/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 { 0 ↦ 0, 1 ↦ 1 }, it remains to prove termination of the 2-rule system { 0 0 0 0 ⟶ 1 0 0 1 , 0 1 0 1 ⟶ 0 0 1 0 } Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols. After renaming modulo the bijection { (0,↑) ↦ 0, (0,↓) ↦ 1, (1,↓) ↦ 2 }, it remains to prove termination of the 7-rule system { 0 1 1 1 ⟶ 0 1 2 , 0 1 1 1 ⟶ 0 2 , 0 2 1 2 ⟶ 0 1 2 1 , 0 2 1 2 ⟶ 0 2 1 , 0 2 1 2 ⟶ 0 , 1 1 1 1 →= 2 1 1 2 , 1 2 1 2 →= 1 1 2 1 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 2 ↦ 1, 1 ↦ 2 }, it remains to prove termination of the 3-rule system { 0 1 2 1 ⟶ 0 2 1 2 , 2 2 2 2 →= 1 2 2 1 , 2 1 2 1 →= 2 2 1 2 } Applying sparse tiling TROC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (3,0) ↦ 0, (0,1) ↦ 1, (1,2) ↦ 2, (2,1) ↦ 3, (1,1) ↦ 4, (0,2) ↦ 5, (2,2) ↦ 6, (1,4) ↦ 7, (2,4) ↦ 8, (3,2) ↦ 9, (3,1) ↦ 10 }, it remains to prove termination of the 27-rule system { 0 1 2 3 4 ⟶ 0 5 3 2 3 , 0 1 2 3 2 ⟶ 0 5 3 2 6 , 0 1 2 3 7 ⟶ 0 5 3 2 8 , 5 6 6 6 3 →= 1 2 6 3 4 , 5 6 6 6 6 →= 1 2 6 3 2 , 5 6 6 6 8 →= 1 2 6 3 7 , 2 6 6 6 3 →= 4 2 6 3 4 , 2 6 6 6 6 →= 4 2 6 3 2 , 2 6 6 6 8 →= 4 2 6 3 7 , 6 6 6 6 3 →= 3 2 6 3 4 , 6 6 6 6 6 →= 3 2 6 3 2 , 6 6 6 6 8 →= 3 2 6 3 7 , 9 6 6 6 3 →= 10 2 6 3 4 , 9 6 6 6 6 →= 10 2 6 3 2 , 9 6 6 6 8 →= 10 2 6 3 7 , 5 3 2 3 4 →= 5 6 3 2 3 , 5 3 2 3 2 →= 5 6 3 2 6 , 5 3 2 3 7 →= 5 6 3 2 8 , 2 3 2 3 4 →= 2 6 3 2 3 , 2 3 2 3 2 →= 2 6 3 2 6 , 2 3 2 3 7 →= 2 6 3 2 8 , 6 3 2 3 4 →= 6 6 3 2 3 , 6 3 2 3 2 →= 6 6 3 2 6 , 6 3 2 3 7 →= 6 6 3 2 8 , 9 3 2 3 4 →= 9 6 3 2 3 , 9 3 2 3 2 →= 9 6 3 2 6 , 9 3 2 3 7 →= 9 6 3 2 8 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 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 ⎟ ⎝ ⎠ 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 }, it remains to prove termination of the 24-rule system { 0 1 2 3 4 ⟶ 0 5 3 2 3 , 0 1 2 3 2 ⟶ 0 5 3 2 6 , 0 1 2 3 7 ⟶ 0 5 3 2 8 , 5 6 6 6 3 →= 1 2 6 3 4 , 5 6 6 6 6 →= 1 2 6 3 2 , 5 6 6 6 8 →= 1 2 6 3 7 , 2 6 6 6 3 →= 4 2 6 3 4 , 2 6 6 6 6 →= 4 2 6 3 2 , 2 6 6 6 8 →= 4 2 6 3 7 , 6 6 6 6 3 →= 3 2 6 3 4 , 6 6 6 6 6 →= 3 2 6 3 2 , 6 6 6 6 8 →= 3 2 6 3 7 , 5 3 2 3 4 →= 5 6 3 2 3 , 5 3 2 3 2 →= 5 6 3 2 6 , 5 3 2 3 7 →= 5 6 3 2 