/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 4-rule system { 0 1 1 1 ⟶ 0 0 1 0 , 1 0 1 0 ⟶ 0 1 0 0 , 1 1 0 1 ⟶ 1 0 1 0 , 0 0 0 0 ⟶ 1 0 0 0 } Applying sparse tiling TRFC(2) after reversal [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { 2 ↦ 0, 1 ↦ 1, 0 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6 }, it remains to prove termination of the 30-rule system { 0 1 2 2 2 ⟶ 0 3 1 0 1 , 3 1 2 2 2 ⟶ 3 3 1 0 1 , 4 1 2 2 2 ⟶ 4 3 1 0 1 , 0 1 2 2 0 ⟶ 0 3 1 0 3 , 3 1 2 2 0 ⟶ 3 3 1 0 3 , 4 1 2 2 0 ⟶ 4 3 1 0 3 , 2 0 1 0 1 ⟶ 0 1 0 3 1 , 1 0 1 0 1 ⟶ 3 1 0 3 1 , 5 0 1 0 1 ⟶ 4 1 0 3 1 , 2 0 1 0 3 ⟶ 0 1 0 3 3 , 1 0 1 0 3 ⟶ 3 1 0 3 3 , 5 0 1 0 3 ⟶ 4 1 0 3 3 , 2 0 1 0 6 ⟶ 0 1 0 3 6 , 1 0 1 0 6 ⟶ 3 1 0 3 6 , 5 0 1 0 6 ⟶ 4 1 0 3 6 , 2 2 0 1 2 ⟶ 2 0 1 0 1 , 1 2 0 1 2 ⟶ 1 0 1 0 1 , 5 2 0 1 2 ⟶ 5 0 1 0 1 , 2 2 0 1 0 ⟶ 2 0 1 0 3 , 1 2 0 1 0 ⟶ 1 0 1 0 3 , 5 2 0 1 0 ⟶ 5 0 1 0 3 , 0 3 3 3 1 ⟶ 2 0 3 3 1 , 3 3 3 3 1 ⟶ 1 0 3 3 1 , 4 3 3 3 1 ⟶ 5 0 3 3 1 , 0 3 3 3 3 ⟶ 2 0 3 3 3 , 3 3 3 3 3 ⟶ 1 0 3 3 3 , 4 3 3 3 3 ⟶ 5 0 3 3 3 , 0 3 3 3 6 ⟶ 2 0 3 3 6 , 3 3 3 3 6 ⟶ 1 0 3 3 6 , 4 3 3 3 6 ⟶ 5 0 3 3 6 } Applying sparse untiling TRFCU(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. 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 27-rule system { 0 1 2 2 2 ⟶ 0 3 1 0 1 , 3 1 2 2 2 ⟶ 3 3 1 0 1 , 4 1 2 2 2 ⟶ 4 3 1 0 1 , 0 1 2 2 0 ⟶ 0 3 1 0 3 , 3 1 2 2 0 ⟶ 3 3 1 0 3 , 4 1 2 2 0 ⟶ 4 3 1 0 3 , 2 0 1 0 1 ⟶ 0 1 0 3 1 , 1 0 1 0 1 ⟶ 3 1 0 3 1 , 5 0 1 0 1 ⟶ 4 1 0 3 1 , 2 0 1 0 3 ⟶ 0 1 0 3 3 , 1 0 1 0 3 ⟶ 3 1 0 3 3 , 5 0 1 0 3 ⟶ 4 1 0 3 3 , 2 2 0 1 2 ⟶ 2 0 1 0 1 , 1 2 0 1 2 ⟶ 1 0 1 0 1 , 5 2 0 1 2 ⟶ 5 0 1 0 1 , 2 2 0 1 0 ⟶ 2 0 1 0 3 , 1 2 0 1 0 ⟶ 1 0 1 0 3 , 5 2 0 1 0 ⟶ 5 0 1 0 3 , 0 3 3 3 1 ⟶ 2 0 3 3 1 , 3 3 3 3 1 ⟶ 1 0 3 3 1 , 4 3 3 3 1 ⟶ 5 0 3 3 1 , 0 3 3 3 3 ⟶ 2 0 3 3 3 , 3 3 3 3 3 ⟶ 1 0 3 3 3 , 4 3 3 3 3 ⟶ 5 0 3 3 3 , 0 3 3 3 6 ⟶ 2 0 3 3 6 , 3 3 3 3 6 ⟶ 1 0 3 3 6 , 4 3 3 3 6 ⟶ 5 0 3 3 6 } 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 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 1 0 0 ⎟ ⎜ 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎜ 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 26-rule system { 0 1 2 2 2 ⟶ 0 3 1 0 1 , 3 1 2 2 2 ⟶ 3 3 1 0 1 , 4 1 2 2 2 ⟶ 4 3 1 0 1 , 0 1 2 2 0 ⟶ 0 3 1 0 3 , 3 1 2 2 0 ⟶ 3 3 1 0 3 , 4 1 2 2 0 ⟶ 4 3 1 0 3 , 2 0 1 0 1 ⟶ 0 1 0 3 1 , 1 0 1 0 1 ⟶ 3 1 0 3 1 , 5 0 1 0 1 ⟶ 4 1 0 3 1 , 2 0 1 0 3 ⟶ 0 1 0 3 3 , 1 0 1 0 3 ⟶ 3 1 0 3 3 , 5 0 1 0 3 ⟶ 4 1 0 3 3 , 2 2 0 1 2 ⟶ 2 0 1 0 1 , 1 2 0 1 2 ⟶ 1 0 1 0 1 , 5 2 0 1 2 ⟶ 5 0 1 0 1 , 2 2 0 1 0 ⟶ 2 0 1 0 3 , 1 2 0 1 0 ⟶ 1 0 1 0 3 , 5 2 0 1 0 ⟶ 5 0 1 0 3 , 0 3 3 3 1 ⟶ 2 0 3 3 1 , 3 3 3 3 1 ⟶ 1 0 3 3 1 , 4 3 3 3 1 ⟶ 5 0 3 3 1 , 0 3 3 3 3 ⟶ 2 0 3 3 3 , 3 3 3 3 3 ⟶ 1 0 3 3 3 , 4 3 3 3 3 ⟶ 5 0 3 3 3 , 0 3 3 3 6 ⟶ 2 0 3 3 6 , 3 3 3 3 6 ⟶ 1 0 3 3 6 } Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols. After renaming modulo the bijection { (0,↑) ↦ 0, (1,↓) ↦ 1, (2,↓) ↦ 2, (3,↓) ↦ 3, (0,↓) ↦ 4, (3,↑) ↦ 5, (1,↑) ↦ 6, (4,↑) ↦ 7, (2,↑) ↦ 8, (5,↑) ↦ 9, (6,↓) ↦ 10, (4,↓) ↦ 11, (5,↓) ↦ 12 }, it remains to prove termination of the 126-rule system { 0 1 2 2 2 ⟶ 0 3 1 4 1 , 0 1 2 2 2 ⟶ 5 1 4 1 , 0 1 2 2 2 ⟶ 6 4 1 , 0 1 2 2 2 ⟶ 0 1 , 0 1 2 2 2 ⟶ 6 , 5 1 2 2 2 ⟶ 5 3 1 4 1 , 5 1 2 2 2 ⟶ 5 1 4 1 , 5 1 2 2 2 ⟶ 6 4 1 , 5 1 2 2 2 ⟶ 0 1 , 5 1 2 2 2 ⟶ 6 , 7 1 2 2 2 ⟶ 7 3 1 4 1 , 7 1 2 2 2 ⟶ 5 1 4 1 , 7 1 2 2 2 ⟶ 6 4 1 , 7 1 2 2 2 ⟶ 0 1 , 7 1 2 2 2 ⟶ 6 , 0 1 2 2 4 ⟶ 0 3 1 4 3 , 0 1 2 2 4 ⟶ 5 1 4 3 , 0 1 2 2 4 ⟶ 6 4 3 , 0 1 2 2 4 ⟶ 0 3 , 0 1 2 2 4 ⟶ 5 , 5 1 2 2 4 ⟶ 5 3 1 4 3 , 5 1 2 2 4 ⟶ 5 1 4 3 , 5 1 2 2 4 ⟶ 6 4 3 , 5 1 2 2 4 ⟶ 0 3 , 5 1 2 2 4 ⟶ 5 , 7 1 2 2 4 ⟶ 7 3 1 4 3 , 7 1 2 2 4 ⟶ 5 1 4 3 , 7 1 2 2 4 ⟶ 6 4 3 , 7 1 2 2 4 ⟶ 0 3 , 7 1 2 2 4 ⟶ 5 , 8 4 1 4 1 ⟶ 0 1 4 3 1 , 8 4 1 4 1 ⟶ 6 4 3 1 , 8 4 1 4 1 ⟶ 0 3 1 , 8 4 1 4 1 ⟶ 5 1 , 6 4 1 4 1 ⟶ 5 1 4 3 1 , 6 4 1 4 1 ⟶ 6 4 3 1 , 6 4 1 4 1 ⟶ 0 3 1 , 6 4 1 4 1 ⟶ 5 1 , 9 4 1 4 1 ⟶ 7 1 4 3 1 , 9 4 1 4 1 ⟶ 6 4 3 1 , 9 4 1 4 1 ⟶ 0 3 1 , 9 4 1 4 1 ⟶ 5 1 , 8 4 1 4 3 ⟶ 0 1 4 3 3 , 8 4 1 4 3 ⟶ 6 4 3 3 , 8 4 1 4 3 ⟶ 0 3 3 , 8 4 1 4 3 ⟶ 5 3 , 6 4 1 4 3 ⟶ 5 1 4 3 3 , 6 4 1 4 3 ⟶ 6 4 3 3 , 6 4 1 4 3 ⟶ 0 3 3 , 6 4 1 4 3 ⟶ 5 3 , 9 4 1 4 3 ⟶ 7 1 4 3 3 , 9 4 1 4 3 ⟶ 6 4 3 3 , 9 4 1 4 3 ⟶ 0 3 3 , 9 4 1 4 3 ⟶ 5 3 , 8 2 4 1 2 ⟶ 8 4 1 4 1 , 8 2 4 1 2 ⟶ 0 1 4 1 , 8 2 4 1 2 ⟶ 6 4 1 , 8 2 4 1 2 ⟶ 0 1 , 8 2 4 1 2 ⟶ 6 , 6 2 4 1 2 ⟶ 6 4 1 4 1 , 6 2 4 1 2 ⟶ 0 1 4 1 , 6 2 4 1 2 ⟶ 6 4 1 , 6 2 4 1 2 ⟶ 0 1 , 6 2 4 1 2 ⟶ 6 , 9 2 4 1 2 ⟶ 9 4 1 4 1 , 9 2 4 1 2 ⟶ 0 1 4 1 , 9 2 4 1 2 ⟶ 6 4 1 , 9 2 4 1 2 ⟶ 0 1 , 9 2 4 1 2 ⟶ 6 , 8 2 4 1 4 ⟶ 8 4 1 4 3 , 8 2 4 1 4 ⟶ 0 1 4 3 , 8 2 4 1 4 ⟶ 6 4 3 , 8 2 4 1 4 ⟶ 0 3 , 8 2 4 1 4 ⟶ 5 , 6 2 4 1 4 ⟶ 6 4 1 4 3 , 6 2 4 1 4 ⟶ 0 1 4 3 , 6 2 4 1 4 ⟶ 6 4 3 , 6 2 4 1 4 ⟶ 0 3 , 6 2 4 1 4 ⟶ 5 , 9 2 4 1 4 ⟶ 9 4 1 4 3 , 9 2 4 1 4 ⟶ 0 1 4 3 , 9 2 4 1 4 ⟶ 6 4 3 , 9 2 4 1 4 ⟶ 0 3 , 9 2 4 1 4 ⟶ 5 , 0 3 3 3 1 ⟶ 8 4 3 3 1 , 0 3 3 3 1 ⟶ 0 3 3 1 , 5 3 3 3 1 ⟶ 6 4 3 3 1 , 5 3 3 3 1 ⟶ 0 3 3 1 , 7 3 3 3 1 ⟶ 9 4 3 3 1 , 7 3 3 3 1 ⟶ 0 3 3 1 , 0 3 3 3 3 ⟶ 8 4 3 3 3 , 0 3 3 3 3 ⟶ 0 3 3 3 , 5 3 3 3 3 ⟶ 6 4 3 3 3 , 5 3 3 3 3 ⟶ 0 3 3 3 , 7 3 3 3 3 ⟶ 9 4 3 3 3 , 7 3 3 3 3 ⟶ 0 3 3 3 , 0 3 3 3 10 ⟶ 8 4 3 3 10 , 0 3 3 3 10 ⟶ 0 3 3 10 , 5 3 3 3 10 ⟶ 6 4 3 3 10 , 5 3 3 3 10 ⟶ 0 3 3 10 , 4 1 2 2 2 →= 4 3 1 4 1 , 3 1 2 2 2 →= 3 3 1 4 1 , 11 1 2 2 2 →= 11 3 1 4 1 , 4 1 2 2 4 →= 4 3 1 4 3 , 3 1 2 2 4 →= 3 3 1 4 3 , 11 1 2 2 4 →= 11 3 1 4 3 , 2 4 1 4 1 →= 4 1 4 3 1 , 1 4 1 4 1 →= 3 1 4 3 1 , 12 4 1 4 1 →= 11 1 4 3 1 , 2 4 1 4 3 →= 4 1 4 3 3 , 1 4 1 4 3 →= 3 1 4 3 3 , 12 4 1 4 3 →= 11 1 4 3 3 , 2 2 4 1 2 →= 2 4 1 4 1 , 1 2 4 1 2 →= 1 4 1 4 1 , 12 2 4 1 2 →= 12 4 1 4 1 , 2 2 4 1 4 →= 2 4 1 4 3 , 1 2 4 1 4 →= 1 4 1 4 3 , 12 2 4 1 4 →= 12 4 1 4 3 , 4 3 3 3 1 →= 2 4 3 3 1 , 3 3 3 3 1 →= 1 4 3 3 1 , 11 3 3 3 1 →= 12 4 3 3 1 , 4 3 3 3 3 →= 2 4 3 3 3 , 3 3 3 3 3 →= 1 4 3 3 3 , 11 3 3 3 3 →= 12 4 3 3 3 , 4 3 3 3 10 →= 2 4 3 3 10 , 3 3 3 3 10 →= 1 4 3 3 10 } 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, 5 ↦ 5, 7 ↦ 6, 8 ↦ 7, 6 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12 }, it remains to prove termination of the 52-rule system { 0 1 2 2 2 ⟶ 0 3 1 4 1 , 5 1 2 2 2 ⟶ 5 3 1 4 1 , 6 1 2 2 2 ⟶ 6 3 1 4 1 , 0 1 2 2 4 ⟶ 0 3 1 4 3 , 5 1 2 2 4 ⟶ 5 3 1 4 3 , 6 1 2 2 4 ⟶ 6 3 1 4 3 , 7 4 1 4 1 ⟶ 0 1 4 3 1 , 8 4 1 4 1 ⟶ 5 1 4 3 1 , 9 4 1 4 1 ⟶ 6 1 4 3 1 , 7 4 1 4 3 ⟶ 0 1 4 3 3 , 8 4 1 4 3 ⟶ 5 1 4 3 3 , 9 4 1 4 3 ⟶ 6 1 4 3 3 , 7 2 4 1 2 ⟶ 7 4 1 4 1 , 8 2 4 1 2 ⟶ 8 4 1 4 1 , 9 2 4 1 2 ⟶ 9 4 1 4 1 , 7 2 4 1 4 ⟶ 7 4 1 4 3 , 8 2 4 1 4 ⟶ 8 4 1 4 3 , 9 2 4 1 4 ⟶ 9 4 1 4 3 , 0 3 3 3 1 ⟶ 7 4 3 3 1 , 5 3 3 3 1 ⟶ 8 4 3 3 1 , 6 3 3 3 1 ⟶ 9 4 3 3 1 , 0 3 3 3 3 ⟶ 7 4 3 3 3 , 5 3 3 3 3 ⟶ 8 4 3 3 3 , 6 3 3 3 3 ⟶ 9 4 3 3 3 , 0 3 3 3 10 ⟶ 7 4 3 3 10 , 5 3 3 3 10 ⟶ 8 4 3 3 10 , 4 1 2 2 2 →= 4 3 1 4 1 , 3 1 2 2 2 →= 3 3 1 4 1 , 11 1 2 2 2 →= 11 3 1 4 1 , 4 1 2 2 4 →= 4 3 1 4 3 , 3 1 2 2 4 →= 3 3 1 4 3 , 11 1 2 2 4 →= 11 3 1 4 3 , 2 4 1 4 1 →= 4 1 4 3 1 , 1 4 1 4 1 →= 3 1 4 3 1 , 12 4 1 4 1 →= 11 1 4 3 1 , 2 4 1 4 3 →= 4 1 4 3 3 , 1 4 1 4 3 →= 3 1 4 3 3 , 12 4 1 4 3 →= 11 1 4 3 3 , 2 2 4 1 2 →= 2 4 1 4 1 , 1 2 4 1 2 →= 1 4 1 4 1 , 12 2 4 1 2 →= 12 4 1 4 1 , 2 2 4 1 4 →= 2 4 1 4 3 , 1 2 4 1 4 →= 1 4 1 4 3 , 12 2 4 1 4 →= 12 4 1 4 3 , 4 3 3 3 1 →= 2 4 3 3 1 , 3 3 3 3 1 →= 1 4 3 3 1 , 11 3 3 3 1 →= 12 4 3 3 1 , 4 3 3 3 3 →= 2 4 3 3 3 , 3 3 3 3 3 →= 1 4 3 3 3 , 11 3 3 3 3 →= 12 4 3 3 3 , 4 3 3 3 10 →= 2 4 3 3 10 , 3 3 3 3 10 →= 1 4 3 3 10 } 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 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 1 0 0 ⎟ ⎜ 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎜ 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 51-rule system { 0 1 2 2 2 ⟶ 0 3 1 4 1 , 5 1 2 2 2 ⟶ 5 3 1 4 1 , 6 1 2 2 2 ⟶ 6 3 1 4 1 , 0 1 2 2 4 ⟶ 0 3 1 4 3 , 5 1 2 2 4 ⟶ 5 3 1 4 3 , 6 1 2 2 4 ⟶ 6 3 1 4 3 , 7 4 1 4 1 ⟶ 0 1 4 3 1 , 8 4 1 4 1 ⟶ 5 1 4 3 1 , 9 4 1 4 1 ⟶ 6 1 4 3 1 , 7 4 1 4 3 ⟶ 0 1 4 3 3 , 8 4 1 4 3 ⟶ 5 1 4 3 3 , 9 4 1 4 3 ⟶ 6 1 4 3 3 , 7 2 4 1 2 ⟶ 7 4 1 4 1 , 8 2 4 1 2 ⟶ 8 4 1 4 1 , 9 2 4 1 2 ⟶ 9 4 1 4 1 , 7 2 4 1 4 ⟶ 7 4 1 4 3 , 8 2 4 1 4 ⟶ 8 4 1 4 3 , 9 2 4 1 4 ⟶ 9 4 1 4 3 , 0 3 3 3 1 ⟶ 7 4 3 3 1 , 5 3 3 3 1 ⟶ 8 4 3 3 1 , 6 3 3 3 1 ⟶ 9 4 3 3 1 , 0 3 3 3 3 ⟶ 7 4 3 3 3 , 5 3 3 3 3 ⟶ 8 4 3 3 3 , 6 3 3 3 3 ⟶ 9 4 3 3 3 , 5 3 3 3 10 ⟶ 8 4 3 3 10 , 4 1 2 2 2 →= 4 3 1 4 1 , 3 1 2 2 2 →= 3 3 1 4 1 , 11 1 2 2 2 →= 11 3 1 4 1 , 4 1 2 2 4 →= 4 3 1 4 3 , 3 1 2 2 4 →= 3 3 1 4 3 , 11 1 2 2 4 →= 11 3 1 4 3 , 2 4 1 4 1 →= 4 1 4 3 1 , 1 4 1 4 1 →= 3 1 4 3 1 , 12 4 1 4 1 →= 11 1 4 3 1 , 2 4 1 4 3 →= 4 1 4 3 3 , 1 4 1 4 3 →= 3 1 4 3 3 , 12 4 1 4 3 →= 11 1 4 3 3 , 2 2 4 1 2 →= 2 4 1 4 1 , 1 2 4 1 2 →= 1 4 1 4 1 , 12 2 4 1 2 →= 12 4 1 4 1 , 2 2 4 1 4 →= 2 4 1 4 3 , 1 2 4 1 4 →= 1 4 1 4 3 , 12 2 4 1 4 →= 12 4 1 4 3 , 4 3 3 3 1 →= 2 4 3 3 1 , 3 3 3 3 1 →= 1 4 3 3 1 , 11 3 3 3 1 →= 12 4 3 3 1 , 4 3 3 3 3 →= 2 4 3 3 3 , 3 3 3 3 3 →= 1 4 3 3 3 , 11 3 3 3 3 →= 12 4 3 3 3 , 4 3 3 3 10 →= 2 4 3 3 10 , 3 3 3 3 10 →= 1 4 3 3 10 } 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 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 1 0 0 ⎟ ⎜ 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎜ 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 50-rule system { 0 1 2 2 2 ⟶ 0 3 1 4 1 , 5 1 2 2 2 ⟶ 5 3 1 4 1 , 6 1 2 2 2 ⟶ 6 3 1 4 1 , 0 1 2 2 4 ⟶ 0 3 1 4 3 , 5 1 2 2 4 ⟶ 5 3 1 4 3 , 6 1 2 2 4 ⟶ 6 3 1 4 3 , 7 4 1 4 1 ⟶ 0 1 4 3 1 , 8 4 1 4 1 ⟶ 5 1 4 3 1 , 9 4 1 4 1 ⟶ 6 1 4 3 1 , 7 4 1 4 3 ⟶ 0 1 4 3 3 , 8 4 1 4 3 ⟶ 5 1 4 3 3 , 9 4 1 4 3 ⟶ 6 1 4 3 3 , 7 2 4 1 2 ⟶ 7 4 1 4 1 , 8 2 4 1 2 ⟶ 8 4 1 4 1 , 9 2 4 1 2 ⟶ 9 4 1 4 1 , 7 2 4 1 4 ⟶ 7 4 1 4 3 , 8 2 4 1 4 ⟶ 8 4 1 4 3 , 9 2 4 1 4 ⟶ 9 4 1 4 3 , 0 3 3 3 1 ⟶ 7 4 3 3 1 , 5 3 3 3 1 ⟶ 8 4 3 3 1 , 6 3 3 3 1 ⟶ 9 4 3 3 1 , 0 3 3 3 3 ⟶ 7 4 3 3 3 , 5 3 3 3 3 ⟶ 8 4 3 3 3 , 6 3 3 3 3 ⟶ 9 4 3 3 3 , 4 1 2 2 2 →= 4 3 1 4 1 , 3 1 2 2 2 →= 3 3 1 4 1 , 10 1 2 2 2 →= 10 3 1 4 1 , 4 1 2 2 4 →= 4 3 1 4 3 , 3 1 2 2 4 →= 3 3 1 4 3 , 10 1 2 2 4 →= 10 3 1 4 3 , 2 4 1 4 1 →= 4 1 4 3 1 , 1 4 1 4 1 →= 3 1 4 3 1 , 11 4 1 4 1 →= 10 1 4 3 1 , 2 4 1 4 3 →= 4 1 4 3 3 , 1 4 1 4 3 →= 3 1 4 3 3 , 11 4 1 4 3 →= 10 1 4 3 3 , 2 2 4 1 2 →= 2 4 1 4 1 , 1 2 4 1 2 →= 1 4 1 4 1 , 11 2 4 1 2 →= 11 4 1 4 1 , 2 2 4 1 4 →= 2 4 1 4 3 , 1 2 4 1 4 →= 1 4 1 4 3 , 11 2 4 1 4 →= 11 4 1 4 3 , 4 3 3 3 1 →= 2 4 3 3 1 , 3 3 3 3 1 →= 1 4 3 3 1 , 10 3 3 3 1 →= 11 4 3 3 1 , 4 3 3 3 3 →= 2 4 3 3 3 , 3 3 3 3 3 →= 1 4 3 3 3 , 10 3 3 3 3 →= 11 4 3 3 3 , 4 3 3 3 12 →= 2 4 3 3 12 , 3 3 3 3 12 →= 1 4 3 3 12 } Applying sparse tiling TROC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (13,0) ↦ 0, (0,1) ↦ 1, (1,2) ↦ 