/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES After renaming modulo the bijection { b ↦ 0, a ↦ 1 }, it remains to prove termination of the 2-rule system { 0 0 0 0 ⟶ 0 0 0 1 0 , 0 1 0 0 1 0 ⟶ 0 0 1 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, (0,3) ↦ 3, (2,0) ↦ 4 }, it remains to prove termination of the 18-rule system { 0 0 0 0 0 ⟶ 0 0 0 1 2 0 , 0 0 0 0 1 ⟶ 0 0 0 1 2 1 , 0 0 0 0 3 ⟶ 0 0 0 1 2 3 , 2 0 0 0 0 ⟶ 2 0 0 1 2 0 , 2 0 0 0 1 ⟶ 2 0 0 1 2 1 , 2 0 0 0 3 ⟶ 2 0 0 1 2 3 , 4 0 0 0 0 ⟶ 4 0 0 1 2 0 , 4 0 0 0 1 ⟶ 4 0 0 1 2 1 , 4 0 0 0 3 ⟶ 4 0 0 1 2 3 , 0 1 2 0 1 2 0 ⟶ 0 0 1 2 0 0 , 0 1 2 0 1 2 1 ⟶ 0 0 1 2 0 1 , 0 1 2 0 1 2 3 ⟶ 0 0 1 2 0 3 , 2 1 2 0 1 2 0 ⟶ 2 0 1 2 0 0 , 2 1 2 0 1 2 1 ⟶ 2 0 1 2 0 1 , 2 1 2 0 1 2 3 ⟶ 2 0 1 2 0 3 , 4 1 2 0 1 2 0 ⟶ 4 0 1 2 0 0 , 4 1 2 0 1 2 1 ⟶ 4 0 1 2 0 1 , 4 1 2 0 1 2 3 ⟶ 4 0 1 2 0 3 } 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 }, it remains to prove termination of the 15-rule system { 0 0 0 0 0 ⟶ 0 0 0 1 2 0 , 0 0 0 0 1 ⟶ 0 0 0 1 2 1 , 0 0 0 0 3 ⟶ 0 0 0 1 2 3 , 2 0 0 0 0 ⟶ 2 0 0 1 2 0 , 2 0 0 0 1 ⟶ 2 0 0 1 2 1 , 2 0 0 0 3 ⟶ 2 0 0 1 2 3 , 4 0 0 0 0 ⟶ 4 0 0 1 2 0 , 4 0 0 0 1 ⟶ 4 0 0 1 2 1 , 4 0 0 0 3 ⟶ 4 0 0 1 2 3 , 0 1 2 0 1 2 0 ⟶ 0 0 1 2 0 0 , 0 1 2 0 1 2 1 ⟶ 0 0 1 2 0 1 , 0 1 2 0 1 2 3 ⟶ 0 0 1 2 0 3 , 2 1 2 0 1 2 0 ⟶ 2 0 1 2 0 0 , 2 1 2 0 1 2 1 ⟶ 2 0 1 2 0 1 , 2 1 2 0 1 2 3 ⟶ 2 0 1 2 0 3 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 6: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 1 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 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4 }, it remains to prove termination of the 14-rule system { 0 0 0 0 0 ⟶ 0 0 0 1 2 0 , 0 0 0 0 1 ⟶ 0 0 0 1 2 1 , 0 0 0 0 3 ⟶ 0 0 0 1 2 3 , 2 0 0 0 0 ⟶ 2 0 0 1 2 0 , 2 0 0 0 1 ⟶ 2 0 0 1 2 1 , 2 0 0 0 3 ⟶ 2 0 0 1 2 3 , 4 0 0 0 0 ⟶ 4 0 0 1 2 0 , 4 0 0 0 1 ⟶ 4 0 0 1 2 1 , 0 1 2 0 1 2 0 ⟶ 0 0 1 2 0 0 , 0 1 2 0 1 2 1 ⟶ 0 0 1 2 0 1 , 0 1 2 0 1 2 3 ⟶ 0 0 1 2 0 3 , 2 1 2 0 1 2 0 ⟶ 2 0 1 2 0 0 , 2 1 2 0 1 2 1 ⟶ 2 0 1 2 0 1 , 2 1 2 0 1 2 3 ⟶ 2 0 1 2 0 3 } The system was reversed. After renaming modulo the bijection { 0 ↦ 0, 2 ↦ 1, 1 ↦ 2, 3 ↦ 3, 4 ↦ 4 }, it remains to prove termination of the 14-rule system { 0 0 0 0 0 ⟶ 0 1 2 0 0 0 , 2 0 0 0 0 ⟶ 2 1 2 0 0 0 , 3 0 0 0 0 ⟶ 3 1 2 0 0 0 , 0 0 0 0 1 ⟶ 0 1 2 0 0 1 , 2 0 0 0 1 ⟶ 2 1 2 0 0 1 , 3 0 0 0 1 ⟶ 3 1 2 0 0 1 , 0 0 0 0 4 ⟶ 0 1 2 0 0 4 , 2 0 0 0 4 ⟶ 2 1 2 0 0 4 , 0 1 2 0 1 2 0 ⟶ 0 0 1 2 0 0 , 2 1 2 0 1 2 0 ⟶ 2 0 1 2 0 0 , 3 1 2 0 1 2 0 ⟶ 3 0 1 2 0 0 , 0 1 2 0 1 2 1 ⟶ 0 0 1 2 0 1 , 2 1 2 0 1 2 1 ⟶ 2 0 1 2 0 1 , 3 1 2 0 1 2 1 ⟶ 3 0 1 2 0 1 } Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols. After renaming modulo the bijection { (0,↑) ↦ 0, (0,↓) ↦ 1, (1,↓) ↦ 2, (2,↓) ↦ 3, (2,↑) ↦ 4, (3,↑) ↦ 5, (4,↓) ↦ 6, (3,↓) ↦ 7 }, it remains to prove termination of the 54-rule system { 0 1 1 1 1 ⟶ 0 2 3 1 1 1 , 0 1 1 1 1 ⟶ 4 1 1 1 , 4 1 1 1 1 ⟶ 4 2 3 1 1 1 , 4 1 1 1 1 ⟶ 4 1 1 1 , 5 1 1 1 1 ⟶ 5 2 3 1 1 1 , 5 1 1 1 1 ⟶ 4 1 1 1 , 0 1 1 1 2 ⟶ 0 2 3 1 1 2 , 0 1 1 1 2 ⟶ 4 1 1 2 , 4 1 1 1 2 ⟶ 4 2 3 1 1 2 , 4 1 1 1 2 ⟶ 4 1 1 2 , 5 1 1 1 2 ⟶ 5 2 3 1 1 2 , 5 1 1 1 2 ⟶ 4 1 1 2 , 0 1 1 1 6 ⟶ 0 2 3 1 1 6 , 0 1 1 1 6 ⟶ 4 1 1 6 , 4 1 1 1 6 ⟶ 4 2 3 1 1 6 , 4 1 1 1 6 ⟶ 4 1 1 6 , 0 2 3 1 2 3 1 ⟶ 0 1 2 3 1 1 , 0 2 3 1 2 3 1 ⟶ 0 2 3 1 1 , 0 2 3 1 2 3 1 ⟶ 4 1 1 , 0 2 3 1 2 3 1 ⟶ 0 1 , 4 2 3 1 2 3 1 ⟶ 4 1 2 3 1 1 , 4 2 3 1 2 3 1 ⟶ 0 2 3 1 1 , 4 2 3 1 2 3 1 ⟶ 4 1 1 , 4 2 3 1 2 3 1 ⟶ 0 1 , 5 2 3 1 2 3 1 ⟶ 5 1 2 3 1 1 , 5 2 3 1 2 3 1 ⟶ 0 2 3 1 1 , 5 2 3 1 2 3 1 ⟶ 4 1 1 , 5 2 3 1 2 3 1 ⟶ 0 1 , 0 2 3 1 2 3 2 ⟶ 0 1 2 3 1 2 , 0 2 3 1 2 3 2 ⟶ 0 2 3 1 2 , 0 2 3 1 2 3 2 ⟶ 4 1 2 , 0 2 3 1 2 3 2 ⟶ 0 2 , 4 2 3 1 2 3 2 ⟶ 4 1 2 3 1 2 , 4 2 3 1 2 3 2 ⟶ 0 2 3 1 2 , 4 2 3 1 2 3 2 ⟶ 4 1 2 , 4 2 3 1 2 3 2 ⟶ 0 2 , 5 2 3 1 2 3 2 ⟶ 5 1 2 3 1 2 , 5 2 3 1 2 3 2 ⟶ 0 2 3 1 2 , 5 2 3 1 2 3 2 ⟶ 4 1 2 , 5 2 3 1 2 3 2 ⟶ 0 2 , 1 1 1 1 1 →= 1 2 3 1 1 1 , 3 1 1 1 1 →= 3 2 3 1 1 1 , 7 1 1 1 1 →= 7 2 3 1 1 1 , 1 1 1 1 2 →= 1 2 3 1 1 2 , 3 1 1 1 2 →= 3 2 3 1 1 2 , 7 1 1 1 2 →= 7 2 3 1 1 2 , 1 1 1 1 6 →= 1 2 3 1 1 6 , 3 1 1 1 6 →= 3 2 3 1 1 6 , 