8 , 2 3 2 3 4 →= 2 6 3 2 3 , 2 3 2 3 2 →= 2 6 3 2 6 , 2 3 2 3 7 →= 2 6 3 2 8 , 6 3 2 3 4 →= 6 6 3 2 3 , 6 3 2 3 2 →= 6 6 3 2 6 , 6 3 2 3 7 →= 6 6 3 2 8 , 9 3 2 3 4 →= 9 6 3 2 3 , 9 3 2 3 2 →= 9 6 3 2 6 , 9 3 2 3 7 →= 9 6 3 2 8 } 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 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 1 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 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 1 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 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 8 ↦ 7, 7 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 23-rule system { 0 1 2 3 4 ⟶ 0 5 3 2 3 , 0 1 2 3 2 ⟶ 0 5 3 2 6 , 5 6 6 6 3 →= 1 2 6 3 4 , 5 6 6 6 6 →= 1 2 6 3 2 , 5 6 6 6 7 →= 1 2 6 3 8 , 2 6 6 6 3 →= 4 2 6 3 4 , 2 6 6 6 6 →= 4 2 6 3 2 , 2 6 6 6 7 →= 4 2 6 3 8 , 6 6 6 6 3 →= 3 2 6 3 4 , 6 6 6 6 6 →= 3 2 6 3 2 , 6 6 6 6 7 →= 3 2 6 3 8 , 5 3 2 3 4 →= 5 6 3 2 3 , 5 3 2 3 2 →= 5 6 3 2 6 , 5 3 2 3 8 →= 5 6 3 2 7 , 2 3 2 3 4 →= 2 6 3 2 3 , 2 3 2 3 2 →= 2 6 3 2 6 , 2 3 2 3 8 →= 2 6 3 2 7 , 6 3 2 3 4 →= 6 6 3 2 3 , 6 3 2 3 2 →= 6 6 3 2 6 , 6 3 2 3 8 →= 6 6 3 2 7 , 9 3 2 3 4 →= 9 6 3 2 3 , 9 3 2 3 2 →= 9 6 3 2 6 , 9 3 2 3 8 →= 9 6 3 2 7 } 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 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 1 0 0 ⎟ ⎜ 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎜ 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 1 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 ⎟ ⎝ ⎠ 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 }, it remains to prove termination of the 22-rule system { 0 1 2 3 4 ⟶ 0 5 3 2 3 , 0 1 2 3 2 ⟶ 0 5 3 2 6 , 5 6 6 6 3 →= 1 2 6 3 4 , 5 6 6 6 6 →= 1 2 6 3 2 , 2 6 6 6 3 →= 4 2 6 3 4 , 2 6 6 6 6 →= 4 2 6 3 2 , 2 6 6 6 7 →= 4 2 6 3 8 , 6 6 6 6 3 →= 3 2 6 3 4 , 6 6 6 6 6 →= 3 2 6 3 2 , 6 6 6 6 7 →= 3 2 6 3 8 , 5 3 2 3 4 →= 5 6 3 2 3 , 5 3 2 3 2 →= 5 6 3 2 6 , 5 3 2 3 8 →= 5 6 3 2 7 , 2 3 2 3 4 →= 2 6 3 2 3 , 2 3 2 3 2 →= 2 6 3 2 6 , 2 3 2 3 8 →= 2 6 3 2 7 , 6 3 2 3 4 →= 6 6 3 2 3 , 6 3 2 3 2 →= 6 6 3 2 6 , 6 3 2 3 8 →= 6 6 3 2 7 , 9 3 2 3 4 →= 9 6 3 2 3 , 9 3 2 3 2 →= 9 6 3 2 6 , 9 3 2 3 8 →= 9 6 3 2 7 } 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 