2, (2,2) ↦ 3, (0,3) ↦ 4, (3,1) ↦ 5, (1,4) ↦ 6, (4,1) ↦ 7, (2,4) ↦ 8, (13,5) ↦ 9, (5,1) ↦ 10, (5,3) ↦ 11, (13,6) ↦ 12, (6,1) ↦ 13, (6,3) ↦ 14, (4,3) ↦ 15, (3,3) ↦ 16, (13,7) ↦ 17, (7,4) ↦ 18, (1,14) ↦ 19, (13,8) ↦ 20, (8,4) ↦ 21, (13,9) ↦ 22, (9,4) ↦ 23, (3,12) ↦ 24, (3,14) ↦ 25, (7,2) ↦ 26, (8,2) ↦ 27, (9,2) ↦ 28, (11,4) ↦ 29, (13,4) ↦ 30, (10,3) ↦ 31, (13,3) ↦ 32, (13,10) ↦ 33, (10,1) ↦ 34, (11,2) ↦ 35, (13,2) ↦ 36, (13,1) ↦ 37, (13,11) ↦ 38, (12,14) ↦ 39 }, it remains to prove termination of the 410-rule system { 0 1 2 3 3 3 ⟶ 0 4 5 6 7 2 , 0 1 2 3 3 8 ⟶ 0 4 5 6 7 6 , 9 10 2 3 3 3 ⟶ 9 11 5 6 7 2 , 9 10 2 3 3 8 ⟶ 9 11 5 6 7 6 , 12 13 2 3 3 3 ⟶ 12 14 5 6 7 2 , 12 13 2 3 3 8 ⟶ 12 14 5 6 7 6 , 0 1 2 3 8 7 ⟶ 0 4 5 6 15 5 , 0 1 2 3 8 15 ⟶ 0 4 5 6 15 16 , 9 10 2 3 8 7 ⟶ 9 11 5 6 15 5 , 9 10 2 3 8 15 ⟶ 9 11 5 6 15 16 , 12 13 2 3 8 7 ⟶ 12 14 5 6 15 5 , 12 13 2 3 8 15 ⟶ 12 14 5 6 15 16 , 17 18 7 6 7 2 ⟶ 0 1 6 15 5 2 , 17 18 7 6 7 6 ⟶ 0 1 6 15 5 6 , 17 18 7 6 7 19 ⟶ 0 1 6 15 5 19 , 20 21 7 6 7 2 ⟶ 9 10 6 15 5 2 , 20 21 7 6 7 6 ⟶ 9 10 6 15 5 6 , 20 21 7 6 7 19 ⟶ 9 10 6 15 5 19 , 22 23 7 6 7 2 ⟶ 12 13 6 15 5 2 , 22 23 7 6 7 6 ⟶ 12 13 6 15 5 6 , 22 23 7 6 7 19 ⟶ 12 13 6 15 5 19 , 17 18 7 6 15 5 ⟶ 0 1 6 15 16 5 , 17 18 7 6 15 16 ⟶ 0 1 6 15 16 16 , 17 18 7 6 15 24 ⟶ 0 1 6 15 16 24 , 17 18 7 6 15 25 ⟶ 0 1 6 15 16 25 , 20 21 7 6 15 5 ⟶ 9 10 6 15 16 5 , 20 21 7 6 15 16 ⟶ 9 10 6 15 16 16 , 20 21 7 6 15 24 ⟶ 9 10 6 15 16 24 , 20 21 7 6 15 25 ⟶ 9 10 6 15 16 25 , 22 23 7 6 15 5 ⟶ 12 13 6 15 16 5 , 22 23 7 6 15 16 ⟶ 12 13 6 15 16 16 , 22 23 7 6 15 24 ⟶ 12 13 6 15 16 24 , 22 23 7 6 15 25 ⟶ 12 13 6 15 16 25 , 17 26 8 7 2 3 ⟶ 17 18 7 6 7 2 , 17 26 8 7 2 8 ⟶ 17 18 7 6 7 6 , 20 27 8 7 2 3 ⟶ 20 21 7 6 7 2 , 20 27 8 7 2 8 ⟶ 20 21 7 6 7 6 , 22 28 8 7 2 3 ⟶ 22 23 7 6 7 2 , 22 28 8 7 2 8 ⟶ 22 23 7 6 7 6 , 17 26 8 7 6 7 ⟶ 17 18 7 6 15 5 , 17 26 8 7 6 15 ⟶ 17 18 7 6 15 16 , 20 27 8 7 6 7 ⟶ 20 21 7 6 15 5 , 20 27 8 7 6 15 ⟶ 20 21 7 6 15 16 , 22 28 8 7 6 7 ⟶ 22 23 7 6 15 5 , 22 28 8 7 6 15 ⟶ 22 23 7 6 15 16 , 0 4 16 16 5 2 ⟶ 17 18 15 16 5 2 , 0 4 16 16 5 6 ⟶ 17 18 15 16 5 6 , 0 4 16 16 5 19 ⟶ 17 18 15 16 5 19 , 9 11 16 16 5 2 ⟶ 20 21 15 16 5 2 , 9 11 16 16 5 6 ⟶ 20 21 15 16 5 6 , 9 11 16 16 5 19 ⟶ 20 21 15 16 5 19 , 12 14 16 16 5 2 ⟶ 22 23 15 16 5 2 , 12 14 16 16 5 6 ⟶ 22 23 15 16 5 6 , 12 14 16 16 5 19 ⟶ 22 23 15 16 5 19 , 0 4 16 16 16 5 ⟶ 17 18 15 16 16 5 , 0 4 16 16 16 16 ⟶ 17 18 15 16 16 16 , 0 4 16 16 16 24 ⟶ 17 18 15 16 16 24 , 0 4 16 16 16 25 ⟶ 17 18 15 16 16 25 , 9 11 16 16 16 5 ⟶ 20 21 15 16 16 5 , 9 11 16 16 16 16 ⟶ 20 21 15 16 16 16 , 9 11 16 16 16 24 ⟶ 20 21 15 16 16 24 , 9 11 16 16 16 25 ⟶ 20 21 15 16 16 25 , 12 14 16 16 16 5 ⟶ 22 23 15 16 16 5 , 12 14 16 16 16 16 ⟶ 22 23 15 16 16 16 , 12 14 16 16 16 24 ⟶ 22 23 15 16 16 24 , 12 14 16 16 16 25 ⟶ 22 23 15 16 16 25 , 6 7 2 3 3 3 →= 6 15 5 6 7 2 , 6 7 2 3 3 8 →= 6 15 5 6 7 6 , 8 7 2 3 3 3 →= 8 15 5 6 7 2 , 8 7 2 3 3 8 →= 8 15 5 6 7 6 , 18 7 2 3 3 3 →= 18 15 5 6 7 2 , 18 7 2 3 3 8 →= 18 15 5 6 7 6 , 21 7 2 3 3 3 →= 21 15 5 6 7 2 , 21 7 2 3 3 8 →= 21 15 5 6 7 6 , 23 7 2 3 3 3 →= 23 15 5 6 7 2 , 23 7 2 3 3 8 →= 23 15 5 6 7 6 , 29 7 2 3 3 3 →= 29 15 5 6 7 2 , 29 7 2 3 3 8 →= 29 15 5 6 7 6 , 30 7 2 3 3 3 →= 30 15 5 6 7 2 , 30 7 2 3 3 8 →= 30 15 5 6 7 6 , 4 5 2 3 3 3 →= 4 16 5 6 7 2 , 4 5 2 3 3 8 →= 4 16 5 6 7 6 , 16 5 2 3 3 3 →= 16 16 5 6 7 2 , 16 5 2 3 3 8 →= 16 16 5 6 7 6 , 15 5 2 3 3 3 →= 15 16 5 6 7 2 , 15 5 2 3 3 8 →= 15 16 5 6 7 6 , 11 5 2 3 3 3 →= 11 16 5 6 7 2 , 11 5 2 3 3 8 →= 11 16 5 6 7 6 , 14 5 2 3 3 3 →= 14 16 5 6 7 2 , 14 5 2 3 3 8 →= 14 16 5 6 7 6 , 31 5 2 3 3 3 →= 31 16 5 6 7 2 , 31 5 2 3 3 8 →= 31 16 5 6 7 6 , 32 5 2 3 3 3 →= 32 16 5 6 7 2 , 32 5 2 3 3 8 →= 32 16 5 6 7 6 , 33 34 2 3 3 3 →= 33 31 5 6 7 2 , 33 34 2 3 3 8 →= 33 31 5 6 7 6 , 6 7 2 3 8 7 →= 6 15 5 6 15 5 , 6 7 2 3 8 15 →= 6 15 5 6 15 16 , 8 7 2 3 8 7 →= 8 15 5 6 15 5 , 8 7 2 3 8 15 →= 8 15 5 6 15 16 , 18 7 2 3 8 7 →= 18 15 5 6 15 5 , 18 7 2 3 8 15 →= 18 15 5 6 15 16 , 21 7 2 3 8 7 →= 21 15 5 6 15 5 , 21 7 2 3 8 15 →= 21 15 5 6 15 16 , 23 7 2 3 8 7 →= 23 15 5 6 15 5 , 23 7 2 3 8 15 →= 23 15 5 6 15 16 , 29 7 2 3 8 7 →= 29 15 5 6 15 5 , 29 7 2 3 8 15 →= 29 15 5 6 15 16 , 30 7 2 3 8 7 →= 30 15 5 6 15 5 , 30 7 2 3 8 15 →= 30 15 5 6 15 16 , 4 5 2 3 8 7 →= 4 16 5 6 15 5 , 4 5 2 3 8 15 →= 4 16 5 6 15 16 , 16 5 2 3 8 7 →= 16 16 5 6 15 5 , 16 5 2 3 8 15 →= 16 16 5 6 15 16 , 15 5 2 3 8 7 →= 15 16 5 6 15 5 , 15 5 2 3 8 15 →= 15 16 5 6 15 16 , 11 5 2 3 8 7 →= 11 16 5 6 15 5 , 11 5 2 3 8 15 →= 11 16 5 6 15 16 , 14 5 2 3 8 7 →= 14 16 5 6 15 5 , 14 5 2 3 8 15 →= 14 16 5 6 15 16 , 31 5 2 3 8 7 →= 31 16 5 6 15 5 , 31 5 2 3 8 15 →= 31 16 5 6 15 16 , 32 5 2 3 8 7 →= 32 16 5 6 15 5 , 32 5 2 3 8 15 →= 32 16 5 6 15 16 , 33 34 2 3 8 7 →= 33 31 5 6 15 5 , 33 34 2 3 8 15 →= 33 31 5 6 15 16 , 2 8 7 6 7 2 →= 6 7 6 15 5 2 , 2 8 7 6 7 6 →= 6 7 6 15 5 6 , 2 8 7 6 7 19 →= 6 7 6 15 5 19 , 3 8 7 6 7 2 →= 8 7 6 15 5 2 , 3 8 7 6 7 6 →= 8 7 6 15 5 6 , 3 8 7 6 7 19 →= 8 7 6 15 5 19 , 26 8 7 6 7 2 →= 18 7 6 15 5 2 , 26 8 7 6 7 6 →= 18 7 6 15 5 6 , 26 8 7 6 7 19 →= 18 7 6 15 5 19 , 27 8 7 6 7 2 →= 21 7 6 15 5 2 , 27 8 7 6 7 6 →= 21 7 6 15 5 6 , 27 8 7 6 7 19 →= 21 7 6 15 5 19 , 28 8 7 6 7 2 →= 23 7 6 15 5 2 , 28 8 7 6 7 6 →= 23 7 6 15 5 6 , 28 8 7 6 7 19 →= 23 7 