1 2 3 1 2 3 1 →= 1 1 2 3 1 1 , 3 2 3 1 2 3 1 →= 3 1 2 3 1 1 , 7 2 3 1 2 3 1 →= 7 1 2 3 1 1 , 1 2 3 1 2 3 2 →= 1 1 2 3 1 2 , 3 2 3 1 2 3 2 →= 3 1 2 3 1 2 , 7 2 3 1 2 3 2 →= 7 1 2 3 1 2 } 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 }, it remains to prove termination of the 46-rule system { 0 1 1 1 1 ⟶ 0 2 3 1 1 1 , 0 1 1 1 1 ⟶ 4 1 1 1 , 4 1 1 1 1 ⟶ 4 2 3 1 1 1 , 4 1 1 1 1 ⟶ 4 1 1 1 , 5 1 1 1 1 ⟶ 5 2 3 1 1 1 , 0 1 1 1 2 ⟶ 0 2 3 1 1 2 , 0 1 1 1 2 ⟶ 4 1 1 2 , 4 1 1 1 2 ⟶ 4 2 3 1 1 2 , 4 1 1 1 2 ⟶ 4 1 1 2 , 5 1 1 1 2 ⟶ 5 2 3 1 1 2 , 0 1 1 1 6 ⟶ 0 2 3 1 1 6 , 0 1 1 1 6 ⟶ 4 1 1 6 , 4 1 1 1 6 ⟶ 4 2 3 1 1 6 , 4 1 1 1 6 ⟶ 4 1 1 6 , 0 2 3 1 2 3 1 ⟶ 0 1 2 3 1 1 , 0 2 3 1 2 3 1 ⟶ 0 2 3 1 1 , 0 2 3 1 2 3 1 ⟶ 4 1 1 , 0 2 3 1 2 3 1 ⟶ 0 1 , 4 2 3 1 2 3 1 ⟶ 4 1 2 3 1 1 , 4 2 3 1 2 3 1 ⟶ 0 2 3 1 1 , 4 2 3 1 2 3 1 ⟶ 4 1 1 , 4 2 3 1 2 3 1 ⟶ 0 1 , 5 2 3 1 2 3 1 ⟶ 5 1 2 3 1 1 , 0 2 3 1 2 3 2 ⟶ 0 1 2 3 1 2 , 0 2 3 1 2 3 2 ⟶ 0 2 3 1 2 , 0 2 3 1 2 3 2 ⟶ 4 1 2 , 0 2 3 1 2 3 2 ⟶ 0 2 , 4 2 3 1 2 3 2 ⟶ 4 1 2 3 1 2 , 4 2 3 1 2 3 2 ⟶ 0 2 3 1 2 , 4 2 3 1 2 3 2 ⟶ 4 1 2 , 4 2 3 1 2 3 2 ⟶ 0 2 , 5 2 3 1 2 3 2 ⟶ 5 1 2 3 1 2 , 1 1 1 1 1 →= 1 2 3 1 1 1 , 3 1 1 1 1 →= 3 2 3 1 1 1 , 7 1 1 1 1 →= 7 2 3 1 1 1 , 1 1 1 1 2 →= 1 2 3 1 1 2 , 3 1 1 1 2 →= 3 2 3 1 1 2 , 7 1 1 1 2 →= 7 2 3 1 1 2 , 1 1 1 1 6 →= 1 2 3 1 1 6 , 3 1 1 1 6 →= 3 2 3 1 1 6 , 1 2 3 1 2 3 1 →= 1 1 2 3 1 1 , 3 2 3 1 2 3 1 →= 3 1 2 3 1 1 , 7 2 3 1 2 3 1 →= 7 1 2 3 1 1 , 1 2 3 1 2 3 2 →= 1 1 2 3 1 2 , 3 2 3 1 2 3 2 →= 3 1 2 3 1 2 , 7 2 3 1 2 3 2 →= 7 1 2 3 1 2 } 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 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, 7 ↦ 7 }, it remains to prove termination of the 28-rule system { 0 1 1 1 1 ⟶ 0 2 3 1 1 1 , 4 1 1 1 1 ⟶ 4 2 3 1 1 1 , 5 1 1 1 1 ⟶ 5 2 3 1 1 1 , 0 1 1 1 2 ⟶ 0 2 3 1 1 2 , 4 1 1 1 2 ⟶ 4 2 3 1 1 2 , 5 1 1 1 2 ⟶ 5 2 3 1 1 2 , 0 1 1 1 6 ⟶ 0 2 3 1 1 6 , 4 1 1 1 6 ⟶ 4 2 3 1 1 6 , 0 2 3 1 2 3 1 ⟶ 0 1 2 3 1 1 , 4 2 3 1 2 3 1 ⟶ 4 1 2 3 1 1 , 5 2 3 1 2 3 1 ⟶ 5 1 2 3 1 1 , 0 2 3 1 2 3 2 ⟶ 0 1 2 3 1 2 , 4 2 3 1 2 3 2 ⟶ 4 1 2 3 1 2 , 5 2 3 1 2 3 2 ⟶ 5 1 2 3 1 2 , 1 1 1 1 1 →= 1 2 3 1 1 1 , 3 1 1 1 1 →= 3 2 3 1 1 1 , 7 1 1 1 1 →= 7 2 3 1 1 1 , 1 1 1 1 2 →= 1 2 3 1 1 2 , 3 1 1 1 2 →= 3 2 3 1 1 2 , 7 1 1 1 2 →= 7 2 3 1 1 2 , 1 1 1 1 6 →= 1 2 3 1 1 6 , 3 1 1 1 6 →= 3 2 3 1 1 6 , 1 2 3 1 2 3 1 →= 1 1 2 3 1 1 , 3 2 3 1 2 3 1 →= 3 1 2 3 1 1 , 7 2 3 1 2 3 1 →= 7 1 2 3 1 1 , 1 2 3 1 2 3 2 →= 1 1 2 3 1 2 , 3 2 3 1 2 3 2 →= 3 1 2 3 1 2 , 7 2 3 1 2 3 2 →= 7 1 2 3 1 2 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 6: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 1 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 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7 }, it remains to prove termination of the 27-rule system { 0 1 1 1 1 ⟶ 0 2 3 1 1 1 , 4 1 1 1 1 ⟶ 4 2 3 1 1 1 , 5 1 1 1 1 ⟶ 5 2 3 1 1 1 , 0 1 1 1 2 ⟶ 0 2 3 1 1 2 , 4 1 1 1 2 ⟶ 4 2 3 1 1 2 , 5 1 1 1 2 ⟶ 5 2 3 1 1 2 , 4 1 1 1 6 ⟶ 4 2 3 1 1 6 , 0 2 3 1 2 3 1 ⟶ 0 1 2 3 1 1 , 4 2 3 1 2 3 1 ⟶ 4 1 2 3 1 1 , 5 2 3 1 2 3 1 ⟶ 5 1 2 3 1 1 , 0 2 3 1 2 3 2 ⟶ 0 1 2 3 1 2 , 4 2 3 1 2 3 2 ⟶ 4 1 2 3 1 2 , 5 2 3 1 2 3 2 ⟶ 5 1 2 3 1 2 , 1 1 1 1 1 →= 1 2 3 1 1 1 , 3 1 1 1 1 →= 3 2 3 1 1 1 , 7 1 1 1 1 →= 7 2 3 1 1 1 , 1 1 1 1 2 →= 1 2 3 1 1 2 , 3 1 1 1 2 →= 3 2 3 1 1 2 , 7 1 1 1 2 →= 7 2 3 1 1 2 , 1 1 1 1 6 →= 1 2 3 1 1 6 , 3 1 1 1 6 →= 3 2 3 1 1 6 , 1 2 3 1 2 3 1 →= 1 1 2 3 1 1 , 3 2 3 1 2 3 1 →= 3 1 2 3 1 1 , 7 2 3 1 2 3 1 →= 7 1 2 3 1 1 , 1 2 3 1 2 3 2 →= 1 1 2 3 1 2 , 3 2 3 1 2 3 2 →= 3 1 2 3 1 2 , 7 2 3 1 2 3 2 →= 7 1 2 3 1 2 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 6: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 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 ⎟ ⎝ ⎠ 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 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 7 ↦ 6, 6 ↦ 7 }, it remains to prove termination of the 26-rule system { 0 1 1 1 1 ⟶ 0 2 3 1 1 1 , 4 1 1 1 1 ⟶ 4 2 3 1 1 1 , 5 1 1 1 1 ⟶ 5 2 3 1 1 1 , 0 1 1 1 2 ⟶ 0 2 3 1 1 2 , 4 1 1 1 2 ⟶ 4 2 3 1 1 2 , 5 1 1 1 2 ⟶ 5 2 3 1 1 2 , 0 2 3 1 2 3 1 ⟶ 0 1 