/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, c ↦ 2 }, it remains to prove termination of the 4-rule system { 0 1 0 ⟶ 1 0 1 , 0 0 1 ⟶ 0 0 0 , 2 1 ⟶ 1 0 2 , 2 0 ⟶ 0 1 2 } Applying sparse untiling TRFCU(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1 }, it remains to prove termination of the 2-rule system { 0 1 0 ⟶ 1 0 1 , 0 0 1 ⟶ 0 0 0 } Applying sparse tiling TRFC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (0,0) ↦ 0, (0,1) ↦ 1, (1,0) ↦ 2, (1,1) ↦ 3, (0,3) ↦ 4, (1,3) ↦ 5, (2,0) ↦ 6, (2,1) ↦ 7 }, it remains to prove termination of the 18-rule system { 0 1 2 0 ⟶ 1 2 1 2 , 0 1 2 1 ⟶ 1 2 1 3 , 0 1 2 4 ⟶ 1 2 1 5 , 2 1 2 0 ⟶ 3 2 1 2 , 2 1 2 1 ⟶ 3 2 1 3 , 2 1 2 4 ⟶ 3 2 1 5 , 6 1 2 0 ⟶ 7 2 1 2 , 6 1 2 1 ⟶ 7 2 1 3 , 6 1 2 4 ⟶ 7 2 1 5 , 0 0 1 2 ⟶ 0 0 0 0 , 0 0 1 3 ⟶ 0 0 0 1 , 0 0 1 5 ⟶ 0 0 0 4 , 2 0 1 2 ⟶ 2 0 0 0 , 2 0 1 3 ⟶ 2 0 0 1 , 2 0 1 5 ⟶ 2 0 0 4 , 6 0 1 2 ⟶ 6 0 0 0 , 6 0 1 3 ⟶ 6 0 0 1 , 6 0 1 5 ⟶ 6 0 0 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 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 2 0 ⟶ 1 2 1 2 , 0 1 2 1 ⟶ 1 2 1 3 , 0 1 2 4 ⟶ 1 2 1 5 , 2 1 2 0 ⟶ 3 2 1 2 , 2 1 2 1 ⟶ 3 2 1 3 , 2 1 2 4 ⟶ 3 2 1 5 , 0 0 1 2 ⟶ 0 0 0 0 , 0 0 1 3 ⟶ 0 0 0 1 , 0 0 1 5 ⟶ 0 0 0 4 , 2 0 1 2 ⟶ 2 0 0 0 , 2 0 1 3 ⟶ 2 0 0 1 , 2 0 1 5 ⟶ 2 0 0 4 , 6 0 1 2 ⟶ 6 0 0 0 , 6 0 1 3 ⟶ 6 0 0 1 , 6 0 1 5 ⟶ 6 0 0 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 5: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6 }, it remains to prove termination of the 14-rule system { 0 1 2 0 ⟶ 1 2 1 2 , 0 1 2 1 ⟶ 1 2 1 3 , 0 1 2 4 ⟶ 1 2 1 5 , 2 1 2 0 ⟶ 3 2 1 2 , 2 1 2 1 ⟶ 3 2 1 3 , 2 1 2 4 ⟶ 3 2 1 5 , 0 0 1 2 ⟶ 0 0 0 0 , 0 0 1 3 ⟶ 0 0 0 1 , 0 0 1 5 ⟶ 0 0 0 4 , 2 0 1 2 ⟶ 2 0 0 0 , 2 0 1 3 ⟶ 2 0 0 1 , 2 0 1 5 ⟶ 2 0 0 4 , 6 0 1 2 ⟶ 6 0 0 0 , 6 0 1 3 ⟶ 6 0 0 1 } The system was reversed. After renaming modulo the bijection { 0 ↦ 0, 2 ↦ 1, 1 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6 }, it remains to prove termination of the 14-rule system { 0 1 2 0 ⟶ 1 2 1 2 , 2 1 2 0 ⟶ 3 2 1 2 , 4 1 2 0 ⟶ 5 2 1 2 , 0 1 2 1 ⟶ 1 2 1 3 , 2 1 2 1 ⟶ 3 2 1 3 , 4 1 2 1 ⟶ 5 2 1 3 , 1 2 0 0 ⟶ 0 0 0 0 , 3 2 0 0 ⟶ 2 0 0 0 , 5 2 0 0 ⟶ 4 0 0 0 , 1 2 0 1 ⟶ 0 0 0 1 , 3 2 0 1 ⟶ 2 0 0 1 , 5 2 0 1 ⟶ 4 0 0 1 , 1 2 0 6 ⟶ 0 0 0 6 , 3 2 0 6 ⟶ 2 0 0 6 } Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols. After renaming modulo the bijection { (0,↑) ↦ 0, (1,↓) ↦ 1, (2,↓) ↦ 2, (0,↓) ↦ 3, (1,↑) ↦ 4, (2,↑) ↦ 5, (3,↑) ↦ 6, (4,↑) ↦ 7, (5,↑) ↦ 8, (3,↓) ↦ 9, (6,↓) ↦ 10, (4,↓) ↦ 11, (5,↓) ↦ 12 }, it remains to prove termination of the 54-rule system { 0 1 2 3 ⟶ 4 2 1 2 , 0 1 2 3 ⟶ 5 1 2 , 0 1 2 3 ⟶ 4 2 , 0 1 2 3 ⟶ 5 , 5 1 2 3 ⟶ 6 2 1 2 , 5 1 2 3 ⟶ 5 1 2 , 5 1 2 3 ⟶ 4 2 , 5 1 2 3 ⟶ 5 , 7 1 2 3 ⟶ 8 2 1 2 , 7 1 2 3 ⟶ 5 1 2 , 7 1 2 3 ⟶ 4 2 , 7 1 2 3 ⟶ 5 , 0 1 2 1 ⟶ 4 2 1 9 , 0 1 2 1 ⟶ 5 1 9 , 0 1 2 1 ⟶ 4 9 , 0 1 2 1 ⟶ 6 , 5 1 2 1 ⟶ 6 2 1 9 , 5 1 2 1 ⟶ 5 1 9 , 5 1 2 1 ⟶ 4 9 , 5 1 2 1 ⟶ 6 , 7 1 2 1 ⟶ 8 2 1 9 , 7 1 2 1 ⟶ 5 1 9 , 7 1 2 1 ⟶ 4 9 , 7 1 2 1 ⟶ 6 , 4 2 3 3 ⟶ 0 3 3 3 , 4 2 3 3 ⟶ 0 3 3 , 6 2 3 3 ⟶ 5 3 3 3 , 6 2 3 3 ⟶ 0 3 3 , 8 2 3 3 ⟶ 7 3 3 3 , 8 2 3 3 ⟶ 0 3 3 , 4 2 3 1 ⟶ 0 3 3 1 , 4 2 3 1 ⟶ 0 3 1 , 6 2 3 1 ⟶ 5 3 3 1 , 6 2 3 1 ⟶ 0 3 1 , 8 2 3 1 ⟶ 7 3 3 1 , 8 2 3 1 ⟶ 0 3 1 , 4 2 3 10 ⟶ 0 3 3 10 , 4 2 3 10 ⟶ 0 3 10 , 6 2 3 10 ⟶ 5 3 3 10 , 6 2 3 10 ⟶ 0 3 10 , 3 1 2 3 →= 1 2 1 2 , 2 1 2 3 →= 9 2 1 2 , 11 1 2 3 →= 12 2 1 2 , 3 1 2 1 →= 1 2 1 9 , 2 1 2 1 →= 9 2 1 9 , 11 