1 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 0 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 1 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 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 }, it remains to prove termination of the 21-rule system { 0 1 2 3 4 ⟶ 0 5 3 2 3 , 0 1 2 3 2 ⟶ 0 5 3 2 6 , 5 6 6 6 3 →= 1 2 6 3 4 , 5 6 6 6 6 →= 1 2 6 3 2 , 2 6 6 6 3 →= 4 2 6 3 4 , 2 6 6 6 6 →= 4 2 6 3 2 , 2 6 6 6 7 →= 4 2 6 3 8 , 6 6 6 6 3 →= 3 2 6 3 4 , 6 6 6 6 6 →= 3 2 6 3 2 , 6 6 6 6 7 →= 3 2 6 3 8 , 5 3 2 3 4 →= 5 6 3 2 3 , 5 3 2 3 2 →= 5 6 3 2 6 , 5 3 2 3 8 →= 5 6 3 2 7 , 2 3 2 3 4 →= 2 6 3 2 3 , 2 3 2 3 2 →= 2 6 3 2 6 , 2 3 2 3 8 →= 2 6 3 2 7 , 6 3 2 3 4 →= 6 6 3 2 3 , 6 3 2 3 2 →= 6 6 3 2 6 , 6 3 2 3 8 →= 6 6 3 2 7 , 9 3 2 3 4 →= 9 6 3 2 3 , 9 3 2 3 2 →= 9 6 3 2 6 } Applying sparse tiling TROC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (10,0) ↦ 0, (0,1) ↦ 1, (1,2) ↦ 2, (2,3) ↦ 3, (3,4) ↦ 4, (4,2) ↦ 5, (0,5) ↦ 6, (5,3) ↦ 7, (3,2) ↦ 8, (4,4) ↦ 9, (4,8) ↦ 10, (3,8) ↦ 11, (4,11) ↦ 12, (3,11) ↦ 13, (2,6) ↦ 14, (6,3) ↦ 15, (6,6) ↦ 16, (2,7) ↦ 17, (6,7) ↦ 18, (2,11) ↦ 19, (6,11) ↦ 20, (5,6) ↦ 21, (10,5) ↦ 22, (10,1) ↦ 23, (1,4) ↦ 24, (10,2) ↦ 25, (10,4) ↦ 26, (7,11) ↦ 27, (8,11) ↦ 28, (9,6) ↦ 29, (9,3) ↦ 30, (10,6) ↦ 31, (10,3) ↦ 32, (10,9) ↦ 33 }, it remains to prove termination of the 212-rule system { 0 1 2 3 4 5 ⟶ 0 6 7 8 3 8 , 0 1 2 3 4 9 ⟶ 0 6 7 8 3 4 , 0 1 2 3 4 10 ⟶ 0 6 7 8 3 11 , 0 1 2 3 4 12 ⟶ 0 6 7 8 3 13 , 0 1 2 3 8 3 ⟶ 0 6 7 8 14 15 , 0 1 2 3 8 14 ⟶ 0 6 7 8 14 16 , 0 1 2 3 8 17 ⟶ 0 6 7 8 14 18 , 0 1 2 3 8 19 ⟶ 0 6 7 8 14 20 , 6 21 16 16 15 8 →= 1 2 14 15 4 5 , 6 21 16 16 15 4 →= 1 2 14 15 4 9 , 6 21 16 16 15 11 →= 1 2 14 15 4 10 , 6 21 16 16 15 13 →= 1 2 14 15 4 12 , 22 21 16 16 15 8 →= 23 2 14 15 4 5 , 22 21 16 16 15 4 →= 23 2 14 15 4 9 , 22 21 16 16 15 11 →= 23 2 14 15 4 10 , 22 21 16 16 15 13 →= 23 2 14 15 4 12 , 6 21 16 16 16 15 →= 1 2 14 15 8 3 , 6 21 16 16 16 16 →= 1 2 14 15 8 14 , 6 21 16 16 16 18 →= 1 2 14 15 8 17 , 6 21 16 16 16 20 →= 1 2 14 15 8 19 , 22 21 16 16 16 15 →= 23 2 14 15 8 3 , 22 21 16 16 16 16 →= 23 2 14 15 8 14 , 22 21 16 16 16 18 →= 23 2 14 15 8 17 , 22 21 16 16 16 20 →= 