2 3 1 1 , 4 2 3 1 2 3 1 ⟶ 4 1 2 3 1 1 , 5 2 3 1 2 3 1 ⟶ 5 1 2 3 1 1 , 0 2 3 1 2 3 2 ⟶ 0 1 2 3 1 2 , 4 2 3 1 2 3 2 ⟶ 4 1 2 3 1 2 , 5 2 3 1 2 3 2 ⟶ 5 1 2 3 1 2 , 1 1 1 1 1 →= 1 2 3 1 1 1 , 3 1 1 1 1 →= 3 2 3 1 1 1 , 6 1 1 1 1 →= 6 2 3 1 1 1 , 1 1 1 1 2 →= 1 2 3 1 1 2 , 3 1 1 1 2 →= 3 2 3 1 1 2 , 6 1 1 1 2 →= 6 2 3 1 1 2 , 1 1 1 1 7 →= 1 2 3 1 1 7 , 3 1 1 1 7 →= 3 2 3 1 1 7 , 1 2 3 1 2 3 1 →= 1 1 2 3 1 1 , 3 2 3 1 2 3 1 →= 3 1 2 3 1 1 , 6 2 3 1 2 3 1 →= 6 1 2 3 1 1 , 1 2 3 1 2 3 2 →= 1 1 2 3 1 2 , 3 2 3 1 2 3 2 →= 3 1 2 3 1 2 , 6 2 3 1 2 3 2 →= 6 1 2 3 1 2 } Applying sparse tiling TROC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (8,0) ↦ 0, (0,1) ↦ 1, (1,1) ↦ 2, (0,2) ↦ 3, (2,3) ↦ 4, (3,1) ↦ 5, (1,2) ↦ 6, (1,7) ↦ 7, (1,9) ↦ 8, (8,4) ↦ 9, (4,1) ↦ 10, (4,2) ↦ 11, (8,5) ↦ 12, (5,1) ↦ 13, (5,2) ↦ 14, (2,9) ↦ 15, (3,2) ↦ 16, (6,1) ↦ 17, (8,1) ↦ 18, (8,3) ↦ 19, (8,6) ↦ 20, (6,2) ↦ 21, (7,9) ↦ 22 }, it remains to prove termination of the 165-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 0 1 2 2 2 7 ⟶ 0 3 4 5 2 2 7 , 0 1 2 2 2 8 ⟶ 0 3 4 5 2 2 8 , 9 10 2 2 2 2 ⟶ 9 11 4 5 2 2 2 , 9 10 2 2 2 6 ⟶ 9 11 4 5 2 2 6 , 9 10 2 2 2 7 ⟶ 9 11 4 5 2 2 7 , 9 10 2 2 2 8 ⟶ 9 11 4 5 2 2 8 , 12 13 2 2 2 2 ⟶ 12 14 4 5 2 2 2 , 12 13 2 2 2 6 ⟶ 12 14 4 5 2 2 6 , 12 13 2 2 2 7 ⟶ 12 14 4 5 2 2 7 , 12 13 2 2 2 8 ⟶ 12 14 4 5 2 2 8 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 0 1 2 2 6 15 ⟶ 0 3 4 5 2 6 15 , 9 10 2 2 6 4 ⟶ 9 11 4 5 2 6 4 , 9 10 2 2 6 15 ⟶ 9 11 4 5 2 6 15 , 12 13 2 2 6 4 ⟶ 12 14 4 5 2 6 4 , 12 13 2 2 6 15 ⟶ 12 14 4 5 2 6 15 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 0 3 4 5 6 4 5 7 ⟶ 0 1 6 4 5 2 7 , 0 3 4 5 6 4 5 8 ⟶ 0 1 6 4 5 2 8 , 9 11 4 5 6 4 5 2 ⟶ 9 10 6 4 5 2 2 , 9 11 4 5 6 4 5 6 ⟶ 9 10 6 4 5 2 6 , 9 11 4 5 6 4 5 7 ⟶ 9 10 6 4 5 2 7 , 9 11 4 5 6 4 5 8 ⟶ 9 10 6 4 5 2 8 , 12 14 4 5 6 4 5 2 ⟶ 12 13 6 4 5 2 2 , 12 14 4 5 6 4 5 6 ⟶ 12 13 6 4 5 2 6 , 12 14 4 5 6 4 5 7 ⟶ 12 13 6 4 5 2 7 , 12 14 4 5 6 4 5 8 ⟶ 12 13 6 4 5 2 8 , 0 3 4 5 6 4 16 4 ⟶ 0 1 6 4 5 6 4 , 0 3 4 5 6 4 16 15 ⟶ 0 1 6 4 5 6 15 , 9 11 4 5 6 4 16 4 ⟶ 9 10 6 4 5 6 4 , 9 11 4 5 6 4 16 15 ⟶ 9 10 6 4 5 6 15 , 12 14 4 5 6 4 16 4 ⟶ 12 13 6 4 5 6 4 , 12 14 4 5 6 4 16 15 ⟶ 12 13 6 4 5 6 15 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 1 2 2 2 2 7 →= 1 6 4 5 2 2 7 , 1 2 2 2 2 8 →= 1 6 4 5 2 2 8 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 7 →= 2 6 4 5 2 2 7 , 2 2 2 2 2 8 →= 2 6 4 5 2 2 8 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 7 →= 5 6 4 5 2 2 7 , 5 2 2 2 2 8 →= 5 6 4 5 2 2 8 , 10 2 2 2 2 2 →= 10 6 4 5 2 2 2 , 10 2 2 2 2 6 →= 10 6 4 5 2 2 6 , 10 2 2 2 2 7 →= 10 6 4 5 2 2 7 , 10 2 2 2 2 8 →= 10 6 4 5 2 2 8 , 13 2 2 2 2 2 →= 13 6 4 5 2 2 2 , 13 2 2 2 2 6 →= 13 6 4 5 2 2 6 , 13 2 2 2 2 7 →= 13 6 4 5 2 2 7 , 13 2 2 2 2 8 →= 13 6 4 5 2 2 8 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 17 2 2 2 2 7 →= 17 6 4 5 2 2 7 , 17 2 2 2 2 8 →= 17 6 4 5 2 2 8 , 18 2 2 2 2 2 →= 18 6 4 5 2 2 2 , 18 2 2 2 2 6 →= 18 6 4 5 2 2 6 , 18 2 2 2 2 7 →= 18 6 4 5 2 2 7 , 18 2 2 2 2 8 →= 18 6 4 5 2 2 8 , 4 5 2 2 2 2 →= 4 16 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 16 4 5 2 2 6 , 4 5 2 2 2 7 →= 4 16 4 5 2 2 7 , 4 5 2 2 2 8 →= 4 16 4 5 2 2 8 , 19 5 2 2 2 2 →= 19 16 4 5 2 2 2 , 19 5 2 2 2 6 →= 19 16 4 5 2 2 6 , 19 5 2 2 2 7 →= 19 16 4 5 2 2 7 , 19 5 2 2 2 8 →= 19 16 4 5 2 2 8 , 20 17 2 2 2 2 →= 20 21 4 5 2 2 2 , 20 17 2 2 2 6 →= 20 21 4 5 2 2 6 , 20 17 2 2 2 7 →= 20 21 4 5 2 2 7 , 20 17 2 2 2 8 →= 20 21 4 5 2 2 8 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 1 2 2 2 6 15 →= 1 6 4 5 2 6 15 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 15 →= 2 6 4 5 2 6 15 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 15 →= 5 6 4 5 2 6 15 , 10 2 2 2 6 4 →= 10 6 4 5 2 6 4 , 10 2 2 2 6 15 →= 10 6 4 5 2 6 15 , 13 2 2 2 6 4 →= 13 6 4 5 2 6 4 , 13 2 2 2 6 15 →= 13 6 4 5 2 6 15 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 15 →= 17 6 4 5 2 6 15 , 18 2 2 2 6 4 →= 18 6 4 5 2 6 4 , 18 2 2 2 6 15 →= 18 6 4 5 2 6 15 , 4 5 2 2 6 4 →= 4 16 4 5 2 6 4 , 4 5 2 2 6 15 →= 4 16 4 5 2 6 15 , 19 5 2 2 