1 2 1 →= 12 2 1 9 , 1 2 3 3 →= 3 3 3 3 , 9 2 3 3 →= 2 3 3 3 , 12 2 3 3 →= 11 3 3 3 , 1 2 3 1 →= 3 3 3 1 , 9 2 3 1 →= 2 3 3 1 , 12 2 3 1 →= 11 3 3 1 , 1 2 3 10 →= 3 3 3 10 , 9 2 3 10 →= 2 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, 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 2 3 ⟶ 4 2 1 2 , 5 1 2 3 ⟶ 6 2 1 2 , 7 1 2 3 ⟶ 8 2 1 2 , 0 1 2 1 ⟶ 4 2 1 9 , 5 1 2 1 ⟶ 6 2 1 9 , 7 1 2 1 ⟶ 8 2 1 9 , 4 2 3 3 ⟶ 0 3 3 3 , 6 2 3 3 ⟶ 5 3 3 3 , 8 2 3 3 ⟶ 7 3 3 3 , 4 2 3 1 ⟶ 0 3 3 1 , 6 2 3 1 ⟶ 5 3 3 1 , 8 2 3 1 ⟶ 7 3 3 1 , 4 2 3 10 ⟶ 0 3 3 10 , 6 2 3 10 ⟶ 5 3 3 10 , 3 1 2 3 →= 1 2 1 2 , 2 1 2 3 →= 9 2 1 2 , 11 1 2 3 →= 12 2 1 2 , 3 1 2 1 →= 1 2 1 9 , 2 1 2 1 →= 9 2 1 9 , 11 1 2 1 →= 12 2 1 9 , 1 2 3 3 →= 3 3 3 3 , 9 2 3 3 →= 2 3 3 3 , 12 2 3 3 →= 11 3 3 3 , 1 2 3 1 →= 3 3 3 1 , 9 2 3 1 →= 2 3 3 1 , 12 2 3 1 →= 11 3 3 1 , 1 2 3 10 →= 3 3 3 10 , 9 2 3 10 →= 2 3 3 10 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 5: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 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 2 3 ⟶ 4 2 1 2 , 5 1 2 3 ⟶ 6 2 1 2 , 7 1 2 3 ⟶ 8 2 1 2 , 0 1 2 1 ⟶ 4 2 1 9 , 5 1 2 1 ⟶ 6 2 1 9 , 7 1 2 1 ⟶ 8 2 1 9 , 4 2 3 3 ⟶ 0 3 3 3 , 6 2 3 3 ⟶ 5 3 3 3 , 8 2 3 3 ⟶ 7 3 3 3 , 4 2 3 1 ⟶ 0 3 3 1 , 6 2 3 1 ⟶ 5 3 3 1 , 8 2 3 1 ⟶ 7 3 3 1 , 6 2 3 10 ⟶ 5 3 3 10 , 3 1 2 3 →= 1 2 1 2 , 2 1 2 3 →= 9 2 1 2 , 11 1 2 3 →= 12 2 1 2 , 3 1 2 1 →= 1 2 1 9 , 2 1 2 1 →= 9 2 1 9 , 11 1 2 1 →= 12 2 1 9 , 1 2 3 3 →= 3 3 3 3 , 9 2 3 3 →= 2 3 3 3 , 12 2 3 3 →= 11 3 3 3 , 1 2 3 1 →= 3 3 3 1 , 9 2 3 1 →= 2 3 3 1 , 12 2 3 1 →= 11 3 3 1 , 1 2 3 10 →= 3 3 3 10 , 9 2 3 10 →= 2 3 3 10 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 5: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 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 2 3 ⟶ 4 2 1 2 , 5 1 2 3 ⟶ 6 2 1 2 , 7 1 2 3 ⟶ 8 2 1 2 , 0 1 2 1 ⟶ 4 2 1 9 , 5 1 2 1 ⟶ 6 2 1 9 , 7 1 2 1 ⟶ 8 2 1 9 , 4 2 3 3 ⟶ 0 3 3 3 , 6 2 3 3 ⟶ 5 3 3 3 , 8 2 3 3 ⟶ 7 3 3 3 , 4 2 3 1 ⟶ 0 3 3 1 , 6 2 3 1 ⟶ 5 3 3 1 , 8 2 3 1 ⟶ 7 3 3 1 , 3 1 2 3 →= 1 2 1 2 , 2 1 2 3 →= 9 2 1 2 , 10 1 2 3 →= 11 2 1 2 , 3 1 2 1 →= 1 2 1 9 , 2 1 2 1 →= 9 2 1 9 , 10 1 2 1 →= 11 2 1 9 , 1 2 3 3 →= 3 3 3 3 , 9 2 3 3 →= 2 3 3 3 , 11 2 3 3 →= 10 3 3 3 , 1 2 3 1 →= 3 3 3 1 , 9 2 3 1 →= 2 3 3 1 , 11 2 3 1 →= 10 3 3 1 , 1 2 3 12 →= 3 3 3 12 , 9 2 3 12 →= 2 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, (2,3) ↦ 3, (3,1) ↦ 4, (13,4) ↦ 5, (4,2) ↦ 6, (2,1) ↦ 7, (3,3) ↦ 8, (3,12) ↦ 9, (2,12) ↦ 10, (3,14) ↦ 11, (2,14) ↦ 12, (13,5) ↦ 13, (5,1) ↦ 14, (13,6) ↦ 15, (6,2) ↦ 16, (13,7) ↦ 17, (7,1) ↦ 18, (13,8) ↦ 19, (8,2) ↦ 20, (1,9) ↦ 21, (9,2) ↦ 22, (9,9) ↦ 23, (1,14) ↦ 24, (9,14) ↦ 25, (0,3) ↦ 26, (5,3) ↦ 27, (7,3) ↦ 28, (10,3) ↦ 29, (10,1) ↦ 30, (13,3) ↦ 31, (13,1) ↦ 32, (4,9) ↦ 33, (6,9) ↦ 34, (8,9) ↦ 35, (11,2) ↦ 36, (11,9) ↦ 37, (13,2) ↦ 38, (13,9) ↦ 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 6 7 2 7 , 0 1 2 3 8 ⟶ 5 6 7 2 3 , 0 1 2 3 9 ⟶ 5 6 7 2 10 , 0 1 2 3 11 ⟶ 5 6 7 2 12 , 13 14 2 3 4 ⟶ 15 16 7 2 7 , 13 14 2 3 8 ⟶ 15 16 7 2 3 , 13 14 2 3 9 ⟶ 15 16 7 2 10 , 13 14 2 3 11 ⟶ 15 16 7 2 12 , 17 18 2 3 4 ⟶ 19 20 7 2 7 , 17 18 2 3 8 ⟶ 19 20 7 2 3 , 17 18 2 3 9 ⟶ 19 20 7 2 10 , 17 18 2 3 11 ⟶ 19 20 7 2 12 , 0 1 2 7 2 ⟶ 5 6 7 21 22 , 0 1 2 7 21 ⟶ 5 6 7 21 23 , 0 1 2 7 24 ⟶ 5 6 7 21 25 , 13 14 2 7 2 ⟶ 15 16 7 21 22 , 13 14 2 7 21 ⟶ 15 16 7 21 23 , 13 14 2 7 24 ⟶ 15 16 7 21 25 , 17 18 2 7 2 ⟶ 19 20 7 21 22 , 17 18 2 