/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES After renaming modulo the bijection { a ↦ 0, b ↦ 1 }, it remains to prove termination of the 2-rule system { 0 0 0 ⟶ 0 0 1 , 1 0 1 ⟶ 0 1 0 } The system was reversed. After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1 }, it remains to prove termination of the 2-rule system { 0 0 0 ⟶ 1 0 0 , 1 0 1 ⟶ 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, (1,↓) ↦ 3 }, it remains to prove termination of the 6-rule system { 0 1 1 ⟶ 2 1 1 , 2 1 3 ⟶ 0 3 1 , 2 1 3 ⟶ 2 1 , 2 1 3 ⟶ 0 , 1 1 1 →= 3 1 1 , 3 1 3 →= 1 3 1 } 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 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3 }, it remains to prove termination of the 4-rule system { 0 1 1 ⟶ 2 1 1 , 2 1 3 ⟶ 0 3 1 , 1 1 1 →= 3 1 1 , 3 1 3 →= 1 3 1 } Applying sparse tiling TROC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (4,0) ↦ 0, (0,1) ↦ 1, (1,1) ↦ 2, (4,2) ↦ 3, (2,1) ↦ 4, (1,3) ↦ 5, (1,5) ↦ 6, (3,1) ↦ 7, (0,3) ↦ 8, (3,3) ↦ 9, (2,3) ↦ 10, (4,1) ↦ 11, (4,3) ↦ 12 }, it remains to prove termination of the 30-rule system { 0 1 2 2 ⟶ 3 4 2 2 , 0 1 2 5 ⟶ 3 4 2 5 , 0 1 2 6 ⟶ 3 4 2 6 , 3 4 5 7 ⟶ 0 8 7 2 , 3 4 5 9 ⟶ 0 8 7 5 , 1 2 2 2 →= 8 7 2 2 , 1 2 2 5 →= 8 7 2 5 , 1 2 2 6 →= 8 7 2 6 , 2 2 2 2 →= 5 7 2 2 , 2 2 2 5 →= 5 7 2 5 , 2 2 2 6 →= 5 7 2 6 , 4 2 2 2 →= 10 7 2 2 , 4 2 2 5 →= 10 7 2 5 , 4 2 2 6 →= 10 7 2 6 , 7 2 2 2 →= 9 7 2 2 , 7 2 2 5 →= 9 7 2 5 , 7 2 2 6 →= 9 7 2 6 , 11 2 2 2 →= 12 7 2 2 , 11 2 2 5 →= 12 7 2 5 , 11 2 2 6 →= 12 7 2 6 , 8 7 5 7 →= 1 5 7 2 , 8 7 5 9 →= 1 5 7 5 , 5 7 5 7 →= 2 5 7 2 , 5 7 5 9 →= 2 5 7 5 , 10 7 5 7 →= 4 5 7 2 , 10 7 5 9 →= 4 5 7 5 , 9 7 5 7 →= 7 5 7 2 , 9 7 5 9 →= 7 5 7 5 , 12 7 5 7 →= 11 5 7 2 , 12 7 5 9 →= 11 5 7 5 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 5: 0 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 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 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 1 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 0 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, 7 ↦ 6, 8 ↦ 7, 9 ↦ 8, 6 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12 }, it remains to prove termination of the 29-rule system { 0 1 2 2 ⟶ 3 4 2 2 , 0 1 2 5 ⟶ 3 4 2 5 , 3 4 5 6 ⟶ 0 7 6 2 , 3 4 5 8 ⟶ 0 7 6 5 , 1 2 2 2 →= 7 6 2 2 , 1 2 2 5 →= 7 6 2 5 , 1 2 2 9 →= 7 6 2 9 , 2 2 2 2 →= 5 6 2 2 , 2 2 2 5 →= 5 6 2 5 , 2 2 2 9 →= 5 6 2 9 , 4 2 2 2 →= 10 6 2 2 , 4 2 2 5 →= 10 6 2 5 , 4 2 2 9 →= 10 6 2 9 , 6 2 2 2 →= 8 6 2 2 , 6 2 2 5 →= 8 6 2 5 , 6 2 2 9 →= 8 6 2 9 , 11 2 2 2 →= 12 6 2 2 , 11 2 2 5 →= 12 6 2 5 , 11 2 2 9 →= 12 6 2 9 , 7 6 5 6 →= 1 5 6 2 , 7 6 5 8 →= 1 5 6 5 , 5 6 5 6 →= 2 5 6 2 , 5 6 5 8 →= 2 5 6 5 , 10 6 5 6 →= 4 5 6 2 , 10 6 5 8 →= 4 5 6 5 , 8 6 5 6 →= 6 5 6 2 , 8 6 