6 4 →= 19 16 4 5 2 6 4 , 19 5 2 2 6 15 →= 19 16 4 5 2 6 15 , 20 17 2 2 6 4 →= 20 21 4 5 2 6 4 , 20 17 2 2 6 15 →= 20 21 4 5 2 6 15 , 1 2 2 2 7 22 →= 1 6 4 5 2 7 22 , 2 2 2 2 7 22 →= 2 6 4 5 2 7 22 , 5 2 2 2 7 22 →= 5 6 4 5 2 7 22 , 10 2 2 2 7 22 →= 10 6 4 5 2 7 22 , 13 2 2 2 7 22 →= 13 6 4 5 2 7 22 , 17 2 2 2 7 22 →= 17 6 4 5 2 7 22 , 18 2 2 2 7 22 →= 18 6 4 5 2 7 22 , 4 5 2 2 7 22 →= 4 16 4 5 2 7 22 , 19 5 2 2 7 22 →= 19 16 4 5 2 7 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 1 6 4 5 6 4 5 7 →= 1 2 6 4 5 2 7 , 1 6 4 5 6 4 5 8 →= 1 2 6 4 5 2 8 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 2 6 4 5 6 4 5 7 →= 2 2 6 4 5 2 7 , 2 6 4 5 6 4 5 8 →= 2 2 6 4 5 2 8 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 5 6 4 5 6 4 5 7 →= 5 2 6 4 5 2 7 , 5 6 4 5 6 4 5 8 →= 5 2 6 4 5 2 8 , 10 6 4 5 6 4 5 2 →= 10 2 6 4 5 2 2 , 10 6 4 5 6 4 5 6 →= 10 2 6 4 5 2 6 , 10 6 4 5 6 4 5 7 →= 10 2 6 4 5 2 7 , 10 6 4 5 6 4 5 8 →= 10 2 6 4 5 2 8 , 13 6 4 5 6 4 5 2 →= 13 2 6 4 5 2 2 , 13 6 4 5 6 4 5 6 →= 13 2 6 4 5 2 6 , 13 6 4 5 6 4 5 7 →= 13 2 6 4 5 2 7 , 13 6 4 5 6 4 5 8 →= 13 2 6 4 5 2 8 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 17 6 4 5 6 4 5 7 →= 17 2 6 4 5 2 7 , 17 6 4 5 6 4 5 8 →= 17 2 6 4 5 2 8 , 18 6 4 5 6 4 5 2 →= 18 2 6 4 5 2 2 , 18 6 4 5 6 4 5 6 →= 18 2 6 4 5 2 6 , 18 6 4 5 6 4 5 7 →= 18 2 6 4 5 2 7 , 18 6 4 5 6 4 5 8 →= 18 2 6 4 5 2 8 , 4 16 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 16 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 4 16 4 5 6 4 5 7 →= 4 5 6 4 5 2 7 , 4 16 4 5 6 4 5 8 →= 4 5 6 4 5 2 8 , 19 16 4 5 6 4 5 2 →= 19 5 6 4 5 2 2 , 19 16 4 5 6 4 5 6 →= 19 5 6 4 5 2 6 , 19 16 4 5 6 4 5 7 →= 19 5 6 4 5 2 7 , 19 16 4 5 6 4 5 8 →= 19 5 6 4 5 2 8 , 20 21 4 5 6 4 5 2 →= 20 17 6 4 5 2 2 , 20 21 4 5 6 4 5 6 →= 20 17 6 4 5 2 6 , 20 21 4 5 6 4 5 7 →= 20 17 6 4 5 2 7 , 20 21 4 5 6 4 5 8 →= 20 17 6 4 5 2 8 , 1 6 4 5 6 4 16 4 →= 1 2 6 4 5 6 4 , 1 6 4 5 6 4 16 15 →= 1 2 6 4 5 6 15 , 2 6 4 5 6 4 16 4 →= 2 2 6 4 5 6 4 , 2 6 4 5 6 4 16 15 →= 2 2 6 4 5 6 15 , 5 6 4 5 6 4 16 4 →= 5 2 6 4 5 6 4 , 5 6 4 5 6 4 16 15 →= 5 2 6 4 5 6 15 , 10 6 4 5 6 4 16 4 →= 10 2 6 4 5 6 4 , 10 6 4 5 6 4 16 15 →= 10 2 6 4 5 6 15 , 13 6 4 5 6 4 16 4 →= 13 2 6 4 5 6 4 , 13 6 4 5 6 4 16 15 →= 13 2 6 4 5 6 15 , 17 6 4 5 6 4 16 4 →= 17 2 6 4 5 6 4 , 17 6 4 5 6 4 16 15 →= 17 2 6 4 5 6 15 , 18 6 4 5 6 4 16 4 →= 18 2 6 4 5 6 4 , 18 6 4 5 6 4 16 15 →= 18 2 6 4 5 6 15 , 4 16 4 5 6 4 16 4 →= 4 5 6 4 5 6 4 , 4 16 4 5 6 4 16 15 →= 4 5 6 4 5 6 15 , 19 16 4 5 6 4 16 4 →= 19 5 6 4 5 6 4 , 19 16 4 5 6 4 16 15 →= 19 5 6 4 5 6 15 , 20 21 4 5 6 4 16 4 →= 20 17 6 4 5 6 4 , 20 21 4 5 6 4 16 15 →= 20 17 6 4 5 6 15 } Applying sparse untiling TROCU(2) after reversal [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { 2 ↦ 0, 1 ↦ 1, 0 ↦ 2, 5 ↦ 3, 4 ↦ 4, 3 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 10 ↦ 9, 9 ↦ 10, 11 ↦ 11, 13 ↦ 12, 12 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 126-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 0 1 2 2 2 7 ⟶ 0 3 4 5 2 2 7 , 0 1 2 2 2 8 ⟶ 0 3 4 5 2 2 8 , 9 10 2 2 2 2 ⟶ 9 11 4 5 2 2 2 , 9 10 2 2 2 6 ⟶ 9 11 4 5 2 2 6 , 9 10 2 2 2 7 ⟶ 9 11 4 5 2 2 7 , 9 10 2 2 2 8 ⟶ 9 11 4 5 2 2 8 , 12 13 2 2 2 2 ⟶ 12 14 4 5 2 2 2 , 12 13 2 2 2 6 ⟶ 12 14 4 5 2 2 6 , 12 13 2 2 2 7 ⟶ 12 14 4 5 2 2 7 , 12 13 2 2 2 8 ⟶ 12 14 4 5 2 2 8 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 0 1 2 2 6 15 ⟶ 0 3 4 5 2 6 15 , 9 10 2 2 6 4 ⟶ 9 11 4 5 2 6 4 , 9 10 2 2 6 15 ⟶ 9 11 4 5 2 6 15 , 12 13 2 2 6 4 ⟶ 12 14 4 5 2 6 4 , 12 13 2 2 6 15 ⟶ 12 14 4 5 2 6 15 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 9 11 4 5 6 4 5 2 ⟶ 9 10 6 4 5 2 2 , 9 11 4 5 6 4 5 6 ⟶ 9 10 6 4 5 2 6 , 12 14 4 5 6 4 5 2 ⟶ 12 13 6 4 5 2 2 , 12 14 4 5 6 4 5 6 ⟶ 12 13 6 4 5 2 6 , 0 3 4 5 6 4 16 4 ⟶ 0 1 6 4 5 6 4 , 9 11 4 5 6 4 16 4 ⟶ 9 10 6 4 5 6 4 , 12 14 4 5 6 4 16 4 ⟶ 12 13 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 1 2 2 2 2 7 →= 1 6 4 5 2 2 7 , 1 2 2 2 2 8 →= 1 6 4 5 2 2 8 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 7 →= 2 6 4 5 2 2 7 , 2 2 2 2 2 8 →= 2 6 4 5 2 2 