7 21 ⟶ 19 20 7 21 23 , 17 18 2 7 24 ⟶ 19 20 7 21 25 , 5 6 3 8 4 ⟶ 0 26 8 8 4 , 5 6 3 8 8 ⟶ 0 26 8 8 8 , 5 6 3 8 9 ⟶ 0 26 8 8 9 , 5 6 3 8 11 ⟶ 0 26 8 8 11 , 15 16 3 8 4 ⟶ 13 27 8 8 4 , 15 16 3 8 8 ⟶ 13 27 8 8 8 , 15 16 3 8 9 ⟶ 13 27 8 8 9 , 15 16 3 8 11 ⟶ 13 27 8 8 11 , 19 20 3 8 4 ⟶ 17 28 8 8 4 , 19 20 3 8 8 ⟶ 17 28 8 8 8 , 19 20 3 8 9 ⟶ 17 28 8 8 9 , 19 20 3 8 11 ⟶ 17 28 8 8 11 , 5 6 3 4 2 ⟶ 0 26 8 4 2 , 5 6 3 4 21 ⟶ 0 26 8 4 21 , 5 6 3 4 24 ⟶ 0 26 8 4 24 , 15 16 3 4 2 ⟶ 13 27 8 4 2 , 15 16 3 4 21 ⟶ 13 27 8 4 21 , 15 16 3 4 24 ⟶ 13 27 8 4 24 , 19 20 3 4 2 ⟶ 17 28 8 4 2 , 19 20 3 4 21 ⟶ 17 28 8 4 21 , 19 20 3 4 24 ⟶ 17 28 8 4 24 , 26 4 2 3 4 →= 1 2 7 2 7 , 26 4 2 3 8 →= 1 2 7 2 3 , 26 4 2 3 9 →= 1 2 7 2 10 , 26 4 2 3 11 →= 1 2 7 2 12 , 3 4 2 3 4 →= 7 2 7 2 7 , 3 4 2 3 8 →= 7 2 7 2 3 , 3 4 2 3 9 →= 7 2 7 2 10 , 3 4 2 3 11 →= 7 2 7 2 12 , 8 4 2 3 4 →= 4 2 7 2 7 , 8 4 2 3 8 →= 4 2 7 2 3 , 8 4 2 3 9 →= 4 2 7 2 10 , 8 4 2 3 11 →= 4 2 7 2 12 , 27 4 2 3 4 →= 14 2 7 2 7 , 27 4 2 3 8 →= 14 2 7 2 3 , 27 4 2 3 9 →= 14 2 7 2 10 , 27 4 2 3 11 →= 14 2 7 2 12 , 28 4 2 3 4 →= 18 2 7 2 7 , 28 4 2 3 8 →= 18 2 7 2 3 , 28 4 2 3 9 →= 18 2 7 2 10 , 28 4 2 3 11 →= 18 2 7 2 12 , 29 4 2 3 4 →= 30 2 7 2 7 , 29 4 2 3 8 →= 30 2 7 2 3 , 29 4 2 3 9 →= 30 2 7 2 10 , 29 4 2 3 11 →= 30 2 7 2 12 , 31 4 2 3 4 →= 32 2 7 2 7 , 31 4 2 3 8 →= 32 2 7 2 3 , 31 4 2 3 9 →= 32 2 7 2 10 , 31 4 2 3 11 →= 32 2 7 2 12 , 2 7 2 3 4 →= 21 22 7 2 7 , 2 7 2 3 8 →= 21 22 7 2 3 , 2 7 2 3 9 →= 21 22 7 2 10 , 2 7 2 3 11 →= 21 22 7 2 12 , 6 7 2 3 4 →= 33 22 7 2 7 , 6 7 2 3 8 →= 33 22 7 2 3 , 6 7 2 3 9 →= 33 22 7 2 10 , 6 7 2 3 11 →= 33 22 7 2 12 , 16 7 2 3 4 →= 34 22 7 2 7 , 16 7 2 3 8 →= 34 22 7 2 3 , 16 7 2 3 9 →= 34 22 7 2 10 , 16 7 2 3 11 →= 34 22 7 2 12 , 20 7 2 3 4 →= 35 22 7 2 7 , 20 7 2 3 8 →= 35 22 7 2 3 , 20 7 2 3 9 →= 35 22 7 2 10 , 20 7 2 3 11 →= 35 22 7 2 12 , 22 7 2 3 4 →= 23 22 7 2 7 , 22 7 2 3 8 →= 23 22 7 2 3 , 22 7 2 3 9 →= 23 22 7 2 10 , 22 7 2 3 11 →= 23 22 7 2 12 , 36 7 2 3 4 →= 37 22 7 2 7 , 36 7 2 3 8 →= 37 22 7 2 3 , 36 7 2 3 9 →= 37 22 7 2 10 , 36 7 2 3 11 →= 37 22 7 2 12 , 38 7 2 3 4 →= 39 22 7 2 7 , 38 7 2 3 8 →= 39 22 7 2 3 , 38 7 2 3 9 →= 39 22 7 2 10 , 38 7 2 3 11 →= 39 22 7 2 12 , 40 30 2 3 4 →= 41 36 7 2 7 , 40 30 2 3 8 →= 41 36 7 2 3 , 40 30 2 3 9 →= 41 36 7 2 10 , 40 30 2 3 11 →= 41 36 7 2 12 , 26 4 2 7 2 →= 1 2 7 21 22 , 26 4 2 7 21 →= 1 2 7 21 23 , 26 4 2 7 24 →= 1 2 7 21 25 , 3 4 2 7 2 →= 7 2 7 21 22 , 3 4 2 7 21 →= 7 2 7 21 23 , 3 4 2 7 24 →= 7 2 7 21 25 , 8 4 2 7 2 →= 4 2 7 21 22 , 8 4 2 7 21 →= 4 2 7 21 23 , 8 4 2 7 24 →= 4 2 7 21 25 , 27 4 2 7 2 →= 14 2 7 21 22 , 27 4 2 7 21 →= 14 2 7 21 23 , 27 4 2 7 24 →= 14 2 7 21 25 , 28 4 2 7 2 →= 18 2 7 21 22 , 28 4 2 7 21 →= 18 2 7 21 23 , 28 4 2 7 24 →= 18 2 7 21 25 , 29 4 2 7 2 →= 30 2 7 21 22 , 29 4 2 7 21 →= 30 2 7 21 23 , 29 4 2 7 24 →= 30 2 7 21 25 , 31 4 2 7 2 →= 32 2 7 21 22 , 31 4 2 7 21 →= 32 2 7 21 23 , 31 4 2 7 24 →= 32 2 7 21 25 , 2 7 2 7 2 →= 21 22 7 21 22 , 2 7 2 7 21 →= 21 22 7 21 23 , 2 7 2 7 24 →= 21 22 7 21 25 , 6 7 2 7 2 →= 33 22 7 21 22 , 6 7 2 7 21 →= 33 22 7 21 23 , 6 7 2 7 24 →= 33 22 7 21 25 , 16 7 2 7 2 →= 34 22 7 21 22 , 16 7 2 7 21 →= 34 22 7 21 23 , 16 7 2 7 24 →= 34 22 7 21 25 , 20 7 2 7 2 →= 35 22 7 21 22 , 20 7 2 7 21 →= 35 22 7 21 23 , 20 7 2 7 24 →= 35 22 7 21 25 , 22 7 2 7 2 →= 23 22 7 21 22 , 22 7 2 7 21 →= 23 22 7 21 23 , 22 7 2 7 24 →= 23 22 7 21 25 , 36 7 2 7 2 →= 37 