 2 2 , 21 6 2 2 2 7 →= 4 28 5 6 2 7 , 21 6 2 2 2 8 →= 4 28 5 6 2 8 , 21 6 2 2 2 9 →= 4 28 5 6 2 9 , 5 6 2 2 2 2 →= 28 28 5 6 2 2 , 5 6 2 2 2 7 →= 28 28 5 6 2 7 , 5 6 2 2 2 8 →= 28 28 5 6 2 8 , 5 6 2 2 2 9 →= 28 28 5 6 2 9 , 24 6 2 2 2 2 →= 13 28 5 6 2 2 , 24 6 2 2 2 7 →= 13 28 5 6 2 7 , 24 6 2 2 2 8 →= 13 28 5 6 2 8 , 24 6 2 2 2 9 →= 13 28 5 6 2 9 , 26 6 2 2 2 2 →= 17 28 5 6 2 2 , 26 6 2 2 2 7 →= 17 28 5 6 2 7 , 26 6 2 2 2 8 →= 17 28 5 6 2 8 , 26 6 2 2 2 9 →= 17 28 5 6 2 9 , 19 6 2 2 2 2 →= 18 28 5 6 2 2 , 19 6 2 2 2 7 →= 18 28 5 6 2 7 , 19 6 2 2 2 8 →= 18 28 5 6 2 8 , 19 6 2 2 2 9 →= 18 28 5 6 2 9 , 33 6 2 2 2 2 →= 34 28 5 6 2 2 , 33 6 2 2 2 7 →= 34 28 5 6 2 7 , 33 6 2 2 2 8 →= 34 28 5 6 2 8 , 33 6 2 2 2 9 →= 34 28 5 6 2 9 , 35 6 2 2 2 2 →= 36 28 5 6 2 2 , 35 6 2 2 2 7 →= 36 28 5 6 2 7 , 35 6 2 2 2 8 →= 36 28 5 6 2 8 , 35 6 2 2 2 9 →= 36 28 5 6 2 9 , 37 29 2 2 2 2 →= 38 34 5 6 2 2 , 37 29 2 2 2 7 →= 38 34 5 6 2 7 , 37 29 2 2 2 8 →= 38 34 5 6 2 8 , 37 29 2 2 2 9 →= 38 34 5 6 2 9 , 1 2 2 2 7 18 →= 23 18 5 6 7 18 , 1 2 2 2 7 19 →= 23 18 5 6 7 19 , 1 2 2 2 7 20 →= 23 18 5 6 7 20 , 2 2 2 2 7 18 →= 7 18 5 6 7 18 , 2 2 2 2 7 19 →= 7 18 5 6 7 19 , 2 2 2 2 7 20 →= 7 18 5 6 7 20 , 6 2 2 2 7 18 →= 22 18 5 6 7 18 , 6 2 2 2 7 19 →= 22 18 5 6 7 19 , 6 2 2 2 7 20 →= 22 18 5 6 7 20 , 11 2 2 2 7 18 →= 25 18 5 6 7 18 , 11 2 2 2 7 19 →= 25 18 5 6 7 19 , 11 2 2 2 7 20 →= 25 18 5 6 7 20 , 15 2 2 2 7 18 →= 27 18 5 6 7 18 , 15 2 2 2 7 19 →= 27 18 5 6 7 19 , 15 2 2 2 7 20 →= 27 18 5 6 7 20 , 29 2 2 2 7 18 →= 30 18 5 6 7 18 , 29 2 2 2 7 19 →= 30 18 5 6 7 19 , 29 2 2 2 7 20 →= 30 18 5 6 7 20 , 31 2 2 2 7 18 →= 32 18 5 6 7 18 , 31 2 2 2 7 19 →= 32 18 5 6 7 19 , 31 2 2 2 7 20 →= 32 18 5 6 7 20 , 21 6 2 2 7 18 →= 4 28 5 6 7 18 , 21 6 2 2 7 19 →= 4 28 5 6 7 19 , 21 6 2 2 7 20 →= 4 28 5 6 7 20 , 5 6 2 2 7 18 →= 28 28 5 6 7 18 , 5 6 2 2 7 19 →= 28 28 5 6 7 19 , 5 6 2 2 7 20 →= 28 28 5 6 7 20 , 24 6 2 2 7 18 →= 13 28 5 6 7 18 , 24 6 2 2 7 19 →= 13 28 5 6 7 19 , 24 6 2 2 7 20 →= 13 28 5 6 7 20 , 26 6 2 2 7 18 →= 17 28 5 6 7 18 , 26 6 2 2 7 19 →= 17 28 5 6 7 19 , 26 