8 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 7 →= 5 6 4 5 2 2 7 , 5 2 2 2 2 8 →= 5 6 4 5 2 2 8 , 10 2 2 2 2 2 →= 10 6 4 5 2 2 2 , 10 2 2 2 2 6 →= 10 6 4 5 2 2 6 , 10 2 2 2 2 7 →= 10 6 4 5 2 2 7 , 10 2 2 2 2 8 →= 10 6 4 5 2 2 8 , 13 2 2 2 2 2 →= 13 6 4 5 2 2 2 , 13 2 2 2 2 6 →= 13 6 4 5 2 2 6 , 13 2 2 2 2 7 →= 13 6 4 5 2 2 7 , 13 2 2 2 2 8 →= 13 6 4 5 2 2 8 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 17 2 2 2 2 7 →= 17 6 4 5 2 2 7 , 17 2 2 2 2 8 →= 17 6 4 5 2 2 8 , 18 2 2 2 2 2 →= 18 6 4 5 2 2 2 , 18 2 2 2 2 6 →= 18 6 4 5 2 2 6 , 18 2 2 2 2 7 →= 18 6 4 5 2 2 7 , 18 2 2 2 2 8 →= 18 6 4 5 2 2 8 , 4 5 2 2 2 2 →= 4 16 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 16 4 5 2 2 6 , 4 5 2 2 2 7 →= 4 16 4 5 2 2 7 , 4 5 2 2 2 8 →= 4 16 4 5 2 2 8 , 19 5 2 2 2 2 →= 19 16 4 5 2 2 2 , 19 5 2 2 2 6 →= 19 16 4 5 2 2 6 , 19 5 2 2 2 7 →= 19 16 4 5 2 2 7 , 19 5 2 2 2 8 →= 19 16 4 5 2 2 8 , 20 17 2 2 2 2 →= 20 21 4 5 2 2 2 , 20 17 2 2 2 6 →= 20 21 4 5 2 2 6 , 20 17 2 2 2 7 →= 20 21 4 5 2 2 7 , 20 17 2 2 2 8 →= 20 21 4 5 2 2 8 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 1 2 2 2 6 15 →= 1 6 4 5 2 6 15 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 15 →= 2 6 4 5 2 6 15 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 15 →= 5 6 4 5 2 6 15 , 10 2 2 2 6 4 →= 10 6 4 5 2 6 4 , 10 2 2 2 6 15 →= 10 6 4 5 2 6 15 , 13 2 2 2 6 4 →= 13 6 4 5 2 6 4 , 13 2 2 2 6 15 →= 13 6 4 5 2 6 15 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 15 →= 17 6 4 5 2 6 15 , 18 2 2 2 6 4 →= 18 6 4 5 2 6 4 , 18 2 2 2 6 15 →= 18 6 4 5 2 6 15 , 4 5 2 2 6 4 →= 4 16 4 5 2 6 4 , 4 5 2 2 6 15 →= 4 16 4 5 2 6 15 , 19 5 2 2 6 4 →= 19 16 4 5 2 6 4 , 19 5 2 2 6 15 →= 19 16 4 5 2 6 15 , 20 17 2 2 6 4 →= 20 21 4 5 2 6 4 , 20 17 2 2 6 15 →= 20 21 4 5 2 6 15 , 1 2 2 2 7 22 →= 1 6 4 5 2 7 22 , 2 2 2 2 7 22 →= 2 6 4 5 2 7 22 , 5 2 2 2 7 22 →= 5 6 4 5 2 7 22 , 10 2 2 2 7 22 →= 10 6 4 5 2 7 22 , 13 2 2 2 7 22 →= 13 6 4 5 2 7 22 , 17 2 2 2 7 22 →= 17 6 4 5 2 7 22 , 18 2 2 2 7 22 →= 18 6 4 5 2 7 22 , 4 5 2 2 7 22 →= 4 16 4 5 2 7 22 , 19 5 2 2 7 22 →= 19 16 4 5 2 7 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 10 6 4 5 6 4 5 2 →= 10 2 6 4 5 2 2 , 10 6 4 5 6 4 5 6 →= 10 2 6 4 5 2 6 , 13 6 4 5 6 4 5 2 →= 13 2 6 4 5 2 2 , 13 6 4 5 6 4 5 6 →= 13 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 18 6 4 5 6 4 5 2 →= 18 2 6 4 5 2 2 , 18 6 4 5 6 4 5 6 →= 18 2 6 4 5 2 6 , 4 16 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 16 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 19 16 4 5 6 4 5 2 →= 19 5 6 4 5 2 2 , 19 16 4 5 6 4 5 6 →= 19 5 6 4 5 2 6 , 20 21 4 5 6 4 5 2 →= 20 17 6 4 5 2 2 , 20 21 4 5 6 4 5 6 →= 20 17 6 4 5 2 6 , 1 6 4 5 6 4 16 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 16 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 16 4 →= 5 2 6 4 5 6 4 , 10 6 4 5 6 4 16 4 →= 10 2 6 4 5 6 4 , 13 6 4 5 6 4 16 4 →= 13 2 6 4 5 6 4 , 17 6 4 5 6 4 16 4 →= 17 2 6 4 5 6 4 , 18 6 4 5 6 4 16 4 →= 18 2 6 4 5 6 4 , 4 16 4 5 6 4 16 4 →= 4 5 6 4 5 6 4 , 19 16 4 5 6 4 16 4 →= 19 5 6 4 5 6 4 , 20 21 4 5 6 4 16 4 →= 20 17 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 0 ↦ ⎛ ⎞ ⎜ 1 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 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 ⎟ ⎜ 0 1 0 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 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 1 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 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 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 9 ↦ 8, 10 ↦ 9, 11 ↦ 10, 7 ↦ 11, 12 ↦ 12, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 125-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 0 1 2 2 2 7 ⟶ 0 3 4 5 2 2 7 , 8 9 2 2 2 2 ⟶ 8 10 4 5 2 2 2 , 8 9 2 2 2 6 ⟶ 8 10 4 5 2 2 6 , 8 9 2 2 2 11 ⟶ 8 10 4 5 2 2 11 , 8 9 2 2 2 7 ⟶ 8 10 4 5 2 2 7 , 12 13 2 2 2 2 ⟶ 12 14 4 5 2 2 2 , 12 13 2 2 2 6 ⟶ 12 14 4 5 2 2 6 , 12 13 2 2 2 11 ⟶ 12 14 4 5 2 2 11 , 12 13 2 2 2 7 ⟶ 12 14 4 5 2 2 7 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 0 1 2 2 6 15 ⟶ 0 3 4 5 2 