22 7 21 22 , 36 7 2 7 21 →= 37 22 7 21 23 , 36 7 2 7 24 →= 37 22 7 21 25 , 38 7 2 7 2 →= 39 22 7 21 22 , 38 7 2 7 21 →= 39 22 7 21 23 , 38 7 2 7 24 →= 39 22 7 21 25 , 40 30 2 7 2 →= 41 36 7 21 22 , 40 30 2 7 21 →= 41 36 7 21 23 , 40 30 2 7 24 →= 41 36 7 21 25 , 1 2 3 8 4 →= 26 8 8 8 4 , 1 2 3 8 8 →= 26 8 8 8 8 , 1 2 3 8 9 →= 26 8 8 8 9 , 1 2 3 8 11 →= 26 8 8 8 11 , 7 2 3 8 4 →= 3 8 8 8 4 , 7 2 3 8 8 →= 3 8 8 8 8 , 7 2 3 8 9 →= 3 8 8 8 9 , 7 2 3 8 11 →= 3 8 8 8 11 , 4 2 3 8 4 →= 8 8 8 8 4 , 4 2 3 8 8 →= 8 8 8 8 8 , 4 2 3 8 9 →= 8 8 8 8 9 , 4 2 3 8 11 →= 8 8 8 8 11 , 14 2 3 8 4 →= 27 8 8 8 4 , 14 2 3 8 8 →= 27 8 8 8 8 , 14 2 3 8 9 →= 27 8 8 8 9 , 14 2 3 8 11 →= 27 8 8 8 11 , 18 2 3 8 4 →= 28 8 8 8 4 , 18 2 3 8 8 →= 28 8 8 8 8 , 18 2 3 8 9 →= 28 8 8 8 9 , 18 2 3 8 11 →= 28 8 8 8 11 , 30 2 3 8 4 →= 29 8 8 8 4 , 30 2 3 8 8 →= 29 8 8 8 8 , 30 2 3 8 9 →= 29 8 8 8 9 , 30 2 3 8 11 →= 29 8 8 8 11 , 32 2 3 8 4 →= 31 8 8 8 4 , 32 2 3 8 8 →= 31 8 8 8 8 , 32 2 3 8 9 →= 31 8 8 8 9 , 32 2 3 8 11 →= 31 8 8 8 11 , 21 22 3 8 4 →= 2 3 8 8 4 , 21 22 3 8 8 →= 2 3 8 8 8 , 21 22 3 8 9 →= 2 3 8 8 9 , 21 22 3 8 11 →= 2 3 8 8 11 , 33 22 3 8 4 →= 6 3 8 8 4 , 33 22 3 8 8 →= 6 3 8 8 8 , 33 22 3 8 9 →= 6 3 8 8 9 , 33 22 3 8 11 →= 6 3 8 8 11 , 34 22 3 8 4 →= 16 3 8 8 4 , 34 22 3 8 8 →= 16 3 8 8 8 , 34 22 3 8 9 →= 16 3 8 8 9 , 34 22 3 8 11 →= 16 3 8 8 11 , 35 22 3 8 4 →= 20 3 8 8 4 , 35 22 3 8 8 →= 20 3 8 8 8 , 35 22 3 8 9 →= 20 3 8 8 9 , 35 22 3 8 11 →= 20 3 8 8 11 , 23 22 3 8 4 →= 22 3 8 8 4 , 23 22 3 8 8 →= 22 3 8 8 8 , 23 22 3 8 9 →= 22 3 8 8 9 , 23 22 3 8 11 →= 22 3 8 8 11 , 37 22 3 8 4 →= 36 3 8 8 4 , 37 22 3 8 8 →= 36 3 8 8 8 , 37 22 3 8 9 →= 36 3 8 8 9 , 37 22 3 8 11 →= 36 3 8 8 11 , 39 22 3 8 4 →= 38 3 8 8 4 , 39 22 3 8 8 →= 38 3 8 8 8 , 39 22 3 8 9 →= 38 3 8 8 9 , 39 22 3 8 11 →= 38 3 8 8 11 , 41 36 3 8 4 →= 40 29 8 8 4 , 41 36 3 8 8 →= 40 29 8 8 8 , 41 36 3 8 9 →= 40 29 8 8 9 , 41 36 3 8 11 →= 40 29 8 8 11 , 1 2 3 4 2 →= 26 8 8 4 2 , 1 2 3 4 21 →= 26 8 8 4 21 , 1 2 3 4 24 →= 26 8 8 4 24 , 7 2 3 4 2 →= 3 8 8 4 2 , 7 2 3 4 21 →= 3 8 8 4 21 , 7 2 3 4 24 →= 3 8 8 4 24 , 4 2 3 4 2 →= 8 8 8 4 2 , 4 2 3 4 21 →= 8 8 8 4 21 , 4 2 3 4 24 →= 8 8 8 4 24 , 14 2 3 4 2 →= 27 8 8 4 2 , 14 2 3 4 21 →= 27 8 8 4 21 , 14 2 3 4 24 →= 27 8 8 4 24 , 18 2 3 4 2 →= 28 8 8 4 2 , 18 2 3 4 21 →= 28 8 8 4 21 , 18 2 3 4 24 →= 28 8 8 4 24 , 30 2 3 4 2 →= 29 8 8 4 2 , 30 2 3 4 21 →= 29 8 8 4 21 , 30 2 3 4 24 →= 29 8 8 4 24 , 32 2 3 4 2 →= 31 8 8 4 2 , 32 2 3 4 21 →= 31 8 8 4 21 , 32 2 3 4 24 →= 31 8 8 4 24 , 21 22 3 4 2 →= 2 3 8 4 2 , 21 22 3 4 21 →= 2 3 8 4 21 , 21 22 3 4 24 →= 2 3 8 4 24 , 33 22 3 4 2 →= 6 3 8 4 2 , 33 22 3 4 21 →= 6 3 8 4 21 , 33 22 3 4 24 →= 6 3 8 4 24 , 34 22 3 4 2 →= 16 3 8 4 2 , 34 22 3 4 21 →= 16 3 8 4 21 , 34 22 3 4 24 →= 16 3 8 4 24 , 35 22 3 4 2 →= 20 3 8 4 2 , 35 22 3 4 21 →= 20 3 8 4 21 , 35 22 3 4 24 →= 20 3 8 4 24 , 23 22 3 4 2 →= 22 3 8 4 2 , 23 22 3 4 21 →= 22 3 8 4 21 , 23 22 3 4 24 →= 22 3 8 4 24 , 37 22 3 4 2 →= 36 3 8 4 2 , 37 22 3 4 21 →= 36 3 8 4 21 , 37 22 3 4 24 →= 36 3 8 4 24 , 39 22 3 4 2 →= 38 3 8 4 2 , 39 22 3 4 21 →= 38 3 8 4 21 , 39 22 3 4 24 →= 38 3 8 4 24 , 41 36 3 4 2 →= 40 29 8 4 2 , 41 36 3 4 21 →= 40 29 8 4 21 , 41 36 3 4 24 →= 40 29 8 4 24 , 1 2 3 9 42 →= 26 8 8 9 42 , 7 2 3 9 42 →= 3 8 8 9 42 , 4 2 3 9 42 →= 8 8 8 9 42 , 14 2 3 9 42 →= 27 8 8 9 42 , 18 2 3 9 42 →= 28 8 8 9 42 , 30 2 3 9 42 →= 29 8 8 9 42 , 32 2 3 9 42 →= 31 