5 8 →= 6 5 6 5 , 12 6 5 6 →= 11 5 6 2 , 12 6 5 8 →= 11 5 6 5 } 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 1 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 1 ⎟ ⎜ 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 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 1 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 0 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 28-rule system { 0 1 2 2 ⟶ 3 4 2 2 , 0 1 2 5 ⟶ 3 4 2 5 , 3 4 5 6 ⟶ 0 7 6 2 , 3 4 5 8 ⟶ 0 7 6 5 , 1 2 2 2 →= 7 6 2 2 , 1 2 2 5 →= 7 6 2 5 , 2 2 2 2 →= 5 6 2 2 , 2 2 2 5 →= 5 6 2 5 , 2 2 2 9 →= 5 6 2 9 , 4 2 2 2 →= 10 6 2 2 , 4 2 2 5 →= 10 6 2 5 , 4 2 2 9 →= 10 6 2 9 , 6 2 2 2 →= 8 6 2 2 , 6 2 2 5 →= 8 6 2 5 , 6 2 2 9 →= 8 6 2 9 , 11 2 2 2 →= 12 6 2 2 , 11 2 2 5 →= 12 6 2 5 , 11 2 2 9 →= 12 6 2 9 , 7 6 5 6 →= 1 5 6 2 , 7 6 5 8 →= 1 5 6 5 , 5 6 5 6 →= 2 5 6 2 , 5 6 5 8 →= 2 5 6 5 , 10 6 5 6 →= 4 5 6 2 , 10 6 5 8 →= 4 5 6 5 , 8 6 5 6 →= 6 5 6 2 , 8 6 5 8 →= 6 5 6 5 , 12 6 5 6 →= 11 5 6 2 , 12 6 5 8 →= 11 5 6 5 } 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 1 ⎟ ⎜ 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 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 1 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 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 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 ⎟ ⎝ ⎠ 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 2 ⟶ 3 4 2 2 , 0 1 2 5 ⟶ 3 4 2 5 , 3 4 5 6 ⟶ 0 7 6 2 , 3 4 5 8 ⟶ 0 7 6 5 , 1 2 2 2 →= 7 6 2 2 , 1 2 2 5 →= 7 6 2 5 , 2 2 2 2 →= 5 6 2 2 , 2 2 2 5 →= 5 6 2 5 , 2 2 2 9 →= 5 6 2 9 , 4 2 2 2 →= 10 6 2 2 , 4 2 2 5 →= 10 6 2 5 , 4 2 2 9 →= 10 6 2 9 , 6 2 2 2 →= 8 6 2 2 , 6 2 2 5 →= 8 6 2 5 , 6 2 2 9 →= 8 6 2 9 , 11 2 2 2 →= 12 6 2 2 , 11 2 2 5 →= 12 6 2 5 , 7 6 5 6 →= 1 5 6 2 , 7 6 5 8 →= 1 5 6 5 , 5 6 5 6 →= 2 5 6 2 , 5 6 5 8 →= 2 5 6 5 , 10 6 5 6 →= 4 5 6 2 , 10 6 5 8 →= 4 5 6 5 , 8 6 5 6 →= 6 5 6 2 , 8 6 5 8 →= 6 5 6 5 , 12 6 5 6 →= 11 5 6 2 , 12 6 5 8 →= 11 5 6 5 } 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, (13,3) ↦ 4, (3,4) ↦ 5, (4,2) ↦ 6, (2,5) ↦ 7, (2,9) ↦ 8, (2,14) ↦ 9, (5,6) ↦ 10, (5,8) ↦ 11, (5,14) ↦ 12, (4,5) ↦ 13, (6,2) ↦ 14, (0,7) ↦ 15, (7,6) ↦ 16, (6,5) ↦ 17, (8,6) ↦ 18, (8,8) ↦ 19, (13,1) ↦ 20, (13,7) ↦ 21, (1,5) ↦ 22, (11,2) ↦ 23, (11,5) ↦ 24, (13,2) ↦ 25, (13,5) ↦ 26, (9,14) ↦ 27, (3,10) ↦ 28, (10,6) ↦ 29, (13,4) ↦ 30, (13,10) ↦ 31, (7,8) ↦ 32, (10,8) ↦ 33, (12,6) ↦ 34, (12,8) ↦ 35, (13,6) ↦ 36, (13,8) ↦ 37, (13,11) ↦ 38, (13,12) ↦ 39 }, it remains to prove termination of the 212-rule system { 0 1 2 3 3 ⟶ 4 5 6 3 3 , 0 1 2 3 7 ⟶ 4 5 6 3 7 , 0 1 2 3 8 ⟶ 4 5 6 3 8 , 0 1 2 3 9 ⟶ 4 5 6 3 9 , 0 1 2 7 10 ⟶ 4 5 6 7 10 , 0 1 2 7 11 ⟶ 4 5 6 7 11 , 0 1 2 7 12 ⟶ 4 5 6 7 12 , 4 5 13 10 14 ⟶ 0 15 16 14 3 , 4 5 13 10 17 ⟶ 0 15 