6 15 , 8 9 2 2 6 4 ⟶ 8 10 4 5 2 6 4 , 8 9 2 2 6 15 ⟶ 8 10 4 5 2 6 15 , 12 13 2 2 6 4 ⟶ 12 14 4 5 2 6 4 , 12 13 2 2 6 15 ⟶ 12 14 4 5 2 6 15 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 8 10 4 5 6 4 5 2 ⟶ 8 9 6 4 5 2 2 , 8 10 4 5 6 4 5 6 ⟶ 8 9 6 4 5 2 6 , 12 14 4 5 6 4 5 2 ⟶ 12 13 6 4 5 2 2 , 12 14 4 5 6 4 5 6 ⟶ 12 13 6 4 5 2 6 , 0 3 4 5 6 4 16 4 ⟶ 0 1 6 4 5 6 4 , 8 10 4 5 6 4 16 4 ⟶ 8 9 6 4 5 6 4 , 12 14 4 5 6 4 16 4 ⟶ 12 13 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 1 2 2 2 2 11 →= 1 6 4 5 2 2 11 , 1 2 2 2 2 7 →= 1 6 4 5 2 2 7 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 11 →= 2 6 4 5 2 2 11 , 2 2 2 2 2 7 →= 2 6 4 5 2 2 7 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 11 →= 5 6 4 5 2 2 11 , 5 2 2 2 2 7 →= 5 6 4 5 2 2 7 , 9 2 2 2 2 2 →= 9 6 4 5 2 2 2 , 9 2 2 2 2 6 →= 9 6 4 5 2 2 6 , 9 2 2 2 2 11 →= 9 6 4 5 2 2 11 , 9 2 2 2 2 7 →= 9 6 4 5 2 2 7 , 13 2 2 2 2 2 →= 13 6 4 5 2 2 2 , 13 2 2 2 2 6 →= 13 6 4 5 2 2 6 , 13 2 2 2 2 11 →= 13 6 4 5 2 2 11 , 13 2 2 2 2 7 →= 13 6 4 5 2 2 7 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 17 2 2 2 2 11 →= 17 6 4 5 2 2 11 , 17 2 2 2 2 7 →= 17 6 4 5 2 2 7 , 18 2 2 2 2 2 →= 18 6 4 5 2 2 2 , 18 2 2 2 2 6 →= 18 6 4 5 2 2 6 , 18 2 2 2 2 11 →= 18 6 4 5 2 2 11 , 18 2 2 2 2 7 →= 18 6 4 5 2 2 7 , 4 5 2 2 2 2 →= 4 16 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 16 4 5 2 2 6 , 4 5 2 2 2 11 →= 4 16 4 5 2 2 11 , 4 5 2 2 2 7 →= 4 16 4 5 2 2 7 , 19 5 2 2 2 2 →= 19 16 4 5 2 2 2 , 19 5 2 2 2 6 →= 19 16 4 5 2 2 6 , 19 5 2 2 2 11 →= 19 16 4 5 2 2 11 , 19 5 2 2 2 7 →= 19 16 4 5 2 2 7 , 20 17 2 2 2 2 →= 20 21 4 5 2 2 2 , 20 17 2 2 2 6 →= 20 21 4 5 2 2 6 , 20 17 2 2 2 11 →= 20 21 4 5 2 2 11 , 20 17 2 2 2 7 →= 20 21 4 5 2 2 7 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 1 2 2 2 6 15 →= 1 6 4 5 2 6 15 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 15 →= 2 6 4 5 2 6 15 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 15 →= 5 6 4 5 2 6 15 , 9 2 2 2 6 4 →= 9 6 4 5 2 6 4 , 9 2 2 2 6 15 →= 9 6 4 5 2 6 15 , 13 2 2 2 6 4 →= 13 6 4 5 2 6 4 , 13 2 2 2 6 15 →= 13 6 4 5 2 6 15 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 15 →= 17 6 4 5 2 6 15 , 18 2 2 2 6 4 →= 18 6 4 5 2 6 4 , 18 2 2 2 6 15 →= 18 6 4 5 2 6 15 , 4 5 2 2 6 4 →= 4 16 4 5 2 6 4 , 4 5 2 2 6 15 →= 4 16 4 5 2 6 15 , 19 5 2 2 6 4 →= 19 16 4 5 2 6 4 , 19 5 2 2 6 15 →= 19 16 4 5 2 6 15 , 20 17 2 2 6 4 →= 20 21 4 5 2 6 4 , 20 17 2 2 6 15 →= 20 21 4 5 2 6 15 , 1 2 2 2 11 22 →= 1 6 4 5 2 11 22 , 2 2 2 2 11 22 →= 2 6 4 5 2 11 22 , 5 2 2 2 11 22 →= 5 6 4 5 2 11 22 , 9 2 2 2 11 22 →= 9 6 4 5 2 11 22 , 13 2 2 2 11 22 →= 13 6 4 5 2 11 22 , 17 2 2 2 11 22 →= 17 6 4 5 2 11 22 , 18 2 2 2 11 22 →= 18 6 4 5 2 11 22 , 4 5 2 2 11 22 →= 4 16 4 5 2 11 22 , 19 5 2 2 11 22 →= 19 16 4 5 2 11 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 9 6 4 5 6 4 5 2 →= 9 2 6 4 5 2 2 , 9 6 4 5 6 4 5 6 →= 9 2 6 4 5 2 6 , 13 6 4 5 6 4 5 2 →= 13 2 6 4 5 2 2 , 13 6 4 5 6 4 5 6 →= 13 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 18 6 4 5 6 4 5 2 →= 18 2 6 4 5 2 2 , 18 6 4 5 6 4 5 6 →= 18 2 6 4 5 2 6 , 4 16 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 16 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 19 16 4 5 6 4 5 2 →= 19 5 6 4 5 2 2 , 19 16 4 5 6 4 5 6 →= 19 5 6 4 5 2 6 , 20 21 4 5 6 4 5 2 →= 20 17 6 4 5 2 2 , 20 21 4 5 6 4 5 6 →= 20 17 6 4 5 2 6 , 1 6 4 5 6 4 16 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 16 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 16 4 →= 5 2 6 4 5 6 4 , 9 6 4 5 6 4 16 4 →= 9 2 6 4 5 6 4 , 13 6 4 5 6 4 16 4 →= 13 2 6 4 5 6 4 , 17 6 4 5 6 4 16 4 →= 17 2 6 4 5 6 4 , 18 6 4 5 6 4 16 4 →= 18 2 6 4 5 6 4 , 4 16 4 5 6 4 16 4 →= 4 5 6 4 5 6 4 , 19 16 4 5 6 4 16 4 →= 19 5 6 4 5 6 4 , 20 21 4 5 6 4 16 4 →= 20 17 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 0 ↦ ⎛ ⎞ ⎜ 1 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 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 ⎟ ⎜ 0 1 0 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 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 1 