6 15 5 19 , 35 8 7 6 7 2 →= 29 7 6 15 5 2 , 35 8 7 6 7 6 →= 29 7 6 15 5 6 , 35 8 7 6 7 19 →= 29 7 6 15 5 19 , 36 8 7 6 7 2 →= 30 7 6 15 5 2 , 36 8 7 6 7 6 →= 30 7 6 15 5 6 , 36 8 7 6 7 19 →= 30 7 6 15 5 19 , 1 6 7 6 7 2 →= 4 5 6 15 5 2 , 1 6 7 6 7 6 →= 4 5 6 15 5 6 , 1 6 7 6 7 19 →= 4 5 6 15 5 19 , 5 6 7 6 7 2 →= 16 5 6 15 5 2 , 5 6 7 6 7 6 →= 16 5 6 15 5 6 , 5 6 7 6 7 19 →= 16 5 6 15 5 19 , 7 6 7 6 7 2 →= 15 5 6 15 5 2 , 7 6 7 6 7 6 →= 15 5 6 15 5 6 , 7 6 7 6 7 19 →= 15 5 6 15 5 19 , 10 6 7 6 7 2 →= 11 5 6 15 5 2 , 10 6 7 6 7 6 →= 11 5 6 15 5 6 , 10 6 7 6 7 19 →= 11 5 6 15 5 19 , 13 6 7 6 7 2 →= 14 5 6 15 5 2 , 13 6 7 6 7 6 →= 14 5 6 15 5 6 , 13 6 7 6 7 19 →= 14 5 6 15 5 19 , 34 6 7 6 7 2 →= 31 5 6 15 5 2 , 34 6 7 6 7 6 →= 31 5 6 15 5 6 , 34 6 7 6 7 19 →= 31 5 6 15 5 19 , 37 6 7 6 7 2 →= 32 5 6 15 5 2 , 37 6 7 6 7 6 →= 32 5 6 15 5 6 , 37 6 7 6 7 19 →= 32 5 6 15 5 19 , 38 29 7 6 7 2 →= 33 34 6 15 5 2 , 38 29 7 6 7 6 →= 33 34 6 15 5 6 , 38 29 7 6 7 19 →= 33 34 6 15 5 19 , 2 8 7 6 15 5 →= 6 7 6 15 16 5 , 2 8 7 6 15 16 →= 6 7 6 15 16 16 , 2 8 7 6 15 24 →= 6 7 6 15 16 24 , 2 8 7 6 15 25 →= 6 7 6 15 16 25 , 3 8 7 6 15 5 →= 8 7 6 15 16 5 , 3 8 7 6 15 16 →= 8 7 6 15 16 16 , 3 8 7 6 15 24 →= 8 7 6 15 16 24 , 3 8 7 6 15 25 →= 8 7 6 15 16 25 , 26 8 7 6 15 5 →= 18 7 6 15 16 5 , 26 8 7 6 15 16 →= 18 7 6 15 16 16 , 26 8 7 6 15 24 →= 18 7 6 15 16 24 , 26 8 7 6 15 25 →= 18 7 6 15 16 25 , 27 8 7 6 15 5 →= 21 7 6 15 16 5 , 27 8 7 6 15 16 →= 21 7 6 15 16 16 , 27 8 7 6 15 24 →= 21 7 6 15 16 24 , 27 8 7 6 15 25 →= 21 7 6 15 16 25 , 28 8 7 6 15 5 →= 23 7 6 15 16 5 , 28 8 7 6 15 16 →= 23 7 6 15 16 16 , 28 8 7 6 15 24 →= 23 7 6 15 16 24 , 28 8 7 6 15 25 →= 23 7 6 15 16 25 , 35 8 7 6 15 5 →= 29 7 6 15 16 5 , 35 8 7 6 15 16 →= 29 7 6 15 16 16 , 35 8 7 6 15 24 →= 29 7 6 15 16 24 , 35 8 7 6 15 25 →= 29 7 6 15 16 25 , 36 8 7 6 15 5 →= 30 7 6 15 16 5 , 36 8 7 6 15 16 →= 30 7 6 15 16 16 , 36 8 7 6 15 24 →= 30 7 6 15 16 24 , 36 8 7 6 15 25 →= 30 7 6 15 16 25 , 1 6 7 6 15 5 →= 4 5 6 15 16 5 , 1 6 7 6 15 16 →= 4 5 6 15 16 16 , 1 6 7 6 15 24 →= 4 5 6 15 16 24 , 1 6 7 6 15 25 →= 4 5 6 15 16 25 , 5 6 7 6 15 5 →= 16 5 6 15 16 5 , 5 6 7 6 15 16 →= 16 5 6 15 16 16 , 5 6 7 6 15 24 →= 16 5 6 15 16 24 , 5 6 7 6 15 25 →= 16 5 6 15 16 25 , 7 6 7 6 15 5 →= 15 5 6 15 16 5 , 7 6 7 6 15 16 →= 15 5 6 15 16 16 , 7 6 7 6 15 24 →= 15 5 6 15 16 24 , 7 6 7 6 15 25 →= 15 5 6 15 16 25 , 10 6 7 6 15 5 →= 11 5 6 15 16 5 , 10 6 7 6 15 16 →= 11 5 6 15 16 16 , 10 6 7 6 15 24 →= 11 5 6 15 16 24 , 10 6 7 6 15 25 →= 11 5 6 15 16 25 , 13 6 7 6 15 5 →= 14 5 6 15 16 5 , 13 6 7 6 15 16 →= 14 5 6 15 16 16 , 13 6 7 6 15 24 →= 14 5 6 15 16 24 , 13 6 7 6 15 25 →= 14 5 6 15 16 25 , 34 6 7 6 15 5 →= 31 5 6 15 16 5 , 34 6 7 6 15 16 →= 31 5 6 15 16 16 , 34 6 7 6 15 24 →= 31 5 6 15 16 24 , 34 6 7 6 15 25 →= 31 5 6 15 16 25 , 37 6 7 6 15 5 →= 32 5 6 15 16 5 , 37 6 7 6 15 16 →= 32 5 6 15 16 16 , 37 6 7 6 15 24 →= 32 5 6 15 16 24 , 37 6 7 6 15 25 →= 32 5 6 15 16 25 , 38 29 7 6 15 5 →= 33 34 6 15 16 5 , 38 29 7 6 15 16 →= 33 34 6 15 16 16 , 38 29 7 6 15 24 →= 33 34 6 15 16 24 , 38 29 7 6 15 25 →= 33 34 6 15 16 25 , 2 3 8 7 2 3 →= 2 8 7 6 7 2 , 2 3 8 7 2 8 →= 2 8 7 6 7 6 , 3 3 8 7 2 3 →= 3 8 7 6 7 2 , 3 3 8 7 2 8 →= 3 8 7 6 7 6 , 26 3 8 7 2 3 →= 26 8 7 6 7 2 , 26 3 8 7 2 8 →= 26 8 7 6 7 6 , 27 3 8 7 2 3 →= 27 8 7 6 7 2 , 27 3 8 7 2 8 →= 27 8 7 6 7 6 , 28 3 8 7 2 3 →= 28 8 7 6 7 2 , 28 3 8 7 2 8 →= 28 8 7 6 7 6 , 35 3 8 7 2 3 →= 35 8 7 6 7 2 , 35 3 8 7 2 8 →= 35 8 7 6 7 6 , 36 3 8 7 2 3 →= 36 8 7 6 7 2 , 36 3 8 7 2 8 →= 36 8 7 6 7 6 , 1 2 8 7 2 3 →= 1 6 7 6 7 2 , 1 2 8 7 2 8 →= 1 6 7 6 7 6 , 5 2 8 7 2 3 →= 5 6 7 6 7 2 , 5 2 8 7 2 8 →= 5 6 7 6 7 6 , 7 2 8 7 2 3 →= 7 6 7 6 7 2 , 7 2 8 7 2 8 →= 7 6 7 6 7 6 , 10 2 8 7 2 3 →= 10 6 7 6 7 2 , 10 2 8 7 2 8 →= 10 6 7 6 7 6 , 13 2 8 7 2 3 →= 13 6 7 6 7 2 , 13 2 8 7 2 8 →= 13 6 7 6 7 6 , 34 2 8 7 2 3 →= 34 6 7 6 7 2 , 34 2 8 7 2 8 →= 34 6 7 6 7 6 , 37 2 8 7 2 3 →= 37 6 7 6 7 2 , 37 2 8 7 2 8 →= 37 6 7 6 7 6 , 38 35 8 7 2 3 →= 38 29 7 6 7 2 , 38 35 8 7 2 8 →= 38 29 7 6 7 6 , 2 3 8 7 6 7 →= 2 8 7 6 15 5 , 2 3 8 7 6 15 →= 2 8 7 6 15 16 , 3 3 8 7 6 7 →= 3 8 7 6 15 5 , 3 3 8 7 6 15 →= 3 8 7 6 15 16 , 26 3 8 7 6 7 →= 26 8 7 6 15 5 , 26 3 8 7 6 15 →= 26 8 7 6 15 16 , 27 3 8 7 6 7 →= 27 8 7 6 15 5 , 27 3 8 7 6 15 →= 27 8 7 6 15 16 , 28 3 8 7 6 7 →= 28 8 7 6 15 5 , 28 3 8 7 6 15 →= 28 8 7 6 15 16 , 35 3 8 7 6 7 →= 35 8 7 6 15 5 , 35 3 8 7 6 15 →= 35 8 7 6 15 16 , 36 3 8 7 6 7 →= 36 8 7 6 15 5 , 36 3 8 7 6 15 →= 36 8 7 6 15 16 , 1 2 8 7 6 7 →= 1 6 7 6 15 5 , 1 2 8 7 6 15 →= 1 6 7 6 15 16 , 5 2 8 7 6 7 →= 5 6 7 6 15 5 , 5 2 8 7 6 15 →= 5 6 7 6 15 16 , 7 2 8 7 6 7 →= 7 6 7 6 15 5 , 7 2 8 7 6 15 →= 7 6 7 6 15 16 , 10 2 8 7 6 7 →= 10 6 7 6 15 5 , 10 2 8 7 6 15 →= 10 6 7 6 15 16 , 13 2 8 7 6 7 →= 13 6 7 6 15 5 , 13 2 8 7 6 15 →= 13 6 7 6 15 16 , 34 2 8 7 6 7 →= 34 6 7 6 15 5 , 34 2 8 7 6 15 →= 34 6 7 6 15 16 , 37 2 8 7 6 7 →= 37 6 7 6 15 5 , 37 2 8 7 6 15 →= 37 6 7 6 15 16 , 38 35 8 7 6 7 →= 38 29 7 6 15 5 , 38 35 8 7 6 15 →= 38 29 7 6 15 16 , 6 15 16 16 5 2 →= 2 8 15 16 5 2 , 6 15 16 16 5 6 →= 2 8 15 16 5 6 , 6 15 16 16 5 19 →= 2 8 15 16 5 19 , 8 15 16 16 5 2 →= 3 8 15 16 5 2 , 8 15 16 16 5 6 →= 3 8 15 16 5 6 , 8 15 16 16 5 19 →= 3 8 15 16 5 19 , 18 15 16 16 5 2 →= 26 8 15 16 