6 2 2 7 20 →= 17 28 5 6 7 20 , 19 6 2 2 7 18 →= 18 28 5 6 7 18 , 19 6 2 2 7 19 →= 18 28 5 6 7 19 , 19 6 2 2 7 20 →= 18 28 5 6 7 20 , 33 6 2 2 7 18 →= 34 28 5 6 7 18 , 33 6 2 2 7 19 →= 34 28 5 6 7 19 , 33 6 2 2 7 20 →= 34 28 5 6 7 20 , 35 6 2 2 7 18 →= 36 28 5 6 7 18 , 35 6 2 2 7 19 →= 36 28 5 6 7 19 , 35 6 2 2 7 20 →= 36 28 5 6 7 20 , 37 29 2 2 7 18 →= 38 34 5 6 7 18 , 37 29 2 2 7 19 →= 38 34 5 6 7 19 , 37 29 2 2 7 20 →= 38 34 5 6 7 20 , 1 2 2 2 8 39 →= 23 18 5 6 8 39 , 2 2 2 2 8 39 →= 7 18 5 6 8 39 , 6 2 2 2 8 39 →= 22 18 5 6 8 39 , 11 2 2 2 8 39 →= 25 18 5 6 8 39 , 15 2 2 2 8 39 →= 27 18 5 6 8 39 , 29 2 2 2 8 39 →= 30 18 5 6 8 39 , 31 2 2 2 8 39 →= 32 18 5 6 8 39 , 21 6 2 2 8 39 →= 4 28 5 6 8 39 , 5 6 2 2 8 39 →= 28 28 5 6 8 39 , 24 6 2 2 8 39 →= 13 28 5 6 8 39 , 26 6 2 2 8 39 →= 17 28 5 6 8 39 , 19 6 2 2 8 39 →= 18 28 5 6 8 39 , 33 6 2 2 8 39 →= 34 28 5 6 8 39 , 35 6 2 2 8 39 →= 36 28 5 6 8 39 , 23 19 22 18 5 6 →= 1 7 18 5 6 2 , 23 19 22 18 5 22 →= 1 7 18 5 6 7 , 7 19 22 18 5 6 →= 2 7 18 5 6 2 , 7 19 22 18 5 22 →= 2 7 18 5 6 7 , 22 19 22 18 5 6 →= 6 7 18 5 6 2 , 22 19 22 18 5 22 →= 6 7 18 5 6 7 , 25 19 22 18 5 6 →= 11 7 18 5 6 2 , 25 19 22 18 5 22 →= 11 7 18 5 6 7 , 27 19 22 18 5 6 →= 15 7 18 5 6 2 , 27 19 22 18 5 22 →= 15 7 18 5 6 7 , 30 19 22 18 5 6 →= 29 7 18 5 6 2 , 30 19 22 18 5 22 →= 29 7 18 5 6 7 , 32 19 22 18 5 6 →= 31 7 18 5 6 2 , 32 19 22 18 5 22 →= 31 7 18 5 6 7 , 4 5 22 18 5 6 →= 21 22 18 5 6 2 , 4 5 22 18 5 22 →= 21 22 18 5 6 7 , 28 5 22 18 5 6 →= 5 22 18 5 6 2 , 28 5 22 18 5 22 →= 5 22 18 5 6 7 , 13 5 22 18 5 6 →= 24 22 18 5 6 2 , 13 5 22 18 5 22 →= 24 22 18 5 6 7 , 17 5 22 18 5 6 →= 26 22 18 5 6 2 , 17 5 22 18 5 22 →= 26 22 18 5 6 7 , 18 5 22 18 5 6 →= 19 22 18 5 6 2 , 18 5 22 18 5 22 →= 19 22 18 5 6 7 , 34 5 22 18 5 6 →= 33 22 18 5 6 2 , 34 5 22 18 5 22 →= 33 22 18 5 6 7 , 36 5 22 18 5 6 →= 35 22 18 5 6 2 , 36 5 22 18 5 22 →= 35 22 18 5 6 7 , 38 33 22 18 5 6 →= 37 30 18 5 6 2 , 38 33 22 18 5 22 →= 37 30 18 5 6 7 , 23 19 22 18 28 28 →= 1 7 18 5 22 18 , 23 19 22 18 28 5 →= 1 7 18 5 22 19 , 7 19 22 18 28 28 →= 2 7 18 5 