16 14 7 , 4 5 13 11 18 ⟶ 0 15 16 17 10 , 4 5 13 11 19 ⟶ 0 15 16 17 11 , 1 2 3 3 3 →= 15 16 14 3 3 , 1 2 3 3 7 →= 15 16 14 3 7 , 1 2 3 3 8 →= 15 16 14 3 8 , 1 2 3 3 9 →= 15 16 14 3 9 , 20 2 3 3 3 →= 21 16 14 3 3 , 20 2 3 3 7 →= 21 16 14 3 7 , 20 2 3 3 8 →= 21 16 14 3 8 , 20 2 3 3 9 →= 21 16 14 3 9 , 1 2 3 7 10 →= 15 16 14 7 10 , 1 2 3 7 11 →= 15 16 14 7 11 , 1 2 3 7 12 →= 15 16 14 7 12 , 20 2 3 7 10 →= 21 16 14 7 10 , 20 2 3 7 11 →= 21 16 14 7 11 , 20 2 3 7 12 →= 21 16 14 7 12 , 2 3 3 3 3 →= 22 10 14 3 3 , 2 3 3 3 7 →= 22 10 14 3 7 , 2 3 3 3 8 →= 22 10 14 3 8 , 2 3 3 3 9 →= 22 10 14 3 9 , 3 3 3 3 3 →= 7 10 14 3 3 , 3 3 3 3 7 →= 7 10 14 3 7 , 3 3 3 3 8 →= 7 10 14 3 8 , 3 3 3 3 9 →= 7 10 14 3 9 , 6 3 3 3 3 →= 13 10 14 3 3 , 6 3 3 3 7 →= 13 10 14 3 7 , 6 3 3 3 8 →= 13 10 14 3 8 , 6 3 3 3 9 →= 13 10 14 3 9 , 14 3 3 3 3 →= 17 10 14 3 3 , 14 3 3 3 7 →= 17 10 14 3 7 , 14 3 3 3 8 →= 17 10 14 3 8 , 14 3 3 3 9 →= 17 10 14 3 9 , 23 3 3 3 3 →= 24 10 14 3 3 , 23 3 3 3 7 →= 24 10 14 3 7 , 23 3 3 3 8 →= 24 10 14 3 8 , 23 3 3 3 9 →= 24 10 14 3 9 , 25 3 3 3 3 →= 26 10 14 3 3 , 25 3 3 3 7 →= 26 10 14 3 7 , 25 3 3 3 8 →= 26 10 14 3 8 , 25 3 3 3 9 →= 26 10 14 3 9 , 2 3 3 7 10 →= 22 10 14 7 10 , 2 3 3 7 11 →= 22 10 14 7 11 , 2 3 3 7 12 →= 22 10 14 7 12 , 3 3 3 7 10 →= 7 10 14 7 10 , 3 3 3 7 11 →= 7 10 14 7 11 , 3 3 3 7 12 →= 7 10 14 7 12 , 6 3 3 7 10 →= 13 10 14 7 10 , 6 3 3 7 11 →= 13 10 14 7 11 , 6 3 3 7 12 →= 13 10 14 7 12 , 14 3 3 7 10 →= 17 10 14 7 10 , 14 3 3 7 11 →= 17 10 14 7 11 , 14 3 3 7 12 →= 17 10 14 7 12 , 23 3 3 7 10 →= 24 10 14 7 10 , 23 3 3 7 11 →= 24 10 14 7 11 , 23 3 3 7 12 →= 24 10 14 7 12 , 25 3 3 7 10 →= 26 10 14 7 10 , 25 3 3 7 11 →= 26 10 14 7 11 , 25 3 3 7 12 →= 26 10 14 7 12 , 2 3 3 8 27 →= 22 10 14 8 27 , 3 3 3 8 27 →= 7 10 14 8 27 , 6 3 3 8 27 →= 13 10 14 8 27 , 14 3 3 8 27 →= 17 10 14 8 27 , 23 3 3 8 27 →= 24 10 14 8 27 , 25 3 3 8 27 →= 26 10 14 8 27 , 5 6 3 3 3 →= 28 29 14 3 3 , 5 6 3 3 7 →= 28 29 14 3 7 , 5 6 3 3 8 →= 28 29 14 3 8 , 5 6 3 3 9 →= 28 29 14 3 9 , 30 6 3 3 3 →= 31 29 14 3 3 , 30 6 3 3 7 →= 31 29 14 3 7 , 30 6 3 3 8 →= 31 29 14 3 8 , 30 6 3 3 9 →= 31 29 14 3 9 , 5 6 3 7 10 →= 28 29 14 7 10 , 5 6 3 7 11 →= 28 29 14 7 11 , 5 6 3 7 12 →= 28 29 14 7 12 , 30 6 3 7 10 →= 31 29 14 7 10 , 30 6 3 7 11 →= 31 29 14 7 11 , 30 6 3 7 12 →= 31 29 14 7 12 , 5 6 3 8 27 →= 28 29 14 8 27 , 30 6 3 8 27 →= 31 29 14 8 27 , 10 14 3 3 3 →= 11 18 14 3 3 , 10 14 3 3 7 →= 11 18 14 3 7 , 10 14 3 3 8 →= 11 18 14 3 8 , 10 14 3 3 9 →= 11 18 14 3 9 , 16 14 3 3 3 →= 32 18 14 3 3 , 16 14 3 3 7 →= 32 18 14 3 7 , 16 14 3 3 8 →= 32 18 14 3 8 , 16 14 3 3 9 →= 32 18 14 3 9 , 18 14 3 3 3 →= 19 18 14 3 3 , 18 14 3 3 7 →= 19 18 14 3 7 , 18 14 3 3 8 →= 19 18 14 3 8 , 18 14 3 3 9 →= 19 18 14 3 9 , 29 14 3 3 3 →= 33 18 14 3 3 , 29 14 3 3 