23 2 14 15 8 19 , 2 14 16 16 15 8 →= 24 5 14 15 4 5 , 2 14 16 16 15 4 →= 24 5 14 15 4 9 , 2 14 16 16 15 11 →= 24 5 14 15 4 10 , 2 14 16 16 15 13 →= 24 5 14 15 4 12 , 8 14 16 16 15 8 →= 4 5 14 15 4 5 , 8 14 16 16 15 4 →= 4 5 14 15 4 9 , 8 14 16 16 15 11 →= 4 5 14 15 4 10 , 8 14 16 16 15 13 →= 4 5 14 15 4 12 , 5 14 16 16 15 8 →= 9 5 14 15 4 5 , 5 14 16 16 15 4 →= 9 5 14 15 4 9 , 5 14 16 16 15 11 →= 9 5 14 15 4 10 , 5 14 16 16 15 13 →= 9 5 14 15 4 12 , 25 14 16 16 15 8 →= 26 5 14 15 4 5 , 25 14 16 16 15 4 →= 26 5 14 15 4 9 , 25 14 16 16 15 11 →= 26 5 14 15 4 10 , 25 14 16 16 15 13 →= 26 5 14 15 4 12 , 2 14 16 16 16 15 →= 24 5 14 15 8 3 , 2 14 16 16 16 16 →= 24 5 14 15 8 14 , 2 14 16 16 16 18 →= 24 5 14 15 8 17 , 2 14 16 16 16 20 →= 24 5 14 15 8 19 , 8 14 16 16 16 15 →= 4 5 14 15 8 3 , 8 14 16 16 16 16 →= 4 5 14 15 8 14 , 8 14 16 16 16 18 →= 4 5 14 15 8 17 , 8 14 16 16 16 20 →= 4 5 14 15 8 19 , 5 14 16 16 16 15 →= 9 5 14 15 8 3 , 5 14 16 16 16 16 →= 9 5 14 15 8 14 , 5 14 16 16 16 18 →= 9 5 14 15 8 17 , 5 14 16 16 16 20 →= 9 5 14 15 8 19 , 25 14 16 16 16 15 →= 26 5 14 15 8 3 , 25 14 16 16 16 16 →= 26 5 14 15 8 14 , 25 14 16 16 16 18 →= 26 5 14 15 8 17 , 25 14 16 16 16 20 →= 26 5 14 15 8 19 , 2 14 16 16 18 27 →= 24 5 14 15 11 28 , 8 14 16 16 18 27 →= 4 5 14 15 11 28 , 5 14 16 16 18 27 →= 9 5 14 15 11 28 , 25 14 16 16 18 27 →= 26 5 14 15 11 28 , 14 16 16 16 15 8 →= 3 8 14 15 4 5 , 14 16 16 16 15 4 →= 3 8 14 15 4 9 , 14 16 16 16 15 11 →= 3 8 14 15 4 10 , 14 16 16 16 15 13 →= 3 8 14 15 4 12 , 21 16 16 16 15 8 →= 7 8 14 15 4 5 , 21 16 16 16 15 4 →= 7 8 14 15 4 9 , 21 16 16 16 15 11 →= 7 8 14 15 4 10 , 21 16 16 16 15 13 →= 7 8 14 15 4 12 , 16 16 16 16 15 8 →= 15 8 14 15 4 5 , 16 16 16 16 15 4 →= 15 8 14 15 4 9 , 16 16 16 16 15 11 →= 15 8 14 15 4 10 , 16 16 16 16 15 13 →= 15 8 14 15 4 12 , 29 16 16 16 15 8 →= 30 8 14 15 4 5 , 29 16 16 16 15 4 →= 30 8 14 15 4 9 , 29 16 16 16 15 11 →= 30 8 14 15 4 10 , 29 16 16 16 15 13 →= 30 8 14 15 4 12 , 31 16 16 16 15 8 →= 32 8 14 15 4 5 , 31 16 16 16 15 4 →= 32 8 14 15 4 9 , 31 16 16 16 15 11 →= 32 8 14 15 4 10 , 31 16 16 16 15 13 →= 32 8 14 15 4 12 , 14 16 16 16 16 15 →= 3 8 14 15 8 3 , 14 16 16 16 16 16 →= 3 8 14 15 8 14 , 14 16 