22 18 , 7 19 22 18 28 5 →= 2 7 18 5 22 19 , 22 19 22 18 28 28 →= 6 7 18 5 22 18 , 22 19 22 18 28 5 →= 6 7 18 5 22 19 , 25 19 22 18 28 28 →= 11 7 18 5 22 18 , 25 19 22 18 28 5 →= 11 7 18 5 22 19 , 27 19 22 18 28 28 →= 15 7 18 5 22 18 , 27 19 22 18 28 5 →= 15 7 18 5 22 19 , 30 19 22 18 28 28 →= 29 7 18 5 22 18 , 30 19 22 18 28 5 →= 29 7 18 5 22 19 , 32 19 22 18 28 28 →= 31 7 18 5 22 18 , 32 19 22 18 28 5 →= 31 7 18 5 22 19 , 4 5 22 18 28 28 →= 21 22 18 5 22 18 , 4 5 22 18 28 5 →= 21 22 18 5 22 19 , 28 5 22 18 28 28 →= 5 22 18 5 22 18 , 28 5 22 18 28 5 →= 5 22 18 5 22 19 , 13 5 22 18 28 28 →= 24 22 18 5 22 18 , 13 5 22 18 28 5 →= 24 22 18 5 22 19 , 17 5 22 18 28 28 →= 26 22 18 5 22 18 , 17 5 22 18 28 5 →= 26 22 18 5 22 19 , 18 5 22 18 28 28 →= 19 22 18 5 22 18 , 18 5 22 18 28 5 →= 19 22 18 5 22 19 , 34 5 22 18 28 28 →= 33 22 18 5 22 18 , 34 5 22 18 28 5 →= 33 22 18 5 22 19 , 36 5 22 18 28 28 →= 35 22 18 5 22 18 , 36 5 22 18 28 5 →= 35 22 18 5 22 19 , 38 33 22 18 28 28 →= 37 30 18 5 22 18 , 38 33 22 18 28 5 →= 37 30 18 5 22 19 } 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 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 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 0 ⎟ ⎜ 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 3 ↦ 0, 21 ↦ 1, 22 ↦ 2, 18 ↦ 3, 5 ↦ 4, 6 ↦ 5, 0 ↦ 6, 23 ↦ 7, 2 ↦ 8, 12 ↦ 9, 24 ↦ 10, 10 ↦ 11, 25 ↦ 12, 16 ↦ 13, 26 ↦ 14, 14 ↦ 15, 27 ↦ 16, 28 ↦ 17, 19 ↦ 18, 7 ↦ 19, 8 ↦ 20, 9 ↦ 21, 4 ↦ 22, 13 ↦ 23, 17 ↦ 24, 33 ↦ 25, 34 ↦ 26, 35 ↦ 27, 36 ↦ 28, 20 ↦ 29, 39 ↦ 30, 38 ↦ 31, 37 ↦ 32, 30 ↦ 33 }, it remains to prove termination of the 100-rule system { 0 1 2 3 4 5 ⟶ 6 7 3 4 5 8 , 9 10 2 3 4 5 ⟶ 11 12 3 4 5 8 , 13 14 2 3 4 5 ⟶ 15 16 3 4 5 8 , 0 1 2 3 17 17 ⟶ 6 7 3 4 2 3 , 0 1 2 3 17 4 ⟶ 6 7 3 4 2 18 , 9 10 2 3 17 17 ⟶ 11 12 3 4 2 3 , 9 10 2 3 17 4 ⟶ 11 12 3 4 2 18 , 13 14 2 3 17 17 ⟶ 15 16 3 4 2 3 , 13 14 2 3 17 4 ⟶ 15 16 3 4 2 18 , 5 8 8 8 8 8 →= 2 3 4 5 8 8 , 5 8 8 8 8 19 →= 2 3 4 5 8 19 , 5 8 8 8 8 20 →= 2 3 4 5 8 20 , 5 8 8 8 8 21 →= 2 3 4 5 8 21 , 1 5 8 8 8 8 →= 22 17 4 5 8 8 , 1 5 8 8 8 19 →= 22 17 