5 2 , 18 15 16 16 5 6 →= 26 8 15 16 5 6 , 18 15 16 16 5 19 →= 26 8 15 16 5 19 , 21 15 16 16 5 2 →= 27 8 15 16 5 2 , 21 15 16 16 5 6 →= 27 8 15 16 5 6 , 21 15 16 16 5 19 →= 27 8 15 16 5 19 , 23 15 16 16 5 2 →= 28 8 15 16 5 2 , 23 15 16 16 5 6 →= 28 8 15 16 5 6 , 23 15 16 16 5 19 →= 28 8 15 16 5 19 , 29 15 16 16 5 2 →= 35 8 15 16 5 2 , 29 15 16 16 5 6 →= 35 8 15 16 5 6 , 29 15 16 16 5 19 →= 35 8 15 16 5 19 , 30 15 16 16 5 2 →= 36 8 15 16 5 2 , 30 15 16 16 5 6 →= 36 8 15 16 5 6 , 30 15 16 16 5 19 →= 36 8 15 16 5 19 , 4 16 16 16 5 2 →= 1 6 15 16 5 2 , 4 16 16 16 5 6 →= 1 6 15 16 5 6 , 4 16 16 16 5 19 →= 1 6 15 16 5 19 , 16 16 16 16 5 2 →= 5 6 15 16 5 2 , 16 16 16 16 5 6 →= 5 6 15 16 5 6 , 16 16 16 16 5 19 →= 5 6 15 16 5 19 , 15 16 16 16 5 2 →= 7 6 15 16 5 2 , 15 16 16 16 5 6 →= 7 6 15 16 5 6 , 15 16 16 16 5 19 →= 7 6 15 16 5 19 , 11 16 16 16 5 2 →= 10 6 15 16 5 2 , 11 16 16 16 5 6 →= 10 6 15 16 5 6 , 11 16 16 16 5 19 →= 10 6 15 16 5 19 , 14 16 16 16 5 2 →= 13 6 15 16 5 2 , 14 16 16 16 5 6 →= 13 6 15 16 5 6 , 14 16 16 16 5 19 →= 13 6 15 16 5 19 , 31 16 16 16 5 2 →= 34 6 15 16 5 2 , 31 16 16 16 5 6 →= 34 6 15 16 5 6 , 31 16 16 16 5 19 →= 34 6 15 16 5 19 , 32 16 16 16 5 2 →= 37 6 15 16 5 2 , 32 16 16 16 5 6 →= 37 6 15 16 5 6 , 32 16 16 16 5 19 →= 37 6 15 16 5 19 , 33 31 16 16 5 2 →= 38 29 15 16 5 2 , 33 31 16 16 5 6 →= 38 29 15 16 5 6 , 33 31 16 16 5 19 →= 38 29 15 16 5 19 , 6 15 16 16 16 5 →= 2 8 15 16 16 5 , 6 15 16 16 16 16 →= 2 8 15 16 16 16 , 6 15 16 16 16 24 →= 2 8 15 16 16 24 , 6 15 16 16 16 25 →= 2 8 15 16 16 25 , 8 15 16 16 16 5 →= 3 8 15 16 16 5 , 8 15 16 16 16 16 →= 3 8 15 16 16 16 , 8 15 16 16 16 24 →= 3 8 15 16 16 24 , 8 15 16 16 16 25 →= 3 8 15 16 16 25 , 18 15 16 16 16 5 →= 26 8 15 16 16 5 , 18 15 16 16 16 16 →= 26 8 15 16 16 16 , 18 15 16 16 16 24 →= 26 8 15 16 16 24 , 18 15 16 16 16 25 →= 26 8 15 16 16 25 , 21 15 16 16 16 5 →= 27 8 15 16 16 5 , 21 15 16 16 16 16 →= 27 8 15 16 16 16 , 21 15 16 16 16 24 →= 27 8 15 16 16 24 , 21 15 16 16 16 25 →= 27 8 15 16 16 25 , 23 15 16 16 16 5 →= 28 8 15 16 16 5 , 23 15 16 16 16 16 →= 28 8 15 16 16 16 , 23 15 16 16 16 24 →= 28 8 15 16 16 24 , 23 15 16 16 16 25 →= 28 8 15 16 16 25 , 29 15 16 16 16 5 →= 35 8 15 16 16 5 , 29 15 16 16 16 16 →= 35 8 15 16 16 16 , 29 15 16 16 16 24 →= 35 8 15 16 16 24 , 29 15 16 16 16 25 →= 35 8 15 16 16 25 , 30 15 16 16 16 5 →= 36 8 15 16 16 5 , 30 15 16 16 16 16 →= 36 8 15 16 16 16 , 30 15 16 16 16 24 →= 36 8 15 16 16 24 , 30 15 16 16 16 25 →= 36 8 15 16 16 25 , 4 16 16 16 16 5 →= 1 6 15 16 16 5 , 4 16 16 16 16 16 →= 1 6 15 16 16 16 , 4 16 16 16 16 24 →= 1 6 15 16 16 24 , 4 16 16 16 16 25 →= 1 6 15 16 16 25 , 16 16 16 16 16 5 →= 5 6 15 16 16 5 , 16 16 16 16 16 16 →= 5 6 15 16 16 16 , 16 16 16 16 16 24 →= 5 6 15 16 16 24 , 16 16 16 16 16 25 →= 5 6 15 16 16 25 , 15 16 16 16 16 5 →= 7 6 15 16 16 5 , 15 16 16 16 16 16 →= 7 6 15 16 16 16 , 15 16 16 16 16 24 →= 7 6 15 16 16 24 , 15 16 16 16 16 25 →= 7 6 15 16 16 25 , 11 16 16 16 16 5 →= 10 6 15 16 16 5 , 11 16 16 16 16 16 →= 10 6 15 16 16 16 , 11 16 16 16 16 24 →= 10 6 15 16 16 24 , 11 16 16 16 16 25 →= 10 6 15 16 16 25 , 14 16 16 16 16 5 →= 13 6 15 16 16 5 , 14 16 16 16 16 16 →= 13 6 15 16 16 16 , 14 16 16 16 16 24 →= 13 6 15 16 16 24 , 14 16 16 16 16 25 →= 13 6 15 16 16 25 , 31 16 16 16 16 5 →= 34 6 15 16 16 5 , 31 16 16 16 16 16 →= 34 6 15 16 16 16 , 31 16 16 16 16 24 →= 34 6 15 16 16 24 , 31 16 16 16 16 25 →= 34 6 15 16 16 25 , 32 16 16 16 16 5 →= 37 6 15 16 16 5 , 32 16 16 16 16 16 →= 37 6 15 16 16 16 , 32 16 16 16 16 24 →= 37 6 15 16 16 24 , 32 16 16 16 16 25 →= 37 6 15 16 16 25 , 33 31 16 16 16 5 →= 38 29 15 16 16 5 , 33 31 16 16 16 16 →= 38 29 15 16 16 16 , 33 31 16 16 16 24 →= 38 29 15 16 16 24 , 33 31 16 16 16 25 →= 38 29 15 16 16 25 , 6 15 16 16 24 39 →= 2 8 15 16 24 39 , 8 15 16 16 24 39 →= 3 8 15 16 24 39 , 18 15 16 16 24 39 →= 26 8 15 16 24 39 , 21 15 16 16 24 39 →= 27 8 15 16 24 39 , 23 15 16 16 24 39 →= 28 8 15 16 24 39 , 29 15 16 16 24 39 →= 35 8 15 16 24 39 , 30 15 16 16 24 39 →= 36 8 15 16 24 39 , 4 16 16 16 24 39 →= 1 6 15 16 24 39 , 16 16 16 16 24 39 →= 5 6 15 16 24 39 , 15 16 16 16 24 39 →= 7 6 15 16 24 39 , 11 16 16 16 24 39 →= 10 6 15 16 24 39 , 14 16 16 16 24 39 →= 13 6 15 16 24 39 , 31 16 16 16 24 39 →= 34 6 15 16 24 39 , 32 16 16 16 24 39 →= 37 6 15 16 24 39 } 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 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 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 23 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 24 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 25 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 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 { 17 ↦ 0, 18 ↦ 1, 7 ↦ 2, 6 ↦ 3, 15 ↦ 4, 5 ↦ 5, 0 ↦ 6, 1 ↦ 7, 16 ↦ 8, 24 ↦ 9, 25 ↦ 10, 20 ↦ 11, 21 ↦ 12, 9 ↦ 13, 10 ↦ 14, 22 ↦ 15, 23 ↦ 16, 12 ↦ 17, 13 ↦ 18, 26 ↦ 19, 8 ↦ 20, 2 ↦ 21, 3 ↦ 22, 27 ↦ 23, 28 ↦ 24, 4 ↦ 25, 11 ↦ 26, 14 ↦ 27, 31 ↦ 28, 32 ↦ 29, 35 ↦ 30, 29 ↦ 31, 36 ↦ 32, 30 ↦ 33, 38 ↦ 34, 33 ↦ 35, 34 ↦ 36, 37 ↦ 37, 19 ↦ 38, 39 ↦ 39 }, it remains to prove termination of the 187-rule system { 0 1 2 3 4 5 ⟶ 6 7 3 4 8 5 , 0 1 2 3 4 8 ⟶ 6 7 3 4 8 8 , 0 1 2 3 4 9 ⟶ 6 7 3 4 8 9 , 0 1 2 3 4 10 ⟶ 6 7 3 4 8 10 , 11 12 2 3 4 5 ⟶ 13 14 3 4 8 5 , 