8 8 9 42 , 21 22 3 9 42 →= 2 3 8 9 42 , 33 22 3 9 42 →= 6 3 8 9 42 , 34 22 3 9 42 →= 16 3 8 9 42 , 35 22 3 9 42 →= 20 3 8 9 42 , 23 22 3 9 42 →= 22 3 8 9 42 , 37 22 3 9 42 →= 36 3 8 9 42 , 39 22 3 9 42 →= 38 3 8 9 42 } 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 23 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 24 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 25 ↦ ⎛ ⎞ ⎜ 1 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 40 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 41 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 42 ↦ ⎛ ⎞ ⎜ 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, 13 ↦ 9, 14 ↦ 10, 15 ↦ 11, 16 ↦ 12, 17 ↦ 13, 18 ↦ 14, 19 ↦ 15, 20 ↦ 16, 21 ↦ 17, 22 ↦ 18, 23 ↦ 19, 26 ↦ 20, 9 ↦ 21, 11 ↦ 22, 27 ↦ 23, 28 ↦ 24, 24 ↦ 25, 29 ↦ 26, 30 ↦ 27, 31 ↦ 28, 32 ↦ 29, 33 ↦ 30, 34 ↦ 31, 35 ↦ 32, 36 ↦ 33, 37 ↦ 34, 38 ↦ 35, 39 ↦ 36, 40 ↦ 37, 41 ↦ 38, 42 ↦ 39 }, it remains to prove termination of the 212-rule system { 0 1 2 3 4 ⟶ 5 6 7 2 7 , 0 1 2 3 8 ⟶ 5 6 7 2 3 , 9 10 2 3 4 ⟶ 11 12 7 2 7 , 9 10 2 3 8 ⟶ 11 12 7 2 3 , 13 14 2 3 4 ⟶ 15 16 7 2 7 , 13 14 2 3 8 ⟶ 15 16 7 2 3 , 0 1 2 7 2 ⟶ 5 6 7 17 18 , 0 1 2 7 17 ⟶ 5 6 7 17 19 , 9 10 2 7 2 ⟶ 11 12 7 17 18 , 9 10 2 7 17 ⟶ 11 12 7 17 19 , 13 14 2 7 2 ⟶ 15 16 7 17 18 , 13 14 2 7 17 ⟶ 15 16 7 17 19 , 5 6 3 8 4 ⟶ 0 20 8 8 4 , 5 6 3 8 8 ⟶ 0 20 8 8 8 , 5 6 3 8 21 ⟶ 0 20 8 8 21 , 5 6 3 8 22 ⟶ 0 20 8 8 22 , 11 12 3 8 4 ⟶ 9 23 8 8 4 , 11 12 3 8 8 ⟶ 9 23 8 8 8 , 11 12 3 8 21 ⟶ 9 23 8 8 21 , 11 12 3 8 22 ⟶ 9 23 8 8 22 , 15 16 3 8 4 ⟶ 13 24 8 8 4 , 15 16 3 8 8 ⟶ 13 24 8 8 8 , 15 16 3 8 21 ⟶ 13 24 8 8 21 , 15 16 3 8 22 ⟶ 13 24 8 8 22 , 5 6 3 4 2 ⟶ 0 20 8 4 2 , 5 6 3 4 17 ⟶ 0 20 8 4 17 , 5 6 3 4 25 ⟶ 0 20 8 4 25 , 11 12 3 4 2 ⟶ 9 23 8 4 2 , 11 12 3 4 17 ⟶ 9 23 8 4 17 , 11 12 3 4 25 ⟶ 9 23 8 4 25 , 15 16 3 4 2 ⟶ 13 24 8 4 2 , 15 16 3 4 17 ⟶ 13 24 8 4 17 , 15 16 3 4 25 ⟶ 13 24 8 4 25 , 20 4 2 3 4 →= 1 2 7 2 7 , 20 4 2 3 8 →= 1 2 7 2 3 , 3 4 2 3 4 →= 7 2 7 2 7 , 3 4 2 3 8 →= 7 2 7 2 3 , 8 4 2 3 4 →= 4 2 7 2 7 , 8 4 2 3 8 →= 4 2 7 2 3 , 23 4 2 3 4 →= 10 2 7 2 7 , 23 4 2 3 8 →= 10 2 7 2 3 , 24 4 2 3 4 →= 14 2 7 2 7 , 24 4 2 3 8 →= 14 2 7 2 3 , 26 4 2 3 4 →= 27 2 7 2 7 , 26 4 2 3 8 →= 27 2 7 2 3 , 28 4 2 3 4 →= 29 2 7 2 7 , 28 4 2 3 8 →= 29 2 7 2 3 , 2 7 2 3 4 →= 17 18 7 2 7 , 2 7 2 3 8 →= 17 18 7 2 3 , 6 7 2 3 4 →= 30 18 7 2 7 , 6 7 2 3 8 →= 30 18 7 2 3 , 12 7 2 3 4 →= 31 18 7 2 7 , 12 7 2 3 8 →= 31 18 7 2 3 , 16 7 2 3 4 →= 32 18 7 2 7 , 16 7 2 3 8 →= 32 18 7 2 3 , 18 7 2 3 4 →= 19 18 7 2 7 , 18 7 2 3 8 →= 19 18 7 2 3 , 33 7 2 3 4 →= 34 18 7 2 7 , 33 7 2 3 8 →= 34 18 7 2 3 , 35 7 2 3 4 →= 36 18 7 2 7 , 35 7 2 3 8 →= 36 18 7 2 3 , 37 27 2 3 4 →= 38 33 7 2 7 , 37 27 2 3 8 →= 38 33 7 2 3 , 20 4 2 7 2 →= 1 2 7 17 18 , 20 4 2 7 17 →= 1 2 7 17 19 , 3 4 2 7 2 →= 7 2 7 17 18 , 3 4 2 7 17 →= 7 2 7 17 19 , 8 4 2 7 2 →= 4 2 7 17 18 , 8 4 2 7 17 →= 4 2 7 17 19 , 23 4 2 7 2 →= 10 2 7 17 18 , 23 4 2 7 17 →= 10 2 7 17 19 , 24 4 2 7 2 →= 14 2 7 17 18 , 24 4 2 7 17 →= 14 2 7 17 19 , 26 4 2 7 2 →= 27 2 7 17 18 , 26 4 2 7 17 →= 27 2 7 17 19 , 28 4 2 7 2 →= 29 2 7 17 18 , 28 4 2 7 17 →= 29 2 7 17 19 , 2 7 2 7 2 →= 17 18 7 17 18 , 2 7 2 7 17 →= 17 18 7 17 19 , 6 7 2 7 2 →= 30 18 7 17 18 , 6 7 2 7 17 →= 30 18 7 17 19 , 12 7 2 7 2 →= 31 18 7 17 18 , 12 7 2 7 17 →= 31 18 7 17 19 , 16 7 2 7 2 →= 32 18 7 17 18 , 16 7 2 7 17 →= 32 18 7 17 19 , 18 7 2 7 2 →= 19 18 7 17 18 , 18 7 2 7 17 →= 19 