7 →= 33 18 14 3 7 , 29 14 3 3 8 →= 33 18 14 3 8 , 29 14 3 3 9 →= 33 18 14 3 9 , 34 14 3 3 3 →= 35 18 14 3 3 , 34 14 3 3 7 →= 35 18 14 3 7 , 34 14 3 3 8 →= 35 18 14 3 8 , 34 14 3 3 9 →= 35 18 14 3 9 , 36 14 3 3 3 →= 37 18 14 3 3 , 36 14 3 3 7 →= 37 18 14 3 7 , 36 14 3 3 8 →= 37 18 14 3 8 , 36 14 3 3 9 →= 37 18 14 3 9 , 10 14 3 7 10 →= 11 18 14 7 10 , 10 14 3 7 11 →= 11 18 14 7 11 , 10 14 3 7 12 →= 11 18 14 7 12 , 16 14 3 7 10 →= 32 18 14 7 10 , 16 14 3 7 11 →= 32 18 14 7 11 , 16 14 3 7 12 →= 32 18 14 7 12 , 18 14 3 7 10 →= 19 18 14 7 10 , 18 14 3 7 11 →= 19 18 14 7 11 , 18 14 3 7 12 →= 19 18 14 7 12 , 29 14 3 7 10 →= 33 18 14 7 10 , 29 14 3 7 11 →= 33 18 14 7 11 , 29 14 3 7 12 →= 33 18 14 7 12 , 34 14 3 7 10 →= 35 18 14 7 10 , 34 14 3 7 11 →= 35 18 14 7 11 , 34 14 3 7 12 →= 35 18 14 7 12 , 36 14 3 7 10 →= 37 18 14 7 10 , 36 14 3 7 11 →= 37 18 14 7 11 , 36 14 3 7 12 →= 37 18 14 7 12 , 10 14 3 8 27 →= 11 18 14 8 27 , 16 14 3 8 27 →= 32 18 14 8 27 , 18 14 3 8 27 →= 19 18 14 8 27 , 29 14 3 8 27 →= 33 18 14 8 27 , 34 14 3 8 27 →= 35 18 14 8 27 , 36 14 3 8 27 →= 37 18 14 8 27 , 38 23 3 3 3 →= 39 34 14 3 3 , 38 23 3 3 7 →= 39 34 14 3 7 , 38 23 3 3 8 →= 39 34 14 3 8 , 38 23 3 3 9 →= 39 34 14 3 9 , 38 23 3 7 10 →= 39 34 14 7 10 , 38 23 3 7 11 →= 39 34 14 7 11 , 38 23 3 7 12 →= 39 34 14 7 12 , 15 16 17 10 14 →= 1 22 10 14 3 , 15 16 17 10 17 →= 1 22 10 14 7 , 21 16 17 10 14 →= 20 22 10 14 3 , 21 16 17 10 17 →= 20 22 10 14 7 , 15 16 17 11 18 →= 1 22 10 17 10 , 15 16 17 11 19 →= 1 22 10 17 11 , 21 16 17 11 18 →= 20 22 10 17 10 , 21 16 17 11 19 →= 20 22 10 17 11 , 22 10 17 10 14 →= 2 7 10 14 3 , 22 10 17 10 17 →= 2 7 10 14 7 , 7 10 17 10 14 →= 3 7 10 14 3 , 7 10 17 10 17 →= 3 7 10 14 7 , 13 10 17 10 14 →= 6 7 10 14 3 , 13 10 17 10 17 →= 6 7 10 14 7 , 17 10 17 10 14 →= 14 7 10 14 3 , 17 10 17 10 17 →= 14 7 10 14 7 , 24 10 17 10 14 →= 23 7 10 14 3 , 24 10 17 10 17 →= 23 7 10 14 7 , 26 10 17 10 14 →= 25 7 10 14 3 , 26 10 17 10 17 →= 25 7 10 14 7 , 22 10 17 11 18 →= 2 7 10 17 10 , 22 10 17 11 19 →= 2 7 10 17 11 , 7 10 17 11 18 →= 3 7 10 17 10 , 7 10 17 11 19 →= 3 7 10 17 11 , 13 10 17 11 18 →= 6 7 10 17 10 , 13 10 17 11 19 →= 6 7 10 17 11 , 17 10 17 11 18 →= 14 7 10 17 10 , 17 10 17 11 19 →= 14 7 10 17 11 , 24 10 17 11 18 →= 23 7 10 17 10 , 24 10 17 11 19 →= 23 7 10 17 11 , 26 10 17 11 18 →= 25 7 10 17 10 , 26 10 17 11 19 →= 25 7 10 17 11 , 28 29 17 10 14 →= 5 13 10 14 3 , 28 29 17 10 17 →= 5 13 10 14 7 , 31 29 17 10 14 →= 30 13 10 14 3 , 31 29 17 10 17 →= 30 13 10 14 7 , 28 29 17 11 18 →= 5 13 10 17 10 , 28 29 17 11 19 →= 5 13 10 17 11 , 31 29 17 11 18 →= 30 13 10 17 10 , 31 29 17 11 19 →= 30 13 10 17 11 , 11 18 17 10 14 →= 10 17 10 14 3 , 11 18 17 10 17 →= 10 17 10 14 7 , 32 18 17 10 14 →= 16 17 10 14 3 , 32 18 17 10 17 →= 16 17 