16 16 16 18 →= 3 8 14 15 8 17 , 14 16 16 16 16 20 →= 3 8 14 15 8 19 , 21 16 16 16 16 15 →= 7 8 14 15 8 3 , 21 16 16 16 16 16 →= 7 8 14 15 8 14 , 21 16 16 16 16 18 →= 7 8 14 15 8 17 , 21 16 16 16 16 20 →= 7 8 14 15 8 19 , 16 16 16 16 16 15 →= 15 8 14 15 8 3 , 16 16 16 16 16 16 →= 15 8 14 15 8 14 , 16 16 16 16 16 18 →= 15 8 14 15 8 17 , 16 16 16 16 16 20 →= 15 8 14 15 8 19 , 29 16 16 16 16 15 →= 30 8 14 15 8 3 , 29 16 16 16 16 16 →= 30 8 14 15 8 14 , 29 16 16 16 16 18 →= 30 8 14 15 8 17 , 29 16 16 16 16 20 →= 30 8 14 15 8 19 , 31 16 16 16 16 15 →= 32 8 14 15 8 3 , 31 16 16 16 16 16 →= 32 8 14 15 8 14 , 31 16 16 16 16 18 →= 32 8 14 15 8 17 , 31 16 16 16 16 20 →= 32 8 14 15 8 19 , 14 16 16 16 18 27 →= 3 8 14 15 11 28 , 21 16 16 16 18 27 →= 7 8 14 15 11 28 , 16 16 16 16 18 27 →= 15 8 14 15 11 28 , 29 16 16 16 18 27 →= 30 8 14 15 11 28 , 31 16 16 16 18 27 →= 32 8 14 15 11 28 , 6 7 8 3 4 5 →= 6 21 15 8 3 8 , 6 7 8 3 4 9 →= 6 21 15 8 3 4 , 6 7 8 3 4 10 →= 6 21 15 8 3 11 , 6 7 8 3 4 12 →= 6 21 15 8 3 13 , 22 7 8 3 4 5 →= 22 21 15 8 3 8 , 22 7 8 3 4 9 →= 22 21 15 8 3 4 , 22 7 8 3 4 10 →= 22 21 15 8 3 11 , 22 7 8 3 4 12 →= 22 21 15 8 3 13 , 6 7 8 3 8 3 →= 6 21 15 8 14 15 , 6 7 8 3 8 14 →= 6 21 15 8 14 16 , 6 7 8 3 8 17 →= 6 21 15 8 14 18 , 6 7 8 3 8 19 →= 6 21 15 8 14 20 , 22 7 8 3 8 3 →= 22 21 15 8 14 15 , 22 7 8 3 8 14 →= 22 21 15 8 14 16 , 22 7 8 3 8 17 →= 22 21 15 8 14 18 , 22 7 8 3 8 19 →= 22 21 15 8 14 20 , 6 7 8 3 11 28 →= 6 21 15 8 17 27 , 22 7 8 3 11 28 →= 22 21 15 8 17 27 , 2 3 8 3 4 5 →= 2 14 15 8 3 8 , 2 3 8 3 4 9 →= 2 14 15 8 3 4 , 2 3 8 3 4 10 →= 2 14 15 8 3 11 , 2 3 8 3 4 12 →= 2 14 15 8 3 13 , 8 3 8 3 4 5 →= 8 14 15 8 3 8 , 8 3 8 3 4 9 →= 8 14 15 8 3 4 , 8 3 8 3 4 10 →= 8 14 15 8 3 11 , 8 3 8 3 4 12 →= 8 14 15 8 3 13 , 5 3 8 3 4 5 →= 5 14 15 8 3 8 , 5 3 8 3 4 9 →= 5 14 15 8 3 4 , 5 3 8 3 4 10 →= 5 14 15 8 3 11 , 5 3 8 3 4 12 →= 5 14 15 8 3 13 , 25 3 8 3 4 5 →= 25 14 15 8 3 8 , 25 3 8 3 4 9 →= 25 14 15 8 3 4 , 25 3 8 3 4 10 →= 25 14 15 8 3 11 , 25 3 8 3 4 12 →= 25 14 15 8 3 13 , 2 3 8 3 8 3 →= 2 14 15 8 14 15 , 2 3 8 3 8 14 →= 2 14 15 8 14 16 , 2 3 8 3 8 17 →= 2 14 15 8 14 18 , 2 3 8 3 8 19 →= 2 14 15 8 14 20 , 8 3 8 3 8 3 →= 8 14 15 8 14 15 , 8 3 8 3 