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 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 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 9 ↦ 8, 10 ↦ 9, 11 ↦ 10, 7 ↦ 11, 12 ↦ 12, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 124-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 7 8 2 2 2 10 ⟶ 7 9 4 5 2 2 10 , 7 8 2 2 2 11 ⟶ 7 9 4 5 2 2 11 , 12 13 2 2 2 2 ⟶ 12 14 4 5 2 2 2 , 12 13 2 2 2 6 ⟶ 12 14 4 5 2 2 6 , 12 13 2 2 2 10 ⟶ 12 14 4 5 2 2 10 , 12 13 2 2 2 11 ⟶ 12 14 4 5 2 2 11 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 0 1 2 2 6 15 ⟶ 0 3 4 5 2 6 15 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 7 8 2 2 6 15 ⟶ 7 9 4 5 2 6 15 , 12 13 2 2 6 4 ⟶ 12 14 4 5 2 6 4 , 12 13 2 2 6 15 ⟶ 12 14 4 5 2 6 15 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 12 14 4 5 6 4 5 2 ⟶ 12 13 6 4 5 2 2 , 12 14 4 5 6 4 5 6 ⟶ 12 13 6 4 5 2 6 , 0 3 4 5 6 4 16 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 16 4 ⟶ 7 8 6 4 5 6 4 , 12 14 4 5 6 4 16 4 ⟶ 12 13 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 1 2 2 2 2 10 →= 1 6 4 5 2 2 10 , 1 2 2 2 2 11 →= 1 6 4 5 2 2 11 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 10 →= 2 6 4 5 2 2 10 , 2 2 2 2 2 11 →= 2 6 4 5 2 2 11 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 10 →= 5 6 4 5 2 2 10 , 5 2 2 2 2 11 →= 5 6 4 5 2 2 11 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 8 2 2 2 2 10 →= 8 6 4 5 2 2 10 , 8 2 2 2 2 11 →= 8 6 4 5 2 2 11 , 13 2 2 2 2 2 →= 13 6 4 5 2 2 2 , 13 2 2 2 2 6 →= 13 6 4 5 2 2 6 , 13 2 2 2 2 10 →= 13 6 4 5 2 2 10 , 13 2 2 2 2 11 →= 13 6 4 5 2 2 11 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 17 2 2 2 2 10 →= 17 6 4 5 2 2 10 , 17 2 2 2 2 11 →= 17 6 4 5 2 2 11 , 18 2 2 2 2 2 →= 18 6 4 5 2 2 2 , 18 2 2 2 2 6 →= 18 6 4 5 2 2 6 , 18 2 2 2 2 10 →= 18 6 4 5 2 2 10 , 18 2 2 2 2 11 →= 18 6 4 5 2 2 11 , 4 5 2 2 2 2 →= 4 16 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 16 4 5 2 2 6 , 4 5 2 2 2 10 →= 4 16 4 5 2 2 10 , 4 5 2 2 2 11 →= 4 16 4 5 2 2 11 , 19 5 2 2 2 2 →= 19 16 4 5 2 2 2 , 19 5 2 2 2 6 →= 19 16 4 5 2 2 6 , 19 5 2 2 2 10 →= 19 16 4 5 2 2 10 , 19 5 2 2 2 11 →= 19 16 4 5 2 2 11 , 20 17 2 2 2 2 →= 20 21 4 5 2 2 2 , 20 17 2 2 2 6 →= 20 21 4 5 2 2 6 , 20 17 2 2 2 10 →= 20 21 4 5 2 2 10 , 20 17 2 2 2 11 →= 20 21 4 5 2 2 11 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 1 2 2 2 6 15 →= 1 6 4 5 2 6 15 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 15 →= 2 6 4 5 2 6 15 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 15 →= 5 6 4 5 2 6 15 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 8 2 2 2 6 15 →= 8 6 4 5 2 6 15 , 13 2 2 2 6 4 →= 13 6 4 5 2 6 4 , 13 2 2 2 6 15 →= 13 6 4 5 2 6 15 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 15 →= 17 6 4 5 2 6 15 , 18 2 2 2 6 4 →= 18 6 4 5 2 6 4 , 18 2 2 2 6 15 →= 18 6 4 5 2 6 15 , 4 5 2 2 6 4 →= 4 16 4 5 2 6 4 , 4 5 2 2 6 15 →= 4 16 4 5 2 6 15 , 19 5 2 2 6 4 →= 19 16 4 5 2 6 4 , 19 5 2 2 6 15 →= 19 16 4 5 2 6 15 , 20 17 2 2 6 4 →= 20 21 4 5 2 6 4 , 20 17 2 2 6 15 →= 20 21 4 5 2 6 15 , 1 2 2 2 10 22 →= 1 6 4 5 2 10 22 , 2 2 2 2 10 22 →= 2 6 4 5 2 10 22 , 5 2 2 2 10 22 →= 5 6 4 5 2 10 22 , 8 2 2 2 10 22 →= 8 6 4 5 2 10 22 , 13 2 2 2 10 22 →= 13 6 4 5 2 10 22 , 17 2 2 2 10 22 →= 17 6 4 5 2 10 22 , 18 2 2 2 10 22 →= 18 6 4 5 2 10 22 , 4 5 2 2 10 22 →= 4 16 4 5 2 10 22 , 19 5 2 2 10 22 →= 19 16 4 5 2 10 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 13 6 4 5 6 4 5 2 →= 13 2 6 4 5 2 2 , 13 6 4 5 6 4 5 6 →= 13 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 18 6 4 5 6 4 5 2 →= 18 2 6 4 5 2 2 , 18 6 4 5 6 4 5 6 →= 18 2 6 4 5 2 6 , 4 16 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 16 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 19 16 4 5 6 4 5 2 →= 19 5 6 4 5 2 2 , 19 16 4 5 6 4 5 6 →= 19 5 6 4 5 2 6 , 20 21 4 5 6 4 5 2 →= 20 17 6 4 5 2 2 , 20 21 4 5 6 4 5 6 →= 20 17 6 4 5 2 6 , 1 6 4 5 6 4 16 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 16 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 16 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 16 4 →= 8 2 6 