10 14 7 , 19 18 17 10 14 →= 18 17 10 14 3 , 19 18 17 10 17 →= 18 17 10 14 7 , 33 18 17 10 14 →= 29 17 10 14 3 , 33 18 17 10 17 →= 29 17 10 14 7 , 35 18 17 10 14 →= 34 17 10 14 3 , 35 18 17 10 17 →= 34 17 10 14 7 , 37 18 17 10 14 →= 36 17 10 14 3 , 37 18 17 10 17 →= 36 17 10 14 7 , 11 18 17 11 18 →= 10 17 10 17 10 , 11 18 17 11 19 →= 10 17 10 17 11 , 32 18 17 11 18 →= 16 17 10 17 10 , 32 18 17 11 19 →= 16 17 10 17 11 , 19 18 17 11 18 →= 18 17 10 17 10 , 19 18 17 11 19 →= 18 17 10 17 11 , 33 18 17 11 18 →= 29 17 10 17 10 , 33 18 17 11 19 →= 29 17 10 17 11 , 35 18 17 11 18 →= 34 17 10 17 10 , 35 18 17 11 19 →= 34 17 10 17 11 , 37 18 17 11 18 →= 36 17 10 17 10 , 37 18 17 11 19 →= 36 17 10 17 11 , 39 34 17 10 14 →= 38 24 10 14 3 , 39 34 17 10 17 →= 38 24 10 14 7 , 39 34 17 11 18 →= 38 24 10 17 10 , 39 34 17 11 19 →= 38 24 10 17 11 } 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 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 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 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 26 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 27 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 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 ⎟ ⎝ ⎠ 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, 15 ↦ 15, 16 ↦ 16, 18 ↦ 17, 17 ↦ 18, 19 ↦ 19, 27 ↦ 20, 32 ↦ 21, 29 ↦ 22, 33 ↦ 23, 34 ↦ 24, 35 ↦ 25, 36 ↦ 26, 37 ↦ 27, 22 ↦ 28, 21 ↦ 29, 20 ↦ 30, 28 ↦ 31, 31 ↦ 32, 30 ↦ 33, 39 ↦ 34, 38 ↦ 35, 24 ↦ 36 }, it remains to prove termination of the 102-rule system { 0 1 2 3 3 ⟶ 4 5 6 3 3 , 0 1 2 3 7 ⟶ 4 5 6 3 7 , 0 1 2 3 8 ⟶ 4 5 6 3 8 , 0 1 2 3 9 ⟶ 4 5 6 3 9 , 0 1 2 7 10 ⟶ 4 5 6 7 10 , 0 1 2 7 11 ⟶ 4 5 6 7 11 , 0 1 2 7 12 ⟶ 4 5 6 7 12 , 4 5 13 10 14 ⟶ 0 15 16 14 3 , 4 5 13 11 17 ⟶ 0 15 16 18 10 , 4 5 13 11 19 ⟶ 0 15 16 18 11 , 14 3 3 3 3 →= 18 10 14 3 3 , 14 3 3 3 7 →= 18 10 14 3 7 , 14 3 3 3 8 →= 18 10 14 3 8 , 14 3 3 3 9 →= 18 10 14 3 9 , 14 3 3 7 10 →= 18 10 14 7 10 , 14 3 3 7 11 →= 18 10 14 7 11 , 14 3 3 7 12 →= 18 10 14 7 12 , 14 3 3 8 20 →= 18 10 14 8 20 , 10 14 3 3 3 →= 11 17 14 3 3 , 10 14 3 3 7 →= 11 17 14 3 7 , 10 14 3 3 8 →= 11 17 14 3 8 , 10 14 3 3 9 →= 11 17 14 3 9 , 16 14 3 3 3 →= 21 17 14 3 3 , 16 14 3 3 7 →= 21 17 14 3 7 , 16 14 3 3 8 →= 21 17 14 3 8 , 16 14 3 3 9 →= 21 17 14 3 9 , 17 14 3 3 3 →= 19 17 14 3 3 , 17 14 3 3 7 →= 19 17 14 3 7 , 17 14 3 3 8 →= 19 17 14 3 8 , 17 14 3 3 9 →= 19 17 14 3 9 , 22 14 3 3 3 →= 23 17 14 3 3 , 22 14 3 3 7 →= 23 17 14 3 7 , 22 14 3 3 8 →= 23 17 14 3 8 , 22 14 3 3 9 →= 23 17 14 3 9 , 24 14 3 3 3 →= 25 17 14 3 3 , 24 14 3 3 7 →= 25 17 14 3 7 , 24 14 3 3 8 →= 25 17 14 3 8 , 24 14 3 3 9 →= 25 17 14 3 9 , 26 14 3 3 3 →= 27 17 14 3 3 , 26 14 3 3 7 →= 27 17 14 3 7 , 26 14 3 3 8 →= 27 17 14 3 8 , 26 14 3 3 9 →= 27 17 14 3 9 , 10 14 3 7 10 →= 11 17 14 7 10 , 10 14 3 7 11 →= 11 17 14 7 11 , 10 14 3 7 12 →= 11 17 14 7 12 , 16 14 3 7 10 →= 21 17 14 7 10 , 16 14 3 7 11 →= 21 17 14 7 11 , 16 14 3 7 12 →= 21 17 14 7 12 , 17 14 3 7 10 →= 19 17 14 7 10 , 17 14 3 7 11 →= 19 17 14 7 11 , 17 14 3 7 12 →= 19 17 14 7 12 , 22 14 3 7 10 →= 23 17 14 7 10 , 22 14 3 7 11 →= 23 17 14 7 11 , 22 14 3 7 12 →= 23 17 14 7 12 , 24 14 3 7 10 →= 25 17 14 7 10 , 24 14 3 7 11 →= 25 17 14 7 11 , 24 14 3 7 12 →= 25 17 14 7 12 , 26 14 3 7 10 →= 27 17 14 7 10 , 26 14 3 7 11 →= 27 17 14 7 11 , 26 14 3 7 12 →= 27 17 14 7 12 , 10 14 3 8 20 →= 11 17 14 8 20 , 16 14 3 8 20 →= 21 17 14 8 20 , 17 14 3 8 20 →= 19 17 14 8 20 , 22 14 3 8 20 →= 23 17 14 8 20 , 24 14 3 8 20 →= 25 17 14 8 20 , 26 14 3 8 20 →= 27 17 14 8 20 , 15 16 18 10 14 →= 1 28 10 14 3 , 29 16 18 10 14 →= 30 28 10 14 3 , 15 16 18 11 17 →= 1 28 10 18 10 , 15 16 18 11 19 →= 1 28 10 18 11 , 29 16 18 11 17 →= 30 28 10 18 10 , 29 16 18 11 19 →= 30 28 10 18 11 , 7 10 18 10 14 →= 3 7 10 14 3 , 7 10 18 11 17 →= 3 7 10 18 10 , 7 10 18 11 19 →= 3 7 10 18 11 , 31 22 18 10 14 →= 5 13 10 14 3 , 32 22 18 10 14 →= 33 13 10 14 3 , 31 22 18 11 17 →= 5 13 10 18 10 , 31 22 18 11 19 →= 5 13 10 18 11 , 32 22 18 11 17 →= 33 13 10 18 10 , 32 22 18 11 19 →= 33 13 10 18 11 , 11 17 18 10 14 →= 10 18 10 14 3 , 21 17 18 10 14 →= 16 18 10 14 3 , 19 17 18 10 14 →= 17 18 10 14 3 , 23 17 18 10 14 →= 22 18 10 14 3 , 25 17 18 10 14 →= 24 18 10 14 3 , 27 17 18 10 14 →= 26 18 10 14 3 , 11 17 18 11 17 →= 10 18 10 18 10 , 11 17 18 11 19 →= 10 18 10 18 11 , 21 17 18 11 17 →= 16 18 10 18 10 , 21 17 18 11 19 →= 16 18 10 18 11 , 19 17 18 11 17 →= 17 18 10 18 10 , 19 17 18 11 19 →= 17 18 10 18 11 , 23 17 18 11 17 →= 22 18 10 18 10 , 23 17 18 11 19 →= 22 18 10 18 11 , 25 17 18 11 17 →= 24 18 10 18 10 , 25 17 18 11 19 →= 24 18 10 18 11 , 27 17 18 11 17 →= 26 18 10 18 10 , 27 17 18 11 19 →= 26 18 10 18 11 , 34 24 18 10 14 →= 35 36 10 14 3 , 34 24 18 11 17 →= 35 36 10 18 10 , 34 24 18 11 19 →= 35 36 10 18 11 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 23 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 24 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 25 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 26 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 27 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 28 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 29 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 30 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 31 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 32 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 33 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 34 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 35 