8 14 →= 8 14 15 8 14 16 , 8 3 8 3 8 17 →= 8 14 15 8 14 18 , 8 3 8 3 8 19 →= 8 14 15 8 14 20 , 5 3 8 3 8 3 →= 5 14 15 8 14 15 , 5 3 8 3 8 14 →= 5 14 15 8 14 16 , 5 3 8 3 8 17 →= 5 14 15 8 14 18 , 5 3 8 3 8 19 →= 5 14 15 8 14 20 , 25 3 8 3 8 3 →= 25 14 15 8 14 15 , 25 3 8 3 8 14 →= 25 14 15 8 14 16 , 25 3 8 3 8 17 →= 25 14 15 8 14 18 , 25 3 8 3 8 19 →= 25 14 15 8 14 20 , 2 3 8 3 11 28 →= 2 14 15 8 17 27 , 8 3 8 3 11 28 →= 8 14 15 8 17 27 , 5 3 8 3 11 28 →= 5 14 15 8 17 27 , 25 3 8 3 11 28 →= 25 14 15 8 17 27 , 14 15 8 3 4 5 →= 14 16 15 8 3 8 , 14 15 8 3 4 9 →= 14 16 15 8 3 4 , 14 15 8 3 4 10 →= 14 16 15 8 3 11 , 14 15 8 3 4 12 →= 14 16 15 8 3 13 , 21 15 8 3 4 5 →= 21 16 15 8 3 8 , 21 15 8 3 4 9 →= 21 16 15 8 3 4 , 21 15 8 3 4 10 →= 21 16 15 8 3 11 , 21 15 8 3 4 12 →= 21 16 15 8 3 13 , 16 15 8 3 4 5 →= 16 16 15 8 3 8 , 16 15 8 3 4 9 →= 16 16 15 8 3 4 , 16 15 8 3 4 10 →= 16 16 15 8 3 11 , 16 15 8 3 4 12 →= 16 16 15 8 3 13 , 29 15 8 3 4 5 →= 29 16 15 8 3 8 , 29 15 8 3 4 9 →= 29 16 15 8 3 4 , 29 15 8 3 4 10 →= 29 16 15 8 3 11 , 29 15 8 3 4 12 →= 29 16 15 8 3 13 , 31 15 8 3 4 5 →= 31 16 15 8 3 8 , 31 15 8 3 4 9 →= 31 16 15 8 3 4 , 31 15 8 3 4 10 →= 31 16 15 8 3 11 , 31 15 8 3 4 12 →= 31 16 15 8 3 13 , 14 15 8 3 8 3 →= 14 16 15 8 14 15 , 14 15 8 3 8 14 →= 14 16 15 8 14 16 , 14 15 8 3 8 17 →= 14 16 15 8 14 18 , 14 15 8 3 8 19 →= 14 16 15 8 14 20 , 21 15 8 3 8 3 →= 21 16 15 8 14 15 , 21 15 8 3 8 14 →= 21 16 15 8 14 16 , 21 15 8 3 8 17 →= 21 16 15 8 14 18 , 21 15 8 3 8 19 →= 21 16 15 8 14 20 , 16 15 8 3 8 3 →= 16 16 15 8 14 15 , 16 15 8 3 8 14 →= 16 16 15 8 14 16 , 16 15 8 3 8 17 →= 16 16 15 8 14 18 , 16 15 8 3 8 19 →= 16 16 15 8 14 20 , 29 15 8 3 8 3 →= 29 16 15 8 14 15 , 29 15 8 3 8 14 →= 29 16 15 8 14 16 , 29 15 8 3 8 17 →= 29 16 15 8 14 18 , 29 15 8 3 8 19 →= 29 16 15 8 14 20 , 31 15 8 3 8 3 →= 31 16 15 8 14 15 , 31 15 8 3 8 14 →= 31 16 15 8 14 16 , 31 15 8 3 8 17 →= 31 16 15 8 14 18 , 31 15 8 3 8 19 →= 31 16 15 8 14 20 , 14 15 8 3 11 28 →= 14 16 15 8 17 27 , 21 15 8 3 11 28 →= 21 16 15 8 17 27 , 16 15 8 3 11 28 →= 16 16 15 8 17 27 , 29 15 8 3 11 28 →= 29 16 15 8 17 27 , 31 15 8 3 11 28 →= 31 16 15 8 17 27 , 33 30 8 3 4 5 →= 33 29 15 8 3 8 , 33 30 8 3 4 9 →= 33 