18 7 17 19 , 33 7 2 7 2 →= 34 18 7 17 18 , 33 7 2 7 17 →= 34 18 7 17 19 , 35 7 2 7 2 →= 36 18 7 17 18 , 35 7 2 7 17 →= 36 18 7 17 19 , 37 27 2 7 2 →= 38 33 7 17 18 , 37 27 2 7 17 →= 38 33 7 17 19 , 1 2 3 8 4 →= 20 8 8 8 4 , 1 2 3 8 8 →= 20 8 8 8 8 , 1 2 3 8 21 →= 20 8 8 8 21 , 1 2 3 8 22 →= 20 8 8 8 22 , 7 2 3 8 4 →= 3 8 8 8 4 , 7 2 3 8 8 →= 3 8 8 8 8 , 7 2 3 8 21 →= 3 8 8 8 21 , 7 2 3 8 22 →= 3 8 8 8 22 , 4 2 3 8 4 →= 8 8 8 8 4 , 4 2 3 8 8 →= 8 8 8 8 8 , 4 2 3 8 21 →= 8 8 8 8 21 , 4 2 3 8 22 →= 8 8 8 8 22 , 10 2 3 8 4 →= 23 8 8 8 4 , 10 2 3 8 8 →= 23 8 8 8 8 , 10 2 3 8 21 →= 23 8 8 8 21 , 10 2 3 8 22 →= 23 8 8 8 22 , 14 2 3 8 4 →= 24 8 8 8 4 , 14 2 3 8 8 →= 24 8 8 8 8 , 14 2 3 8 21 →= 24 8 8 8 21 , 14 2 3 8 22 →= 24 8 8 8 22 , 27 2 3 8 4 →= 26 8 8 8 4 , 27 2 3 8 8 →= 26 8 8 8 8 , 27 2 3 8 21 →= 26 8 8 8 21 , 27 2 3 8 22 →= 26 8 8 8 22 , 29 2 3 8 4 →= 28 8 8 8 4 , 29 2 3 8 8 →= 28 8 8 8 8 , 29 2 3 8 21 →= 28 8 8 8 21 , 29 2 3 8 22 →= 28 8 8 8 22 , 17 18 3 8 4 →= 2 3 8 8 4 , 17 18 3 8 8 →= 2 3 8 8 8 , 17 18 3 8 21 →= 2 3 8 8 21 , 17 18 3 8 22 →= 2 3 8 8 22 , 30 18 3 8 4 →= 6 3 8 8 4 , 30 18 3 8 8 →= 6 3 8 8 8 , 30 18 3 8 21 →= 6 3 8 8 21 , 30 18 3 8 22 →= 6 3 8 8 22 , 31 18 3 8 4 →= 12 3 8 8 4 , 31 18 3 8 8 →= 12 3 8 8 8 , 31 18 3 8 21 →= 12 3 8 8 21 , 31 18 3 8 22 →= 12 3 8 8 22 , 32 18 3 8 4 →= 16 3 8 8 4 , 32 18 3 8 8 →= 16 3 8 8 8 , 32 18 3 8 21 →= 16 3 8 8 21 , 32 18 3 8 22 →= 16 3 8 8 22 , 19 18 3 8 4 →= 18 3 8 8 4 , 19 18 3 8 8 →= 18 3 8 8 8 , 19 18 3 8 21 →= 18 3 8 8 21 , 19 18 3 8 22 →= 18 3 8 8 22 , 34 18 3 8 4 →= 33 3 8 8 4 , 34 18 3 8 8 →= 33 3 8 8 8 , 34 18 3 8 21 →= 33 3 8 8 21 , 34 18 3 8 22 →= 33 3 8 8 22 , 36 18 3 8 4 →= 35 3 8 8 4 , 36 18 3 8 8 →= 35 3 8 8 8 , 36 18 3 8 21 →= 35 3 8 8 21 , 36 18 3 8 22 →= 35 3 8 8 22 , 38 33 3 8 4 →= 37 26 8 8 4 , 38 33 3 8 8 →= 37 26 8 8 8 , 38 33 3 8 21 →= 37 26 8 8 21 , 38 33 3 8 22 →= 37 26 8 8 22 , 1 2 3 4 2 →= 20 8 8 4 2 , 1 2 3 4 17 →= 20 8 8 4 17 , 1 2 3 4 25 →= 20 8 8 4 25 , 7 2 3 4 2 →= 3 8 8 4 2 , 7 2 3 4 17 →= 3 8 8 4 17 , 7 2 3 4 25 →= 3 8 8 4 25 , 4 2 3 4 2 →= 8 8 8 4 2 , 4 2 3 4 17 →= 8 8 8 4 17 , 4 2 3 4 25 →= 8 8 8 4 25 , 10 2 3 4 2 →= 23 8 8 4 2 , 10 2 3 4 17 →= 23 8 8 4 17 , 10 2 3 4 25 →= 23 8 8 4 25 , 14 2 3 4 2 →= 24 8 8 4 2 , 14 2 3 4 17 →= 24 8 8 4 17 , 14 2 3 4 25 →= 24 8 8 4 25 , 27 2 3 4 2 →= 26 8 8 4 2 , 27 2 3 4 17 →= 26 8 8 4 17 , 27 2 3 4 25 →= 26 8 8 4 25 , 29 2 3 4 2 →= 28 8 8 4 2 , 29 2 3 4 17 →= 28 8 8 4 17 , 29 2 3 4 25 →= 28 8 8 4 25 , 17 18 3 4 2 →= 2 3 8 4 2 , 17 18 3 4 17 →= 2 3 8 4 17 , 17 18 3 4 25 →= 2 3 8 4 25 , 30 18 3 4 2 →= 6 3 8 4 2 , 30 18 3 4 17 →= 6 3 8 4 17 , 30 18 3 4 25 →= 6 3 8 4 25 , 31 18 3 4 2 →= 12 3 8 4 2 , 31 18 3 4 17 →= 12 3 8 4 17 , 31 18 3 4 25 →= 12 3 8 4 25 , 32 18 3 4 2 →= 16 3 8 4 2 , 32 18 3 4 17 →= 16 3 8 4 17 , 32 18 3 4 25 →= 16 3 8 4 25 , 19 18 3 4 2 →= 18 3 8 4 2 , 19 18 3 4 17 →= 18 3 8 4 17 , 19 18 3 4 25 →= 18 3 8 4 25 , 34 18 3 4 2 →= 33 3 8 4 2 , 34 18 3 4 17 →= 33 3 8 4 17 , 34 18 3 4 25 →= 33 3 8 4 25 , 36 18 3 4 2 →= 35 3 8 4 2 , 36 18 3 4 17 →= 35 3 8 4 17 , 36 18 3 4 25 →= 35 3 8 4 25 , 38 33 3 4 2 →= 37 26 8 4 2 , 38 33 3 4 17 →= 37 26 8 4 17 , 38 33 3 4 25 →= 37 26 8 4 25 , 1 2 3 21 39 →= 20 8 8 21 39 , 7 2 3 21 39 →= 3 8 8 21 39 , 4 2 3 21 39 →= 8 8 8 21 39 , 10 2 3 21 39 →= 23 8 8 21 39 , 14 2 3 21 39 →= 24 8 8 21 39 , 27 2 3 21 39 →= 26 8 8 21 39 , 29 2 3 21 39 →= 28 8 8 21 39 , 17 18 3 21 