4 5 8 19 , 1 5 8 8 8 20 →= 22 17 4 5 8 20 , 1 5 8 8 8 21 →= 22 17 4 5 8 21 , 4 5 8 8 8 8 →= 17 17 4 5 8 8 , 4 5 8 8 8 19 →= 17 17 4 5 8 19 , 4 5 8 8 8 20 →= 17 17 4 5 8 20 , 4 5 8 8 8 21 →= 17 17 4 5 8 21 , 10 5 8 8 8 8 →= 23 17 4 5 8 8 , 10 5 8 8 8 19 →= 23 17 4 5 8 19 , 10 5 8 8 8 20 →= 23 17 4 5 8 20 , 10 5 8 8 8 21 →= 23 17 4 5 8 21 , 14 5 8 8 8 8 →= 24 17 4 5 8 8 , 14 5 8 8 8 19 →= 24 17 4 5 8 19 , 14 5 8 8 8 20 →= 24 17 4 5 8 20 , 14 5 8 8 8 21 →= 24 17 4 5 8 21 , 18 5 8 8 8 8 →= 3 17 4 5 8 8 , 18 5 8 8 8 19 →= 3 17 4 5 8 19 , 18 5 8 8 8 20 →= 3 17 4 5 8 20 , 18 5 8 8 8 21 →= 3 17 4 5 8 21 , 25 5 8 8 8 8 →= 26 17 4 5 8 8 , 25 5 8 8 8 19 →= 26 17 4 5 8 19 , 25 5 8 8 8 20 →= 26 17 4 5 8 20 , 25 5 8 8 8 21 →= 26 17 4 5 8 21 , 27 5 8 8 8 8 →= 28 17 4 5 8 8 , 27 5 8 8 8 19 →= 28 17 4 5 8 19 , 27 5 8 8 8 20 →= 28 17 4 5 8 20 , 27 5 8 8 8 21 →= 28 17 4 5 8 21 , 5 8 8 8 19 3 →= 2 3 4 5 19 3 , 5 8 8 8 19 18 →= 2 3 4 5 19 18 , 5 8 8 8 19 29 →= 2 3 4 5 19 29 , 1 5 8 8 19 3 →= 22 17 4 5 19 3 , 1 5 8 8 19 18 →= 22 17 4 5 19 18 , 1 5 8 8 19 29 →= 22 17 4 5 19 29 , 4 5 8 8 19 3 →= 17 17 4 5 19 3 , 4 5 8 8 19 18 →= 17 17 4 5 19 18 , 4 5 8 8 19 29 →= 17 17 4 5 19 29 , 10 5 8 8 19 3 →= 23 17 4 5 19 3 , 10 5 8 8 19 18 →= 23 17 4 5 19 18 , 10 5 8 8 19 29 →= 23 17 4 5 19 29 , 14 5 8 8 19 3 →= 24 17 4 5 19 3 , 14 5 8 8 19 18 →= 24 17 4 5 19 18 , 14 5 8 8 19 29 →= 24 17 4 5 19 29 , 18 5 8 8 19 3 →= 3 17 4 5 19 3 , 18 5 8 8 19 18 →= 3 17 4 5 19 18 , 18 5 8 8 19 29 →= 3 17 4 5 19 29 , 25 5 8 8 19 3 →= 26 17 4 5 19 3 , 25 5 8 8 19 18 →= 26 17 4 5 19 18 , 25 5 8 8 19 29 →= 26 17 4 5 19 29 , 27 5 8 8 19 3 →= 28 17 4 5 19 3 , 27 5 8 8 19 18 →= 28 17 4 5 19 18 , 27 5 8 8 19 29 →= 28 17 4 5 19 29 , 5 8 8 8 20 30 →= 2 3 4 5 20 30 , 1 5 8 8 20 30 →= 22 17 4 5 20 30 , 4 5 8 8 20 30 →= 17 17 4 5 20 30 , 10 5 8 8 20 30 →= 23 17 4 5 20 30 , 14 5 8 8 20 30 →= 24 17 4 5 20 30 , 18 5 8 8 20 30 →= 3 17 4 5 20 30 , 25 5 8 8 20 30 →= 26 17 4 5 20 30 , 27 5 8 8 20 30 →= 28 17 4 5 20 30 , 19 18 2 3 4 5 →= 8 19 3 4 5 8 , 22 4 2 3 4 5 →= 1 2 3 4 5 8 , 17 4 2 3 4 5 →= 4 2 3 4 