29 15 8 3 4 , 33 30 8 3 4 10 →= 33 29 15 8 3 11 , 33 30 8 3 4 12 →= 33 29 15 8 3 13 , 33 30 8 3 8 3 →= 33 29 15 8 14 15 , 33 30 8 3 8 14 →= 33 29 15 8 14 16 , 33 30 8 3 8 17 →= 33 29 15 8 14 18 , 33 30 8 3 8 19 →= 33 29 15 8 14 20 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 24 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 25 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 26 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 27 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 28 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 29 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 30 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 31 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 32 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 33 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13, 14 ↦ 14, 16 ↦ 15, 15 ↦ 16, 21 ↦ 17, 29 ↦ 18, 31 ↦ 19, 28 ↦ 20, 17 ↦ 21, 27 ↦ 22 }, it remains to prove termination of the 50-rule system { 0 1 2 3 4 5 ⟶ 0 6 7 8 3 8 , 0 1 2 3 4 9 ⟶ 0 6 7 8 3 4 , 0 1 2 3 4 10 ⟶ 0 6 7 8 3 11 , 0 1 2 3 4 12 ⟶ 0 6 7 8 3 13 , 0 1 2 3 8 14 ⟶ 0 6 7 8 14 15 , 8 14 15 15 16 8 →= 4 5 14 16 4 5 , 8 14 15 15 16 4 →= 4 5 14 16 4 9 , 8 14 15 15 16 11 →= 4 5 14 16 4 10 , 8 14 15 15 16 13 →= 4 5 14 16 4 12 , 5 14 15 15 16 8 →= 9 5 14 16 4 5 , 5 14 15 15 16 4 →= 9 5 14 16 4 9 , 5 14 15 15 16 11 →= 9 5 14 16 4 10 , 5 14 15 15 16 13 →= 9 5 14 16 4 12 , 8 14 15 15 15 16 →= 4 5 14 16 8 3 , 5 14 15 15 15 16 →= 9 5 14 16 8 3 , 14 15 15 15 16 8 →= 3 8 14 16 4 5 , 14 15 15 15 16 4 →= 3 8 14 16 4 9 , 14 15 15 15 16 11 →= 3 8 14 16 4 10 , 14 15 15 15 16 13 →= 3 8 14 16 4 12 , 14 15 15 15 15 16 →= 3 8 14 16 8 3 , 14 16 8 3 4 5 →= 14 15 16 8 3 8 , 14 16 8 3 4 9 →= 14 15 16 8 3 4 , 14 16 8 3 4 10 →= 14 15 16 8 3 11 , 14 16 8 3 4 12 →= 14 15 16 8 3 13 , 17 16 8 3 4 5 →= 17 15 16 8 3 8 , 17 16 8 3 4 9 →= 17 15 16 8 3 4 , 17 16 8 3 4 10 →= 17 15 16 8 3 11 , 17 16 8 3 4 12 →= 17 15 16 8 3 13 , 15 16 8 3 4 5 →= 15 15 16 8 3 8 , 15 16 8 3 4 9 →= 15 15 16 8 3 4 , 15 16 8 3 4 10 →= 15 15 16 8 3 11 , 15 16 8 3 4 12 →= 15 15 16 8 3 13 , 18 16 8 3 4 5 →= 18 15 16 8 3 8 , 18 16 8 3 4 9 →= 18 15 16 8 3 4 , 18 16 8 3 4 10 →= 