11 12 2 3 4 8 ⟶ 13 14 3 4 8 8 , 11 12 2 3 4 9 ⟶ 13 14 3 4 8 9 , 11 12 2 3 4 10 ⟶ 13 14 3 4 8 10 , 15 16 2 3 4 5 ⟶ 17 18 3 4 8 5 , 15 16 2 3 4 8 ⟶ 17 18 3 4 8 8 , 15 16 2 3 4 9 ⟶ 17 18 3 4 8 9 , 15 16 2 3 4 10 ⟶ 17 18 3 4 8 10 , 0 19 20 2 21 22 ⟶ 0 1 2 3 2 21 , 0 19 20 2 21 20 ⟶ 0 1 2 3 2 3 , 11 23 20 2 21 22 ⟶ 11 12 2 3 2 21 , 11 23 20 2 21 20 ⟶ 11 12 2 3 2 3 , 15 24 20 2 21 22 ⟶ 15 16 2 3 2 21 , 15 24 20 2 21 20 ⟶ 15 16 2 3 2 3 , 0 19 20 2 3 4 ⟶ 0 1 2 3 4 8 , 11 23 20 2 3 4 ⟶ 11 12 2 3 4 8 , 15 24 20 2 3 4 ⟶ 15 16 2 3 4 8 , 25 5 21 22 22 22 →= 25 8 5 3 2 21 , 25 5 21 22 22 20 →= 25 8 5 3 2 3 , 8 5 21 22 22 22 →= 8 8 5 3 2 21 , 8 5 21 22 22 20 →= 8 8 5 3 2 3 , 4 5 21 22 22 22 →= 4 8 5 3 2 21 , 4 5 21 22 22 20 →= 4 8 5 3 2 3 , 26 5 21 22 22 22 →= 26 8 5 3 2 21 , 26 5 21 22 22 20 →= 26 8 5 3 2 3 , 27 5 21 22 22 22 →= 27 8 5 3 2 21 , 27 5 21 22 22 20 →= 27 8 5 3 2 3 , 28 5 21 22 22 22 →= 28 8 5 3 2 21 , 28 5 21 22 22 20 →= 28 8 5 3 2 3 , 29 5 21 22 22 22 →= 29 8 5 3 2 21 , 29 5 21 22 22 20 →= 29 8 5 3 2 3 , 25 5 21 22 20 4 →= 25 8 5 3 4 8 , 8 5 21 22 20 4 →= 8 8 5 3 4 8 , 4 5 21 22 20 4 →= 4 8 5 3 4 8 , 26 5 21 22 20 4 →= 26 8 5 3 4 8 , 27 5 21 22 20 4 →= 27 8 5 3 4 8 , 28 5 21 22 20 4 →= 28 8 5 3 4 8 , 29 5 21 22 20 4 →= 29 8 5 3 4 8 , 21 20 2 3 4 5 →= 3 2 3 4 8 5 , 21 20 2 3 4 8 →= 3 2 3 4 8 8 , 21 20 2 3 4 9 →= 3 2 3 4 8 9 , 21 20 2 3 4 10 →= 3 2 3 4 8 10 , 22 20 2 3 4 5 →= 20 2 3 4 8 5 , 22 20 2 3 4 8 →= 20 2 3 4 8 8 , 22 20 2 3 4 9 →= 20 2 3 4 8 9 , 22 20 2 3 4 10 →= 20 2 3 4 8 10 , 19 20 2 3 4 5 →= 1 2 3 4 8 5 , 19 20 2 3 4 8 →= 1 2 3 4 8 8 , 19 20 2 3 4 9 →= 1 2 3 4 8 9 , 19 20 2 3 4 10 →= 1 2 3 4 8 10 , 23 20 2 3 4 5 →= 12 2 3 4 8 5 , 23 20 2 3 4 8 →= 12 2 3 4 8 8 , 23 20 2 3 4 9 →= 12 2 3 4 8 9 , 23 20 2 3 4 10 →= 12 2 3 4 8 10 , 24 20 2 3 4 5 →= 16 2 3 4 8 5 , 24 20 2 3 4 8 →= 16 2 3 4 8 8 , 24 20 2 3 4 9 →= 16 2 3 4 8 9 , 24 20 2 3 4 10 →= 16 2 3 4 8 10 , 30 20 2 3 4 5 →= 31 2 3 4 8 5 , 30 20 2 3 4 8 →= 31 2 3 4 8 8 , 30 20 2 3 4 9 →= 31 2 3 4 8 9 , 30 20 2 3 4 10 →= 31 2 3 4 8 10 , 32 20 2 3 4 5 →= 33 2 3 4 8 5 , 32 20 2 3 4 8 →= 33 2 3 4 8 8 , 32 20 2 3 4 9 →= 33 2 3 4 8 9 , 32 20 2 3 4 10 →= 33 2 3 4 8 10 , 5 3 2 3 4 5 →= 8 5 3 4 8 5 , 5 3 2 3 4 8 →= 8 5 3 4 8 8 , 5 3 2 3 4 9 →= 8 5 3 4 8 9 , 5 3 2 3 4 10 →= 8 5 3 4 8 10 , 34 31 2 3 4 5 →= 35 36 3 4 8 5 , 34 31 2 3 4 8 →= 35 36 3 4 8 8 , 34 31 2 3 4 9 →= 35 36 3 4 8 9 , 34 31 2 3 4 10 →= 35 36 3 4 8 10 , 21 22 20 2 21 22 →= 21 20 2 3 2 21 , 21 22 20 2 21 20 →= 21 20 2 3 2 3 , 22 22 20 2 21 22 →= 22 20 2 3 2 21 , 22 22 20 2 21 20 →= 22 20 2 3 2 3 , 19 22 20 2 21 22 →= 19 20 2 3 2 21 , 19 22 20 2 21 20 →= 19 20 2 3 2 3 , 23 22 20 2 21 22 →= 23 20 2 3 2 21 , 23 22 20 2 21 20 →= 23 20 2 3 2 3 , 24 22 20 2 21 22 →= 24 20 2 3 2 21 , 24 22 20 2 21 20 →= 24 20 2 3 2 3 , 30 22 20 2 21 22 →= 30 20 2 3 2 21 , 30 22 20 2 21 20 →= 30 20 2 3 2 3 , 32 22 20 2 21 22 →= 32 20 2 3 2 21 , 32 22 20 2 21 20 →= 32 20 2 3 2 3 , 7 21 20 2 21 22 →= 7 3 2 3 2 21 , 7 21 20 2 21 20 →= 7 3 2 3 2 3 , 5 21 20 2 21 22 →= 5 3 2 3 2 21 , 5 21 20 2 21 20 →= 5 3 2 3 2 3 , 2 21 20 2 21 22 →= 2 3 2 3 2 21 , 2 21 20 2 21 20 →= 2 3 2 3 2 3 , 14 21 20 2 21 22 →= 14 3 2 3 2 21 , 14 21 20 2 21 20 →= 14 3 2 3 2 3 , 18 21 20 2 21 22 →= 18 3 2 3 2 21 , 18 21 20 2 21 20 →= 18 3 2 3 2 3 , 36 21 20 2 21 22 →= 36 3 2 3 2 21 , 36 21 20 2 21 20 →= 36 3 2 3 2 3 , 37 21 20 2 21 22 →= 37 3 2 3 2 21 , 37 21 20 2 21 20 →= 37 3 2 3 2 3 , 34 30 20 2 21 22 →= 34 31 2 3 2 21 , 34 30 20 2 21 20 →= 34 31 2 3 2 3 , 21 22 20 2 3 4 →= 21 20 2 3 4 8 , 22 22 20 2 3 4 →= 22 20 2 3 4 8 , 19 22 20 2 3 4 →= 19 20 2 3 4 8 , 23 22 20 2 3 4 →= 23 20 2 3 4 8 , 24 22 20 2 3 4 →= 24 20 2 3 4 8 , 30 22 20 2 3 4 →= 30 20 2 3 4 8 , 32 22 20 2 3 4 →= 32 20 2 3 4 8 , 7 21 20 2 3 4 →= 7 3 2 3 4 8 , 5 21 20 2 3 4 →= 5 3 2 3 4 8 , 2 21 20 2 3 4 →= 2 3 2 3 4 8 , 14 21 20 2 3 4 →= 14 3 2 3 4 8 , 18 21 20 2 3 4 →= 18 3 2 3 4 8 , 36 21 20 2 3 4 →= 36 3 2 3 4 8 , 37 21 20 2 3 4 →= 37 3 2 3 4 8 , 34 30 20 2 3 4 →= 34 31 2 3 4 8 , 3 4 8 8 5 21 →= 21 20 4 8 5 21 , 3 4 8 8 5 3 →= 21 20 4 8 5 3 , 3 4 8 8 5 38 →= 21 20 4 8 5 38 , 20 4 8 8 5 21 →= 22 20 4 8 5 21 , 20 4 8 8 5 3 →= 22 20 4 8 5 3 , 20 4 8 8 5 38 →= 22 20 4 8 5 38 , 1 4 8 8 5 21 →= 19 20 4 8 5 21 , 1 4 8 8 5 3 →= 19 20 4 8 5 3 , 1 4 8 8 5 38 →= 19 20 4 8 5 38 , 12 4 8 8 5 21 →= 23 20 4 8 5 21 , 12 4 8 8 5 3 →= 23 20 4 8 5 3 , 12 4 8 8 5 38 →= 23 20 4 8 5 38 , 16 4 8 8 5 21 →= 24 20 4 8 5 21 , 16 4 8 8 5 3 →= 24 20 4 8 5 3 , 16 4 8 8 5 38 →= 24 20 4 8 5 38 , 31 4 8 8 5 21 →= 30 20 4 8 5 21 , 31 4 8 8 5 3 →= 30 20 4 8 5 3 , 31 4 8 8 5 38 →= 30 20 4 8 5 38 , 33 4 8 8 5 21 →= 32 20 4 8 5 21 , 33 4 8 8 5 3 →= 32 20 4 8 5 3 , 33 4 8 8 5 38 →= 32 20 4 8 5 38 , 4 8 8 8 5 21 →= 2 3 4 8 5 21 , 4 8 8 8 5 3 →= 2 3 4 8 5 3 , 4 8 8 8 5 38 →= 2 3 4 8 5 38 , 3 4 8 8 8 5 →= 21 20 4 8 8 5 , 3 4 8 8 8 8 →= 21 20 4 8 8 8 , 3 4 8 8 8 9 →= 21 20 4 8 8 9 , 3 4 8 8 8 10 →= 21 20 4 8 8 10 , 20 4 8 8 8 5 →= 22 20 4 8 8 5 , 20 4 8 8 8 8 →= 22 20 4 8 8 8 , 20 4 8 8 8 9 →= 22 20 4 8 8 9 , 20 4 8 8 8 10 →= 22 20 4 8 8 10 , 1 4 8 8 8 5 →= 19 20 4 8 8 5 , 1 4 8 8 8 8 →= 19 20 4 8 8 8 , 1 4 8 8 8 9 →= 19 20 4 8 8 9 , 1 4 8 8 8 10 →= 19 20 4 8 8 10 , 12 4 8 8 8 5 →= 23 20 4 8 8 5 , 12 4 8 8 8 8 →= 23 20 4 8 8 8 , 12 4 8 8 8 9 →= 23 20 4 8 8 9 , 12 4 8 8 8 10 →= 23 20 4 8 8 10 , 16 4 8 8 8 5 →= 24 20 4 8 8 5 , 16 4 8 8 8 8 →= 24 20 4 8 8 8 , 16 4 8 8 8 9 →= 24 20 4 8 8 9 , 16 4 8 8 8 10 →= 24 20 4 8 8 10 , 31 4 8 8 8 5 →= 30 20 4 8 8 5 , 31 4 8 8 8 8 →= 30 20 4 8 8 8 , 31 4 8 8 8 9 →= 30 20 4 8 8 9 , 31 4 8 8 8 10 →= 30 20 4 8 8 10 , 33 4 8 8 8 5 →= 32 20 4 8 8 5 , 33 4 8 8 8 8 →= 32 20 4 8 8 8 , 33 4 8 8 8 9 →= 32 20 4 8 8 9 , 33 4 8 8 8 10 →= 32 20 4 8 8 10 , 4 8 8 8 8 5 →= 2 3 4 8 8 5 , 4 8 8 8 8 8 →= 2 3 4 8 8 8 , 4 8 8 8 8 9 →= 2 3 4 8 8 9 , 4 8 8 8 8 10 →= 2 3 4 8 8 10 , 3 4 8 8 9 39 →= 21 20 4 8 9 39 , 20 4 8 8 9 39 →= 22 20 4 8 9 39 , 1 4 8 8 9 39 →= 19 20 4 8 9 39 , 12 4 8 8 9 39 →= 23 20 4 8 9 39 , 16 4 8 8 9 39 →= 24 20 4 8 9 39 , 31 4 8 8 9 39 →= 30 20 4 8 9 39 , 33 4 8 8 9 39 →= 32 20 4 8 9 39 , 4 8 8 8 9 39 →= 2 3 4 8 9 39 } 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 23 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 24 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 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 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 36 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 37 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 38 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 39 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 19 ↦ 1, 20 ↦ 2, 2 ↦ 3, 21 ↦ 4, 22 ↦ 5, 1 ↦ 6, 3 ↦ 7, 11 ↦ 8, 23 ↦ 9, 12 ↦ 10, 15 ↦ 11, 24 ↦ 12, 16 ↦ 13, 4 ↦ 14, 8 ↦ 15, 25 ↦ 16, 5 ↦ 17, 26 ↦ 18, 27 ↦ 19, 28 ↦ 20, 29 ↦ 21, 9 ↦ 22, 10 ↦ 23, 30 ↦ 24, 31 ↦ 25, 32 ↦ 26, 33 ↦ 27, 7 ↦ 28, 14 ↦ 29, 18 ↦ 30, 36 ↦ 31, 37 ↦ 32, 34 ↦ 33, 38 ↦ 34, 39 ↦ 35 }, it remains to prove termination of the 171-rule system { 0 1 2 3 4 5 ⟶ 0 6 3 7 3 4 , 0 1 2 3 4 2 ⟶ 0 6 3 7 3 7 , 8 9 2 3 4 5 ⟶ 8 10 3 7 3 4 , 8 9 2 3 4 2 ⟶ 8 10 3 7 3 7 , 11 12 2 3 4 5 ⟶ 11 13 3 7 3 4 , 11 12 2 3 4 2 ⟶ 11 13 3 7 3 7 , 0 1 2 3 7 14 ⟶ 0 6 3 7 14 15 , 8 9 2 3 7 14 ⟶ 8 10 3 7 14 15 , 11 12 2 3 7 14 ⟶ 11 13 3 7 14 15 , 16 17 4 5 5 5 →= 16 15 17 7 3 4 , 16 17 4 5 5 2 →= 16 15 17 7 3 7 , 15 17 4 5 5 5 →= 15 15 17 7 3 4 , 15 17 4 5 5 2 →= 15 15 17 7 3 7 , 14 17 4 5 5 5 →= 14 15 17 7 3 4 , 14 17 4 5 5 2 →= 14 15 17 7 3 7 , 18 17 4 5 5 5 →= 18 15 17 7 3 4 , 18 17 4 5 5 2 →= 18 15 17 7 3 7 , 19 17 4 5 5 5 →= 19 15 17 7 3 4 , 19 17 4 5 5 2 →= 19 15 17 7 3 7 , 20 17 4 5 5 5 →= 20 15 17 7 3 4 , 20 17 4 5 5 2 →= 20 15 17 7 3 7 , 21 17 4 5 5 5 →= 21 15 17 7 3 4 , 21 17 4 5 5 2 →= 21 15 17 7 3 7 , 16 17 4 5 2 14 →= 16 15 17 7 14 15 , 15 17 4 5 2 14 →= 15 15 17 7 14 15 , 14 17 4 5 2 14 →= 14 15 17 7 14 15 , 18 17 4 5 2 14 →= 18 15 17 7 14 15 , 19 17 4 5 2 14 →= 19 15 17 7 14 15 , 20 17 4 5 2 14 →= 20 15 17 7 14 15 , 21 17 4 5 2 14 →= 21 15 17 7 14 15 , 4 2 3 7 14 17 →= 7 3 7 14 15 17 , 4 2 3 7 14 15 →= 7 3 7 14 15 15 , 4 2 3 7 14 22 →= 7 3 7 14 15 22 , 4 2 3 7 14 23 →= 7 3 7 14 15 23 , 5 2 3 7 14 17 →= 2 3 7 14 15 17 , 5 2 3 7 14 15 →= 2 3 7 14 15 15 , 5 2 3 7 14 22 →= 2 3 7 14 15 22 , 5 2 3 7 14 23 →= 2 3 7 14 15 23 , 1 2 3 7 14 17 →= 6 3 7 14 15 17 , 1 2 3 7 14 15 →= 6 3 7 14 15 15 , 1 2 3 7 14 22 →= 6 3 7 14 15 22 , 1 2 3 7 14 23 →= 6 3 7 14 15 23 , 9 2 3 7 14 17 →= 10 3 7 14 15 17 , 9 2 3 7 14 15 →= 10 3 7 14 15 15 , 9 2 3 7 14 22 →= 10 3 7 14 15 22 , 9 2 3 7 14 23 →= 10 3 7 14 15 23 , 12 2 3 7 14 17 →= 13 3 7 14 15 17 , 12 2 3 7 14 15 →= 13 3 7 14 15 15 , 12 2 3 7 14 22 →= 13 3 7 14 15 22 , 12 2 3 7 14 23 →= 13 3 7 14 15 23 , 24 2 3 7 14 17 →= 25 3 7 14 15 17 , 24 2 3 7 14 15 →= 25 3 7 14 15 15 , 24 2 3 7 14 22 →= 25 3 7 14 15 22 , 24 2 3 7 14 23 →= 25 3 7 14 15 23 , 26 2 3 7 14 17 →= 27 3 7 14 15 17 , 26 2 3 7 14 15 →= 27 3 7 14 15 15 , 26 2 3 7 14 22 →= 27 3 7 14 15 22 , 26 2 3 7 14 23 →= 27 3 7 14 15 23 , 17 7 3 7 14 17 →= 15 17 7 14 15 17 , 17 7 3 7 14 15 →= 15 17 7 14 15 15 , 17 7 3 7 14 22 →= 15 17 7 14 15 22 , 17 7 3 7 14 23 →= 15 17 7 14 15 23 , 4 5 2 3 4 5 →= 4 2 3 7 3 4 , 4 5 2 3 4 2 →= 4 2 3 7 3 7 , 5 5 2 3 4 5 →= 5 2 3 7 3 4 , 5 5 2 3 4 2 →= 5 2 3 7 3 7 , 1 5 2 3 4 5 →= 1 2 3 7 3 4 , 1 5 2 3 4 2 →= 1 2 3 7 3 7 , 9 5 2 3 4 5 →= 9 2 3 7 3 4 , 9 5 2 3 4 2 →= 9 2 3 7 3 7 , 12 5 2 3 4 5 →= 12 2 3 7 3 4 , 12 5 2 3 4 2 →= 12 2 3 7 3 7 , 24 5 2 3 4 5 →= 24 2 3 7 3 4 , 24 5 2 3 4 2 →= 24 2 3 7 3 7 , 26 5 2 3 4 5 →= 26 2 3 7 3 4 , 26 5 2 3 4 2 →= 26 2 3 7 3 7 , 28 4 2 3 4 5 →= 28 7 3 7 3 4 , 28 4 2 3 4 2 →= 28 7 3 7 3 7 , 17 4 2 3 4 5 →= 17 7 3 7 3 4 , 17 4 2 3 4 2 →= 17 7 3 7 3 7 , 3 4 2 3 4 5 →= 3 7 3 7 3 4 , 3 4 2 3 4 2 →= 3 7 3 7 3 7 , 29 4 2 3 4 5 →= 29 7 3 7 3 4 , 29 4 2 3 4 2 →= 29 7 3 7 3 7 , 30 4 2 3 4 5 →= 30 7 3 7 3 4 , 30 4 2 3 4 2 →= 30 7 3 7 3 7 , 31 4 2 3 4 5 →= 31 7 3 7 3 4 , 31 4 2 3 4 2 →= 31 7 3 7 3 7 , 32 4 2 3 4 5 →= 32 7 3 7 3 4 , 32 4 2 3 4 2 →= 32 7 3 7 3 7 , 33 24 2 3 4 5 →= 33 25 3 7 3 4 , 33 24 2 3 4 2 →= 33 25 3 7 3 7 , 4 5 2 3 7 14 →= 4 2 3 7 14 15 , 5 5 2 3 7 14 →= 5 2 3 7 14 15 , 1 5 2 3 7 14 →= 1 2 3 7 14 15 , 9 5 2 3 7 14 →= 9 2 3 7 14 15 , 12 5 2 3 7 14 →= 12 2 3 7 14 15 , 24 5 2 3 7 14 →= 24 2 3 7 14 15 , 26 5 2 3 7 14 →= 26 2 3 7 14 15 , 28 4 2 3 7 14 →= 28 7 3 7 14 15 , 17 4 2 3 7 14 →= 17 7 3 7 14 15 , 3 4 2 3 7 14 →= 3 7 3 7 14 15 , 29 4 2 3 7 14 →= 29 7 3 7 14 15 , 30 4 2 3 7 14 →= 30 7 3 7 14 15 , 31 4 2 3 7 14 →= 31 7 3 7 14 15 , 32 4 2 3 7 14 →= 32 7 3 7 14 15 , 33 24 2 3 7 