5 8 , 23 4 2 3 4 5 →= 10 2 3 4 5 8 , 24 4 2 3 4 5 →= 14 2 3 4 5 8 , 3 4 2 3 4 5 →= 18 2 3 4 5 8 , 26 4 2 3 4 5 →= 25 2 3 4 5 8 , 28 4 2 3 4 5 →= 27 2 3 4 5 8 , 31 25 2 3 4 5 →= 32 33 3 4 5 8 , 19 18 2 3 17 17 →= 8 19 3 4 2 3 , 19 18 2 3 17 4 →= 8 19 3 4 2 18 , 22 4 2 3 17 17 →= 1 2 3 4 2 3 , 22 4 2 3 17 4 →= 1 2 3 4 2 18 , 17 4 2 3 17 17 →= 4 2 3 4 2 3 , 17 4 2 3 17 4 →= 4 2 3 4 2 18 , 23 4 2 3 17 17 →= 10 2 3 4 2 3 , 23 4 2 3 17 4 →= 10 2 3 4 2 18 , 24 4 2 3 17 17 →= 14 2 3 4 2 3 , 24 4 2 3 17 4 →= 14 2 3 4 2 18 , 3 4 2 3 17 17 →= 18 2 3 4 2 3 , 3 4 2 3 17 4 →= 18 2 3 4 2 18 , 26 4 2 3 17 17 →= 25 2 3 4 2 3 , 26 4 2 3 17 4 →= 25 2 3 4 2 18 , 28 4 2 3 17 17 →= 27 2 3 4 2 3 , 28 4 2 3 17 4 →= 27 2 3 4 2 18 , 31 25 2 3 17 17 →= 32 33 3 4 2 3 , 31 25 2 3 17 4 →= 32 33 3 4 2 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 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 30 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 31 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 32 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 33 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 5 ↦ 0, 8 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 19 ↦ 5, 20 ↦ 6, 21 ↦ 7, 1 ↦ 8, 22 ↦ 9, 17 ↦ 10, 10 ↦ 11, 23 ↦ 12, 14 ↦ 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 1 1 1 1 →= 2 3 4 0 1 1 , 0 1 1 1 1 5 →= 2 3 4 0 1 5 , 0 1 1 1 1 6 →= 2 3 4 0 1 6 , 0 1 1 1 1 7 →= 2 3 4 0 1 7 , 8 0 1 1 1 1 →= 9 10 4 0 1 1 , 8 0 1 1 1 5 →= 9 10 4 0 1 5 , 8 0 1 1 1 6 →= 9 10 4 0 1 6 , 8 0 1 1 1 7 →= 9 10 4 0 1 7 , 4 0 1 1 1 1 →= 10 10 4 0 1 1 , 4 0 1 1 1 5 →= 10 10 4 0 1 5 , 4 0 1 1 1 6 →= 10 10 4 0 1 6 , 4 0 1 1 1 7 →= 10 10 4 0 1 7 , 11 0 1 1 1 1 →= 12 10 4 0 1 1 , 11 0 1 1 1 5 →= 12 10 4 0 1 5 , 11 0 1 1 1 6 →= 12 10 4 0 1 6 , 11 0 1 1 1 7 →= 12 10 4 0 1 7 , 13 0 1 1 1 1 →= 14 10 4 0 1 1 , 13 0 1 1 1 5 →= 14 10 4 0 1 5 , 13 0 1 1 1 6 →= 14 10 4 0 1 6 , 13 0 1 1 1 7 →= 14 10 4 0 1 7 , 15 0 1 1 1 1 →= 3 10 4 0 1 1 , 15 0 1 1 1 5 →= 3 10 4 0 1 5 , 15 0 1 1 1 6 →= 3 10 4 0 1 6 , 15 0 1 1 1 7 →= 3 10 4 0 1 7 , 16 0 1 1 1 1 →= 17 10 4 0 1 