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 36 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 4 ↦ 0, 5 ↦ 1, 13 ↦ 2, 10 ↦ 3, 14 ↦ 4, 0 ↦ 5, 15 ↦ 6, 16 ↦ 7, 3 ↦ 8, 11 ↦ 9, 17 ↦ 10, 18 ↦ 11, 19 ↦ 12, 7 ↦ 13, 8 ↦ 14, 9 ↦ 15, 12 ↦ 16, 20 ↦ 17, 21 ↦ 18, 22 ↦ 19, 23 ↦ 20, 24 ↦ 21, 25 ↦ 22, 26 ↦ 23, 27 ↦ 24, 31 ↦ 25 }, it remains to prove termination of the 83-rule system { 0 1 2 3 4 ⟶ 5 6 7 4 8 , 0 1 2 9 10 ⟶ 5 6 7 11 3 , 0 1 2 9 12 ⟶ 5 6 7 11 9 , 4 8 8 8 8 →= 11 3 4 8 8 , 4 8 8 8 13 →= 11 3 4 8 13 , 4 8 8 8 14 →= 11 3 4 8 14 , 4 8 8 8 15 →= 11 3 4 8 15 , 4 8 8 13 3 →= 11 3 4 13 3 , 4 8 8 13 9 →= 11 3 4 13 9 , 4 8 8 13 16 →= 11 3 4 13 16 , 4 8 8 14 17 →= 11 3 4 14 17 , 3 4 8 8 8 →= 9 10 4 8 8 , 3 4 8 8 13 →= 9 10 4 8 13 , 3 4 8 8 14 →= 9 10 4 8 14 , 3 4 8 8 15 →= 9 10 4 8 15 , 7 4 8 8 8 →= 18 10 4 8 8 , 7 4 8 8 13 →= 18 10 4 8 13 , 7 4 8 8 14 →= 18 10 4 8 14 , 7 4 8 8 15 →= 18 10 4 8 15 , 10 4 8 8 8 →= 12 10 4 8 8 , 10 4 8 8 13 →= 12 10 4 8 13 , 10 4 8 8 14 →= 12 10 4 8 14 , 10 4 8 8 15 →= 12 10 4 8 15 , 19 4 8 8 8 →= 20 10 4 8 8 , 19 4 8 8 13 →= 20 10 4 8 13 , 19 4 8 8 14 →= 20 10 4 8 14 , 19 4 8 8 15 →= 20 10 4 8 15 , 21 4 8 8 8 →= 22 10 4 8 8 , 21 4 8 8 13 →= 22 10 4 8 13 , 21 4 8 8 14 →= 22 10 4 8 14 , 21 4 8 8 15 →= 22 10 4 8 15 , 23 4 8 8 8 →= 24 10 4 8 8 , 23 4 8 8 13 →= 24 10 4 8 13 , 23 4 8 8 14 →= 24 10 4 8 14 , 23 4 8 8 15 →= 24 10 4 8 15 , 3 4 8 13 3 →= 9 10 4 13 3 , 3 4 8 13 9 →= 9 10 4 13 9 , 3 4 8 13 16 →= 9 10 4 13 16 , 7 4 8 13 3 →= 18 10 4 13 3 , 7 4 8 13 9 →= 18 10 4 13 9 , 7 4 8 13 16 →= 18 10 4 13 16 , 10 4 8 13 3 →= 12 10 4 13 3 , 10 4 8 13 9 →= 12 10 4 13 9 , 10 4 8 13 16 →= 12 10 4 13 16 , 19 4 8 13 3 →= 20 10 4 13 3 , 19 4 8 13 9 →= 20 10 4 13 9 , 19 4 8 13 16 →= 20 10 4 13 16 , 21 4 8 13 3 →= 22 10 4 13 3 , 21 4 8 13 9 →= 22 10 4 13 9 , 21 4 8 13 16 →= 22 10 4 13 16 , 23 4 8 13 3 →= 24 10 4 13 3 , 23 4 8 13 9 →= 24 10 4 13 9 , 23 4 8 13 16 →= 24 10 4 13 16 , 3 4 8 14 17 →= 9 10 4 14 17 , 7 4 8 14 17 →= 18 10 4 14 17 , 10 4 8 14 17 →= 12 10 4 14 17 , 19 4 8 14 17 →= 20 10 4 14 17 , 21 4 8 14 17 →= 22 10 4 14 17 , 23 4 8 14 17 →= 24 10 4 14 17 , 13 3 11 3 4 →= 8 13 3 4 8 , 13 3 11 9 10 →= 8 13 3 11 3 , 13 3 11 9 12 →= 8 13 3 11 9 , 25 19 11 3 4 →= 1 2 3 4 8 , 25 19 11 9 10 →= 1 2 3 11 3 , 25 19 11 9 12 →= 1 2 3 11 9 , 9 10 11 3 4 →= 3 11 3 4 8 , 18 10 11 3 4 →= 7 11 3 4 8 , 12 10 11 3 4 →= 10 11 3 4 8 , 20 10 11 3 4 →= 19 11 3 4 8 , 22 10 11 3 4 →= 21 11 3 4 8 , 24 10 11 3 4 →= 23 11 3 4 8 , 9 10 11 9 10 →= 3 11 3 11 3 , 9 10 11 9 12 →= 3 11 3 11 9 , 18 10 11 9 10 →= 7 11 3 11 3 , 18 10 11 9 12 →= 7 11 3 11 9 , 12 10 11 9 10 →= 10 11 3 11 3 , 12 10 11 9 12 →= 10 11 3 11 9 , 20 10 11 9 10 →= 19 11 3 11 3 , 20 10 11 9 12 →= 19 11 3 11 9 , 22 10 11 9 10 →= 21 11 3 11 3 , 22 10 11 9 12 →= 21 11 3 11 9 , 24 