14 →= 33 25 3 7 14 15 , 7 14 15 15 17 4 →= 4 2 14 15 17 4 , 7 14 15 15 17 7 →= 4 2 14 15 17 7 , 7 14 15 15 17 34 →= 4 2 14 15 17 34 , 2 14 15 15 17 4 →= 5 2 14 15 17 4 , 2 14 15 15 17 7 →= 5 2 14 15 17 7 , 2 14 15 15 17 34 →= 5 2 14 15 17 34 , 6 14 15 15 17 4 →= 1 2 14 15 17 4 , 6 14 15 15 17 7 →= 1 2 14 15 17 7 , 6 14 15 15 17 34 →= 1 2 14 15 17 34 , 10 14 15 15 17 4 →= 9 2 14 15 17 4 , 10 14 15 15 17 7 →= 9 2 14 15 17 7 , 10 14 15 15 17 34 →= 9 2 14 15 17 34 , 13 14 15 15 17 4 →= 12 2 14 15 17 4 , 13 14 15 15 17 7 →= 12 2 14 15 17 7 , 13 14 15 15 17 34 →= 12 2 14 15 17 34 , 25 14 15 15 17 4 →= 24 2 14 15 17 4 , 25 14 15 15 17 7 →= 24 2 14 15 17 7 , 25 14 15 15 17 34 →= 24 2 14 15 17 34 , 27 14 15 15 17 4 →= 26 2 14 15 17 4 , 27 14 15 15 17 7 →= 26 2 14 15 17 7 , 27 14 15 15 17 34 →= 26 2 14 15 17 34 , 14 15 15 15 17 4 →= 3 7 14 15 17 4 , 14 15 15 15 17 7 →= 3 7 14 15 17 7 , 14 15 15 15 17 34 →= 3 7 14 15 17 34 , 7 14 15 15 15 17 →= 4 2 14 15 15 17 , 7 14 15 15 15 15 →= 4 2 14 15 15 15 , 7 14 15 15 15 22 →= 4 2 14 15 15 22 , 7 14 15 15 15 23 →= 4 2 14 15 15 23 , 2 14 15 15 15 17 →= 5 2 14 15 15 17 , 2 14 15 15 15 15 →= 5 2 14 15 15 15 , 2 14 15 15 15 22 →= 5 2 14 15 15 22 , 2 14 15 15 15 23 →= 5 2 14 15 15 23 , 6 14 15 15 15 17 →= 1 2 14 15 15 17 , 6 14 15 15 15 15 →= 1 2 14 15 15 15 , 6 14 15 15 15 22 →= 1 2 14 15 15 22 , 6 14 15 15 15 23 →= 1 2 14 15 15 23 , 10 14 15 15 15 17 →= 9 2 14 15 15 17 , 10 14 15 15 15 15 →= 9 2 14 15 15 15 , 10 14 15 15 15 22 →= 9 2 14 15 15 22 , 10 14 15 15 15 23 →= 9 2 14 15 15 23 , 13 14 15 15 15 17 →= 12 2 14 15 15 17 , 13 14 15 15 15 15 →= 12 2 14 15 15 15 , 13 14 15 15 15 22 →= 12 2 14 15 15 22 , 13 14 15 15 15 23 →= 12 2 14 15 15 23 , 25 14 15 15 15 17 →= 24 2 14 15 15 17 , 25 14 15 15 15 15 →= 24 2 14 15 15 15 , 25 14 15 15 15 22 →= 24 2 14 15 15 22 , 25 14 15 15 15 23 →= 24 2 14 15 15 23 , 27 14 15 15 15 17 →= 26 2 14 15 15 17 , 27 14 15 15 15 15 →= 26 2 14 15 15 15 , 27 14 15 15 15 22 →= 26 2 14 15 15 22 , 27 14 15 15 15 23 →= 26 2 14 15 15 23 , 14 15 15 15 15 17 →= 3 7 14 15 15 17 , 14 15 15 15 15 15 →= 3 7 14 15 15 15 , 14 15 15 15 15 22 →= 3 7 14 15 15 22 , 14 15 15 15 15 23 →= 3 7 14 15 15 23 , 7 14 15 15 22 35 →= 4 2 14 15 22 35 , 2 14 15 15 22 35 →= 5 2 14 15 22 35 , 6 14 15 15 22 35 →= 1 2 14 15 22 35 , 10 14 15 15 22 35 →= 9 2 14 15 22 35 , 13 14 15 15 22 35 →= 12 2 14 15 22 35 , 25 14 15 15 22 35 →= 24 2 14 15 22 35 , 27 14 15 15 22 35 →= 26 2 14 15 22 35 , 14 15 15 15 22 35 →= 3 7 14 15 22 35 } Applying sparse untiling TROCU(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { 15 ↦ 0, 17 ↦ 1, 4 ↦ 2, 5 ↦ 3, 7 ↦ 4, 3 ↦ 5, 2 ↦ 6, 14 ↦ 7, 1 ↦ 8, 6 ↦ 9, 9 ↦ 10, 10 ↦ 11, 12 ↦ 12, 13 ↦ 13, 24 ↦ 14, 25 ↦ 15, 26 ↦ 16, 27 ↦ 17, 28 ↦ 18, 29 ↦ 19, 30 ↦ 20, 31 ↦ 21, 32 ↦ 22, 34 ↦ 23, 22 ↦ 24, 23 ↦ 25, 35 ↦ 26 }, it remains to prove termination of the 77-rule system { 0 1 2 3 3 3 →= 0 0 1 4 5 2 , 0 1 2 3 3 6 →= 0 0 1 4 5 4 , 0 1 2 3 6 7 →= 0 0 1 4 7 0 , 2 6 5 4 7 0 →= 4 5 4 7 0 0 , 3 6 5 4 7 0 →= 6 5 4 7 0 0 , 8 6 5 4 7 0 →= 9 5 4 7 0 0 , 10 6 5 4 7 0 →= 11 5 4 7 0 0 , 12 6 5 4 7 0 →= 13 5 4 7 0 0 , 14 6 5 4 7 0 →= 15 5 4 7 0 0 , 16 6 5 4 7 0 →= 17 5 4 7 0 0 , 1 4 5 4 7 0 →= 0 1 4 7 0 0 , 2 3 6 5 2 3 →= 2 6 5 4 5 2 , 2 3 6 5 2 6 →= 2 6 5 4 5 4 , 3 3 6 5 2 3 →= 3 6 5 4 5 2 , 3 3 6 5 2 6 →= 3 6 5 4 5 4 , 8 3 6 5 2 3 →= 8 6 5 4 5 2 , 8 3 6 5 2 6 →= 8 6 5 4 5 4 , 10 3 6 5 2 3 →= 10 6 5 4 5 2 , 10 3 6 5 2 6 →= 10 6 5 4 5 4 , 12 3 6 5 2 3 →= 12 6 5 4 5 2 , 12 3 6 5 2 6 →= 12 6 5 4 5 4 , 14 3 6 5 2 3 →= 14 6 5 4 5 2 , 14 3 6 5 2 6 →= 14 6 5 4 5 4 , 16 3 6 5 2 3 →= 16 6 5 4 5 2 , 16 3 6 5 2 6 →= 16 6 5 4 5 4 , 18 2 6 5 2 3 →= 18 4 5 4 5 2 , 18 2 6 5 2 6 →= 18 4 5 4 5 4 , 1 2 6 5 2 3 →= 1 4 5 4 5 2 , 1 2 6 5 2 6 →= 1 4 5 4 5 4 , 5 2 6 5 2 3 →= 5 4 5 4 5 2 , 5 2 6 5 2 6 →= 5 4 5 4 5 4 , 19 2 6 5 2 3 →= 19 4 5 4 5 2 , 19 2 6 5 2 6 →= 19 4 5 4 5 4 , 20 2 6 5 2 3 →= 20 4 5 4 5 2 , 20 2 6 5 2 6 →= 20 4 5 4 5 4 , 21 2 6 5 2 3 →= 21 4 5 4 5 2 , 21 2 6 5 2 6 →= 21 4 5 4 5 4 , 22 2 6 5 2 3 →= 22 4 5 4 5 2 , 22 2 6 5 2 6 →= 22 4 5 4 5 4 , 2 3 6 5 4 7 →= 2 6 5 4 7 0 , 3 3 6 5 4 7 →= 3 6 5 4 7 0 , 8 3 6 5 4 7 →= 8 6 5 4 7 0 , 10 3 6 5 4 7 →= 10 6 5 4 7 0 , 12 3 6 5 4 7 →= 12 6 5 4 7 0 , 14 3 6 5 4 7 →= 14 6 5 4 7 0 , 16 3 6 5 4 7 →= 16 6 5 4 7 0 , 18 2 6 5 4 7 →= 18 4 5 4 7 0 , 1 2 6 5 4 7 →= 1 4 5 4 7 0 , 5 2 6 5 4 7 →= 5 4 5 4 7 0 , 19 2 6 5 4 7 →= 19 4 5 4 7 0 , 20 2 6 5 4 7 →= 20 4 5 4 7 0 , 21 2 6 5 4 7 →= 21 4 5 4 7 0 , 22 2 6 5 4 7 →= 22 4 5 4 7 0 , 4 7 0 0 1 2 →= 2 6 7 0 1 2 , 4 7 0 0 1 4 →= 2 6 7 0 1 4 , 4 7 0 0 1 23 →= 2 6 7 0 1 23 , 6 7 0 0 1 2 →= 3 6 7 0 1 2 , 6 7 0 0 1 4 →= 3 6 7 0 1 4 , 6 7 0 0 1 23 →= 3 6 7 0 1 23 , 7 0 0 0 1 2 →= 5 4 7 0 1 2 , 7 0 0 0 1 4 →= 5 4 7 0 1 4 , 7 0 0 0 1 23 →= 5 4 7 0 1 23 , 4 7 0 0 0 1 →= 2 6 7 0 0 1 , 4 7 0 0 0 0 →= 2 6 7 0 0 0 , 4 7 0 0 0 24 →= 2 6 7 0 0 24 , 4 7 0 0 0 25 →= 2 6 7 0 0 25 , 6 7 0 0 0 1 →= 3 6 7 0 0 1 , 6 7 0 0 0 0 →= 3 6 7 0 0 0 , 6 7 0 0 0 24 →= 3 6 7 0 0 24 , 6 7 0 0 0 25 →= 3 6 7 0 0 25 , 7 0 0 0 0 1 →= 5 4 7 0 0 1 , 7 0 0 0 0 0 →= 5 4 7 0 0 0 , 7 0 0 0 0 24 →= 5 4 7 0 0 24 , 7 0 0 0 0 25 →= 5 4 7 0 0 25 , 4 7 0 0 24 26 →= 2 6 7 0 24 26 , 6 7 0 0 24 26 →= 3 6 7 0 24 26 , 7 0 0 0 24 26 →= 5 4 7 0 24 26 } The system is trivially terminating.