18 15 16 8 3 11 , 18 16 8 3 4 12 →= 18 15 16 8 3 13 , 19 16 8 3 4 5 →= 19 15 16 8 3 8 , 19 16 8 3 4 9 →= 19 15 16 8 3 4 , 19 16 8 3 4 10 →= 19 15 16 8 3 11 , 19 16 8 3 4 12 →= 19 15 16 8 3 13 , 14 16 8 3 8 14 →= 14 15 16 8 14 15 , 17 16 8 3 8 14 →= 17 15 16 8 14 15 , 15 16 8 3 8 14 →= 15 15 16 8 14 15 , 18 16 8 3 8 14 →= 18 15 16 8 14 15 , 19 16 8 3 8 14 →= 19 15 16 8 14 15 , 14 16 8 3 11 20 →= 14 15 16 8 21 22 , 17 16 8 3 11 20 →= 17 15 16 8 21 22 , 15 16 8 3 11 20 →= 15 15 16 8 21 22 , 18 16 8 3 11 20 →= 18 15 16 8 21 22 , 19 16 8 3 11 20 →= 19 15 16 8 21 22 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 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 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 8 ↦ 0, 14 ↦ 1, 15 ↦ 2, 16 ↦ 3, 4 ↦ 4, 5 ↦ 5, 9 ↦ 6, 11 ↦ 7, 10 ↦ 8, 13 ↦ 9, 12 ↦ 10, 3 ↦ 11, 17 ↦ 12, 18 ↦ 13, 19 ↦ 14 }, it remains to prove termination of the 40-rule system { 0 1 2 2 3 0 →= 4 5 1 3 4 5 , 0 1 2 2 3 4 →= 4 5 1 3 4 6 , 0 1 2 2 3 7 →= 4 5 1 3 4 8 , 0 1 2 2 3 9 →= 4 5 1 3 4 10 , 5 1 2 2 3 0 →= 6 5 1 3 4 5 , 5 1 2 2 3 4 →= 6 5 1 3 4 6 , 5 1 2 2 3 7 →= 6 5 1 3 4 8 , 5 1 2 2 3 9 →= 6 5 1 3 4 10 , 0 1 2 2 2 3 →= 4 5 1 3 0 11 , 5 1 2 2 2 3 →= 6 5 1 3 0 11 , 1 2 2 2 3 0 →= 11 0 1 3 4 5 , 1 2 2 2 3 4 →= 11 0 1 3 4 6 , 1 2 2 2 3 7 →= 11 0 1 3 4 8 , 1 2 2 2 3 9 →= 11 0 1 3 4 10 , 1 2 2 2 2 3 →= 11 0 1 3 0 11 , 1 3 0 11 4 5 →= 1 2 3 0 11 0 , 1 3 0 11 4 6 →= 1 2 3 0 11 4 , 1 3 0 11 4 8 →= 1 2 3 0 11 7 , 1 3 0 11 4 10 →= 1 2 3 0 11 9 , 12 3 0 11 4 5 →= 12 2 3 0 11 0 , 12 3 0 11 4 6 →= 12 2 3 0 11 4 , 12 3 0 11 4 8 →= 12 2 3 0 11 7 , 12 3 0 11 4 10 →= 12 2 3 0 11 9 , 2 3 0 11 4 5 →= 2 2 3 0 11 0 , 2 3 0 11 4 6 →= 2 2 3 0 11 4 , 2 3 0 11 4 8 →= 2 2 3 0 11 7 , 2 3 0 11 4 10 →= 2 2 3 0 11 9 , 13 3 0 11 4 5 →= 13 2 3 0 11 0 , 13 3 0 11 4 6 →= 13 2 3 0 11 4 , 13 3 0 11 4 8 →= 13 2 3 0 11 7 , 13 3 0 11 4 10 →= 13 2 3 0 11 9 , 14 3 0 11 4 5 →= 14 2 3 0 11 0 , 14 3 0 11 4 6 →= 14 2 3 0 11 4 , 14 3 0 11 4 8 →= 14 2 3 0 11 7 , 14 3 0 11 4 10 →= 14 2 3 0 11 9 , 1 3 0 11 0 1 →= 1 2 3 0 1 2 , 12 3 0 11 0 1 →= 12 2 3 0 1 2 , 2 3 0 11 0 1 →= 2 2 3 0 1 2 , 13 3 0 11 0 1 →= 13 2 3 0 1 2 , 14 3 0 11 0 1 →= 14 2 3 0 1 2 } The system is trivially terminating.