39 →= 2 3 8 21 39 , 30 18 3 21 39 →= 6 3 8 21 39 , 31 18 3 21 39 →= 12 3 8 21 39 , 32 18 3 21 39 →= 16 3 8 21 39 , 19 18 3 21 39 →= 18 3 8 21 39 , 34 18 3 21 39 →= 33 3 8 21 39 , 36 18 3 21 39 →= 35 3 8 21 39 } 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 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 23 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 24 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 25 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 8 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 9 ↦ 8, 10 ↦ 9, 11 ↦ 10, 12 ↦ 11, 13 ↦ 12, 14 ↦ 13, 15 ↦ 14, 16 ↦ 15, 17 ↦ 16, 18 ↦ 17, 19 ↦ 18, 4 ↦ 19, 30 ↦ 20, 31 ↦ 21, 32 ↦ 22, 33 ↦ 23, 34 ↦ 24, 35 ↦ 25, 36 ↦ 26, 37 ↦ 27, 27 ↦ 28, 38 ↦ 29, 21 ↦ 30, 22 ↦ 31, 25 ↦ 32, 39 ↦ 33 }, it remains to prove termination of the 100-rule system { 0 1 2 3 4 ⟶ 5 6 7 2 3 , 8 9 2 3 4 ⟶ 10 11 7 2 3 , 12 13 2 3 4 ⟶ 14 15 7 2 3 , 0 1 2 7 2 ⟶ 5 6 7 16 17 , 0 1 2 7 16 ⟶ 5 6 7 16 18 , 8 9 2 7 2 ⟶ 10 11 7 16 17 , 8 9 2 7 16 ⟶ 10 11 7 16 18 , 12 13 2 7 2 ⟶ 14 15 7 16 17 , 12 13 2 7 16 ⟶ 14 15 7 16 18 , 4 19 2 3 4 →= 19 2 7 2 3 , 2 7 2 3 4 →= 16 17 7 2 3 , 6 7 2 3 4 →= 20 17 7 2 3 , 11 7 2 3 4 →= 21 17 7 2 3 , 15 7 2 3 4 →= 22 17 7 2 3 , 17 7 2 3 4 →= 18 17 7 2 3 , 23 7 2 3 4 →= 24 17 7 2 3 , 25 7 2 3 4 →= 26 17 7 2 3 , 27 28 2 3 4 →= 29 23 7 2 3 , 4 19 2 7 2 →= 19 2 7 16 17 , 4 19 2 7 16 →= 19 2 7 16 18 , 2 7 2 7 2 →= 16 17 7 16 17 , 2 7 2 7 16 →= 16 17 7 16 18 , 6 7 2 7 2 →= 20 17 7 16 17 , 6 7 2 7 16 →= 20 17 7 16 18 , 11 7 2 7 2 →= 21 17 7 16 17 , 11 7 2 7 16 →= 21 17 7 16 18 , 15 7 2 7 2 →= 22 17 7 16 17 , 15 7 2 7 16 →= 22 17 7 16 18 , 17 7 2 7 2 →= 18 17 7 16 17 , 17 7 2 7 16 →= 18 17 7 16 18 , 23 7 2 7 2 →= 24 17 7 16 17 , 23 7 2 7 16 →= 24 17 7 16 18 , 25 7 2 7 2 →= 26 17 7 16 17 , 25 7 2 7 16 →= 26 17 7 16 18 , 27 28 2 7 2 →= 29 23 7 16 17 , 27 28 2 7 16 →= 29 23 7 16 18 , 7 2 3 4 19 →= 3 4 4 4 19 , 7 2 3 4 4 →= 3 4 4 4 4 , 7 2 3 4 30 →= 3 4 4 4 30 , 7 2 3 4 31 →= 3 4 4 4 31 , 16 17 3 4 19 →= 2 3 4 4 19 , 16 17 3 4 4 →= 2 3 4 4 4 , 16 17 3 4 30 →= 2 3 4 4 30 , 16 17 3 4 31 →= 2 3 4 4 31 , 20 17 3 4 19 →= 6 3 4 4 19 , 20 17 3 4 4 →= 6 3 4 4 4 , 20 17 3 4 30 →= 6 3 4 4 30 , 20 17 3 4 31 →= 6 3 4 4 31 , 21 17 3 4 19 →= 11 3 4 4 19 , 21 17 3 4 4 →= 11 3 4 4 4 , 21 17 3 4 30 →= 11 3 4 4 30 , 21 17 3 4 31 →= 11 3 4 4 31 , 22 17 3 4 19 →= 15 3 4 4 19 , 22 17 3 4 4 →= 15 3 4 4 4 , 22 17 3 4 30 →= 15 3 4 4 30 , 22 17 3 4 31 →= 15 3 4 4 31 , 18 17 3 4 19 →= 17 3 4 4 19 , 18 17 3 4 4 →= 17 3 4 4 4 , 18 17 3 4 30 →= 17 3 4 4 30 , 18 17 3 4 31 →= 17 3 4 4 31 , 24 17 3 4 19 →= 23 3 4 4 19 , 24 17 3 4 4 →= 23 3 4 4 4 , 24 17 3 4 30 →= 23 3 4 4 30 , 24 17 3 4 31 →= 23 3 4 4 31 , 26 17 3 4 19 →= 25 3 4 4 19 , 26 17 3 4 4 →= 25 3 4 4 4 , 26 17 3 4 30 →= 25 3 4 4 30 , 26 17 3 4 31 →= 25 3 4 4 31 , 7 2 3 19 2 →= 3 4 4 19 2 , 7 2 3 19 16 →= 3 4 4 19 16 , 7 2 3 19 32 →= 3 4 4 19 32 , 16 17 3 19 2 →= 2 3 4 19 2 , 16 17 3 19 16 →= 2 3 4 19 16 , 16 17 3 19 32 →= 2 3 4 19 32 , 20 17 3 19 2 →= 6 3 4 19 2 , 20 17 3 19 16 →= 6 3 4 19 16 , 20 17 3 19 32 →= 6 3 4 19 32 , 21 17 3 19 2 →= 11 3 4 19 2 , 21 17 3 19 16 →= 11 3 4 19 16 , 21 17 3 19 32 →= 11 3 4 19 32 , 22 17 3 19 2 →= 15 3 4 19 2 , 22 17 3 19 16 →= 15 3 4 19 16 , 22 17 3 19 32 →= 15 3 4 19 32 , 18 17 3 19 2 →= 17 3 4 19 2 , 18 17 3 19 16 →= 17 3 4 19 16 , 18 17 3 19 32 →= 17 3 4 19 32 , 24 17 3 19 2 →= 23 3 4 19 2 , 24 17 3 