10 11 9 10 →= 23 11 3 11 3 , 24 10 11 9 12 →= 23 11 3 11 9 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 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 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 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 4 ↦ 0, 8 ↦ 1, 11 ↦ 2, 3 ↦ 3, 13 ↦ 4, 14 ↦ 5, 15 ↦ 6, 9 ↦ 7, 16 ↦ 8, 17 ↦ 9, 10 ↦ 10, 7 ↦ 11, 18 ↦ 12, 12 ↦ 13, 19 ↦ 14, 20 ↦ 15, 21 ↦ 16, 22 ↦ 17, 23 ↦ 18, 24 ↦ 19, 25 ↦ 20, 1 ↦ 21, 2 ↦ 22 }, it remains to prove termination of the 80-rule system { 0 1 1 1 1 →= 2 3 0 1 1 , 0 1 1 1 4 →= 2 3 0 1 4 , 0 1 1 1 5 →= 2 3 0 1 5 , 0 1 1 1 6 →= 2 3 0 1 6 , 0 1 1 4 3 →= 2 3 0 4 3 , 0 1 1 4 7 →= 2 3 0 4 7 , 0 1 1 4 8 →= 2 3 0 4 8 , 0 1 1 5 9 →= 2 3 0 5 9 , 3 0 1 1 1 →= 7 10 0 1 1 , 3 0 1 1 4 →= 7 10 0 1 4 , 3 0 1 1 5 →= 7 10 0 1 5 , 3 0 1 1 6 →= 7 10 0 1 6 , 11 0 1 1 1 →= 12 10 0 1 1 , 11 0 1 1 4 →= 12 10 0 1 4 , 11 0 1 1 5 →= 12 10 0 1 5 , 11 0 1 1 6 →= 12 10 0 1 6 , 10 0 1 1 1 →= 13 10 0 1 1 , 10 0 1 1 4 →= 13 10 0 1 4 , 10 0 1 1 5 →= 13 10 0 1 5 , 10 0 1 1 6 →= 13 10 0 1 6 , 14 0 1 1 1 →= 15 10 0 1 1 , 14 0 1 1 4 →= 15 10 0 1 4 , 14 0 1 1 5 →= 15 10 0 1 5 , 14 0 1 1 6 →= 15 10 0 1 6 , 16 0 1 1 1 →= 17 10 0 1 1 , 16 0 1 1 4 →= 17 10 0 1 4 , 16 0 1 1 5 →= 17 10 0 1 5 , 16 0 1 1 6 →= 17 10 0 1 6 , 18 0 1 1 1 →= 19 10 0 1 1 , 18 0 1 1 4 →= 19 10 0 1 4 , 18 0 1 1 5 →= 19 10 0 1 5 , 18 0 1 1 6 →= 19 10 0 1 6 , 3 0 1 4 3 →= 7 10 0 4 3 , 3 0 1 4 7 →= 7 10 0 4 7 , 3 0 1 4 8 →= 7 10 0 4 8 , 11 0 1 4 3 →= 12 10 0 4 3 , 11 0 1 4 7 →= 12 10 0 4 7 , 11 0 1 4 8 →= 12 10 0 4 8 , 10 0 1 4 3 →= 13 10 0 4 3 , 10 0 1 4 7 →= 13 10 0 4 7 , 10 0 1 4 8 →= 13 10 0 4 8 , 14 0 1 4 3 →= 15 10 0 4 3 , 14 0 1 4 7 →= 15 10 0 4 7 , 14 0 1 4 8 →= 15 10 0 4 8 , 16 0 1 4 3 →= 17 10 0 4 3 , 16 0 1 4 7 →= 17 10 0 4 7 , 16 0 1 4 8 →= 17 10 0 4 8 , 18 0 1 4 3 →= 19 10 0 4 3 , 18 0 1 4 7 →= 19 10 0 4 7 , 18 0 1 4 8 →= 19 10 0 4 8 , 3 0 1 5 9 →= 7 10 0 5 9 , 11 0 1 5 9 →= 12 10 0 5 9 , 10 0 1 5 9 →= 13 10 0 5 9 , 14 0 1 5 9 →= 15 10 0 5 9 , 16 0 1 5 9 →= 17 10 0 5 9 , 18 0 1 5 9 →= 19 10 0 5 9 , 4 3 2 3 0 →= 1 4 3 0 1 , 4 3 2 7 10 →= 1 4 3 2 3 , 4 3 2 7 13 →= 1 4 3 2 7 , 20 14 2 3 0 →= 21 22 3 0 1 , 20 14 2 7 10 →= 21 22 3 2 3 , 20 14 2 7 13 →= 21 22 3 2 7 , 7 10 2 3 0 →= 3 2 3 0 1 , 12 10 2 3 0 →= 11 2 3 0 1 , 13 10 2 3 0 →= 10 2 3 0 1 , 15 10 2 3 0 →= 14 2 3 0 1 , 17 10 2 3 0 →= 16 2 3 0 1 , 19 10 2 3 0 →= 18 2 3 0 1 , 7 10 2 7 10 →= 3 2 3 2 3 , 7 10 2 7 13 →= 3 2 3 2 7 , 12 10 2 7 10 →= 11 2 3 2 3 , 12 10 2 7 13 →= 11 2 3 2 7 , 13 10 2 7 10 →= 10 2 3 2 3 , 13 10 2 7 13 →= 10 2 3 2 7 , 15 10 2 7 10 →= 14 2 3 2 3 , 15 10 2 7 13 →= 14 2 3 2 7 , 17 10 2 7 10 →= 16 2 3 2 3 , 17 10 2 7 13 →= 16 2 3 2 7 , 19 10 2 7 10 →= 18 2 3 2 3 , 19 10 2 7 13 →= 18 2 3 2 7 } The system is trivially terminating.