1 , 16 0 1 1 1 5 →= 17 10 4 0 1 5 , 16 0 1 1 1 6 →= 17 10 4 0 1 6 , 16 0 1 1 1 7 →= 17 10 4 0 1 7 , 18 0 1 1 1 1 →= 19 10 4 0 1 1 , 18 0 1 1 1 5 →= 19 10 4 0 1 5 , 18 0 1 1 1 6 →= 19 10 4 0 1 6 , 18 0 1 1 1 7 →= 19 10 4 0 1 7 , 0 1 1 1 5 3 →= 2 3 4 0 5 3 , 0 1 1 1 5 15 →= 2 3 4 0 5 15 , 0 1 1 1 5 20 →= 2 3 4 0 5 20 , 8 0 1 1 5 3 →= 9 10 4 0 5 3 , 8 0 1 1 5 15 →= 9 10 4 0 5 15 , 8 0 1 1 5 20 →= 9 10 4 0 5 20 , 4 0 1 1 5 3 →= 10 10 4 0 5 3 , 4 0 1 1 5 15 →= 10 10 4 0 5 15 , 4 0 1 1 5 20 →= 10 10 4 0 5 20 , 11 0 1 1 5 3 →= 12 10 4 0 5 3 , 11 0 1 1 5 15 →= 12 10 4 0 5 15 , 11 0 1 1 5 20 →= 12 10 4 0 5 20 , 13 0 1 1 5 3 →= 14 10 4 0 5 3 , 13 0 1 1 5 15 →= 14 10 4 0 5 15 , 13 0 1 1 5 20 →= 14 10 4 0 5 20 , 15 0 1 1 5 3 →= 3 10 4 0 5 3 , 15 0 1 1 5 15 →= 3 10 4 0 5 15 , 15 0 1 1 5 20 →= 3 10 4 0 5 20 , 16 0 1 1 5 3 →= 17 10 4 0 5 3 , 16 0 1 1 5 15 →= 17 10 4 0 5 15 , 16 0 1 1 5 20 →= 17 10 4 0 5 20 , 18 0 1 1 5 3 →= 19 10 4 0 5 3 , 18 0 1 1 5 15 →= 19 10 4 0 5 15 , 18 0 1 1 5 20 →= 19 10 4 0 5 20 , 0 1 1 1 6 21 →= 2 3 4 0 6 21 , 8 0 1 1 6 21 →= 9 10 4 0 6 21 , 4 0 1 1 6 21 →= 10 10 4 0 6 21 , 11 0 1 1 6 21 →= 12 10 4 0 6 21 , 13 0 1 1 6 21 →= 14 10 4 0 6 21 , 15 0 1 1 6 21 →= 3 10 4 0 6 21 , 16 0 1 1 6 21 →= 17 10 4 0 6 21 , 18 0 1 1 6 21 →= 19 10 4 0 6 21 , 5 15 2 3 4 0 →= 1 5 3 4 0 1 , 9 4 2 3 4 0 →= 8 2 3 4 0 1 , 10 4 2 3 4 0 →= 4 2 3 4 0 1 , 12 4 2 3 4 0 →= 11 2 3 4 0 1 , 14 4 2 3 4 0 →= 13 2 3 4 0 1 , 3 4 2 3 4 0 →= 15 2 3 4 0 1 , 17 4 2 3 4 0 →= 16 2 3 4 0 1 , 19 4 2 3 4 0 →= 18 2 3 4 0 1 , 5 15 2 3 10 10 →= 1 5 3 4 2 3 , 5 15 2 3 10 4 →= 1 5 3 4 2 15 , 9 4 2 3 10 10 →= 8 2 3 4 2 3 , 9 4 2 3 10 4 →= 8 2 3 4 2 15 , 10 4 2 3 10 10 →= 4 2 3 4 2 3 , 10 4 2 3 10 4 →= 4 2 3 4 2 15 , 12 4 2 3 10 10 →= 11 2 3 4 2 3 , 12 4 2 3 10 4 →= 11 2 3 4 2 15 , 14 4 2 3 10 10 →= 13 2 3 4 2 3 , 14 4 2 3 10 4 →= 13 2 3 4 2 15 , 3 4 2 3 10 10 →= 15 2 3 4 2 3 , 3 4 2 3 10 4 →= 15 2 3 4 2 15 , 17 4 2 3 10 10 →= 16 2 3 4 2 3 , 17 4 2 3 10 4 →= 16 2 3 4 2 15 , 19 4 2 3 10 10 →= 18 2 3 4 2 3 , 19 4 2 3 10 4 →= 18 2 3 4 2 15 } The system is trivially terminating.