19 16 →= 23 3 4 19 16 , 24 17 3 19 32 →= 23 3 4 19 32 , 26 17 3 19 2 →= 25 3 4 19 2 , 26 17 3 19 16 →= 25 3 4 19 16 , 26 17 3 19 32 →= 25 3 4 19 32 , 7 2 3 30 33 →= 3 4 4 30 33 , 16 17 3 30 33 →= 2 3 4 30 33 , 20 17 3 30 33 →= 6 3 4 30 33 , 21 17 3 30 33 →= 11 3 4 30 33 , 22 17 3 30 33 →= 15 3 4 30 33 , 18 17 3 30 33 →= 17 3 4 30 33 , 24 17 3 30 33 →= 23 3 4 30 33 , 26 17 3 30 33 →= 25 3 4 30 33 } 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 0 ⎟ ⎜ 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 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 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 32 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 33 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 4 ↦ 0, 19 ↦ 1, 2 ↦ 2, 3 ↦ 3, 7 ↦ 4, 16 ↦ 5, 17 ↦ 6, 6 ↦ 7, 20 ↦ 8, 11 ↦ 9, 21 ↦ 10, 15 ↦ 11, 22 ↦ 12, 18 ↦ 13, 23 ↦ 14, 24 ↦ 15, 25 ↦ 16, 26 ↦ 17, 30 ↦ 18, 31 ↦ 19, 32 ↦ 20, 33 ↦ 21 }, it remains to prove termination of the 88-rule system { 0 1 2 3 0 →= 1 2 4 2 3 , 2 4 2 3 0 →= 5 6 4 2 3 , 7 4 2 3 0 →= 8 6 4 2 3 , 9 4 2 3 0 →= 10 6 4 2 3 , 11 4 2 3 0 →= 12 6 4 2 3 , 6 4 2 3 0 →= 13 6 4 2 3 , 14 4 2 3 0 →= 15 6 4 2 3 , 16 4 2 3 0 →= 17 6 4 2 3 , 0 1 2 4 2 →= 1 2 4 5 6 , 0 1 2 4 5 →= 1 2 4 5 13 , 2 4 2 4 2 →= 5 6 4 5 6 , 2 4 2 4 5 →= 5 6 4 5 13 , 7 4 2 4 2 →= 8 6 4 5 6 , 7 4 2 4 5 →= 8 6 4 5 13 , 9 4 2 4 2 →= 10 6 4 5 6 , 9 4 2 4 5 →= 10 6 4 5 13 , 11 4 2 4 2 →= 12 6 4 5 6 , 11 4 2 4 5 →= 12 6 4 5 13 , 6 4 2 4 2 →= 13 6 4 5 6 , 6 4 2 4 5 →= 13 6 4 5 13 , 14 4 2 4 2 →= 15 6 4 5 6 , 14 4 2 4 5 →= 15 6 4 5 13 , 16 4 2 4 2 →= 17 6 4 5 6 , 16 4 2 4 5 →= 17 6 4 5 13 , 4 2 3 0 1 →= 3 0 0 0 1 , 4 2 3 0 0 →= 3 0 0 0 0 , 4 2 3 0 18 →= 3 0 0 0 18 , 4 2 3 0 19 →= 3 0 0 0 19 , 5 6 3 0 1 →= 2 3 0 0 1 , 5 6 3 0 0 →= 2 3 0 0 0 , 5 6 3 0 18 →= 2 3 0 0 18 , 5 6 3 0 19 →= 2 3 0 0 19 , 8 6 3 0 1 →= 7 3 0 0 1 , 8 6 3 0 0 →= 7 3 0 0 0 , 8 6 3 0 18 →= 7 3 0 0 18 , 8 6 3 0 19 →= 7 3 0 0 19 , 10 6 3 0 1 →= 9 3 0 0 1 , 10 6 3 0 0 →= 9 3 0 0 0 , 10 6 3 0 18 →= 9 3 0 0 18 , 10 6 3 0 19 →= 9 3 0 0 19 , 12 6 3 0 1 →= 11 3 0 0 1 , 12 6 3 0 0 →= 11 3 0 0 0 , 12 6 3 0 18 →= 11 3 0 0 18 , 12 6 3 0 19 →= 11 3 0 0 19 , 13 6 3 0 1 →= 6 3 0 0 1 , 13 6 3 0 0 →= 6 3 0 0 0 , 13 6 3 0 18 →= 6 3 0 0 18 , 13 6 3 0 19 →= 6 3 0 0 19 , 15 6 3 0 1 →= 14 3 0 0 1 , 15 6 3 0 0 →= 14 3 0 0 0 , 15 6 3 0 18 →= 14 3 0 0 18 , 15 6 3 0 19 →= 14 3 0 0 19 , 17 6 3 0 1 →= 16 3 0 0 1 , 17 6 3 0 0 →= 16 3 0 0 0 , 17 6 3 0 18 →= 16 3 0 0 18 , 17 6 3 0 19 →= 16 3 0 0 19 , 4 2 3 1 2 →= 3 0 0 1 2 , 4 2 3 1 5 →= 3 0 0 1 5 , 4 2 3 1 20 →= 3 0 0 1 20 , 5 6 3 1 2 →= 2 3 0 1 2 , 5 6 3 1 5 →= 2 3 0 1 5 , 5 6 3 1 20 →= 2 3 0 1 20 , 8 6 3 1 2 →= 7 3 0 1 2 , 8 6 3 1 5 →= 7 3 0 1 5 , 8 6 3 1 20 →= 7 3 0 1 20 , 10 6 3 1 2 →= 9 3 0 1 2 , 10 6 3 1 5 →= 9 3 0 1 5 , 10 6 3 1 20 →= 9 3 0 1 20 , 12 6 3 1 2 →= 11 3 0 1 2 , 12 6 3 1 5 →= 11 3 0 1 5 , 12 6 3 1 20 →= 11 3 0 1 20 , 13 6 3 1 2 →= 6 3 0 1 2 , 13 6 3 1 5 →= 6 3 0 1 5 , 13 6 3 1 20 →= 6 3 0 1 20 , 15 6 3 1 2 →= 14 3 0 1 2 , 15 6 3 1 5 →= 14 3 0 1 5 , 15 6 3 1 20 →= 14 3 0 1 20 , 17 6 3 1 2 →= 16 3 0 1 2 , 17 6 3 1 5 →= 16 3 0 1 5 , 17 6 3 1 20 →= 16 3 0 1 20 , 4 2 3 18 21 →= 3 0 0 18 21 , 5 6 3 18 21 →= 2 3 0 18 21 , 8 6 3 18 21 →= 7 3 0 18 21 , 10 6 3 18 21 →= 9 3 0 18 21 , 12 6 3 18 21 →= 11 3 0 18 21 , 13 6 3 18 21 →= 6 3 0 18 21 , 15 6 3 18 21 →= 14 3 0 18 21 , 17 6 3 18 21 →= 16 3 0 18 21 } The system is trivially terminating.