4 5 6 4 , 13 6 4 5 6 4 16 4 →= 13 2 6 4 5 6 4 , 17 6 4 5 6 4 16 4 →= 17 2 6 4 5 6 4 , 18 6 4 5 6 4 16 4 →= 18 2 6 4 5 6 4 , 4 16 4 5 6 4 16 4 →= 4 5 6 4 5 6 4 , 19 16 4 5 6 4 16 4 →= 19 5 6 4 5 6 4 , 20 21 4 5 6 4 16 4 →= 20 17 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 1 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 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 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 ⎟ ⎜ 0 1 0 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 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 13 ↦ 12, 14 ↦ 13, 10 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 123-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 7 8 2 2 2 10 ⟶ 7 9 4 5 2 2 10 , 11 12 2 2 2 2 ⟶ 11 13 4 5 2 2 2 , 11 12 2 2 2 6 ⟶ 11 13 4 5 2 2 6 , 11 12 2 2 2 14 ⟶ 11 13 4 5 2 2 14 , 11 12 2 2 2 10 ⟶ 11 13 4 5 2 2 10 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 0 1 2 2 6 15 ⟶ 0 3 4 5 2 6 15 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 7 8 2 2 6 15 ⟶ 7 9 4 5 2 6 15 , 11 12 2 2 6 4 ⟶ 11 13 4 5 2 6 4 , 11 12 2 2 6 15 ⟶ 11 13 4 5 2 6 15 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 11 13 4 5 6 4 5 2 ⟶ 11 12 6 4 5 2 2 , 11 13 4 5 6 4 5 6 ⟶ 11 12 6 4 5 2 6 , 0 3 4 5 6 4 16 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 16 4 ⟶ 7 8 6 4 5 6 4 , 11 13 4 5 6 4 16 4 ⟶ 11 12 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 1 2 2 2 2 14 →= 1 6 4 5 2 2 14 , 1 2 2 2 2 10 →= 1 6 4 5 2 2 10 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 10 →= 2 6 4 5 2 2 10 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 10 →= 5 6 4 5 2 2 10 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 8 2 2 2 2 14 →= 8 6 4 5 2 2 14 , 8 2 2 2 2 10 →= 8 6 4 5 2 2 10 , 12 2 2 2 2 2 →= 12 6 4 5 2 2 2 , 12 2 2 2 2 6 →= 12 6 4 5 2 2 6 , 12 2 2 2 2 14 →= 12 6 4 5 2 2 14 , 12 2 2 2 2 10 →= 12 6 4 5 2 2 10 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 17 2 2 2 2 14 →= 17 6 4 5 2 2 14 , 17 2 2 2 2 10 →= 17 6 4 5 2 2 10 , 18 2 2 2 2 2 →= 18 6 4 5 2 2 2 , 18 2 2 2 2 6 →= 18 6 4 5 2 2 6 , 18 2 2 2 2 14 →= 18 6 4 5 2 2 14 , 18 2 2 2 2 10 →= 18 6 4 5 2 2 10 , 4 5 2 2 2 2 →= 4 16 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 16 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 16 4 5 2 2 14 , 4 5 2 2 2 10 →= 4 16 4 5 2 2 10 , 19 5 2 2 2 2 →= 19 16 4 5 2 2 2 , 19 5 2 2 2 6 →= 19 16 4 5 2 2 6 , 19 5 2 2 2 14 →= 19 16 4 5 2 2 14 , 19 5 2 2 2 10 →= 19 16 4 5 2 2 10 , 20 17 2 2 2 2 →= 20 21 4 5 2 2 2 , 20 17 2 2 2 6 →= 20 21 4 5 2 2 6 , 20 17 2 2 2 14 →= 20 21 4 5 2 2 14 , 20 17 2 2 2 10 →= 20 21 4 5 2 2 10 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 1 2 2 2 6 15 →= 1 6 4 5 2 6 15 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 15 →= 2 6 4 5 2 6 15 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 15 →= 5 6 4 5 2 6 15 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 8 2 2 2 6 15 →= 8 6 4 5 2 6 15 , 12 2 2 2 6 4 →= 12 6 4 5 2 6 4 , 12 2 2 2 6 15 →= 12 6 4 5 2 6 15 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 15 →= 17 6 4 5 2 6 15 , 18 2 2 2 6 4 →= 18 6 4 5 2 6 4 , 18 2 2 2 6 15 →= 18 6 4 5 2 6 15 , 4 5 2 2 6 4 →= 4 16 4 5 2 6 4 , 4 5 2 2 6 15 →= 4 16 4 5 2 6 15 , 19 5 2 2 6 4 →= 19 16 4 5 2 6 4 , 19 5 2 2 6 15 →= 19 16 4 5 2 6 15 , 20 17 2 2 6 4 →= 20 21 4 5 2 6 4 , 20 17 2 2 6 15 →= 20 21 4 5 2 6 15 , 1 2 2 2 14 22 →= 1 6 4 5 2 14 22 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 8 2 2 2 14 22 →= 8 6 4 5 2 14 22 , 12 2 2 2 14 22 →= 12 6 4 5 2 14 22 , 17 2 2 2 14 22 →= 17 6 4 5 2 14 22 , 18 2 2 2 14 22 →= 18 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 16 4 5 2 14 22 , 19 5 2 2 14 22 →= 19 16 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 12 6 4 5 6 4 5 2 →= 12 2 6 4 5 2 2 , 12 6 4 5 6 4 5 6 →= 12 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 18 6 4 5 6 4 5 2 →= 18 2 6 4 5 2 2 , 18 6 4 5 6 4 5 6 →= 18 2 6 4 5 2 6 , 4 16 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 16 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 19 16 4 5 6 4 5 2 →= 19 5 6 4 5 2 2 , 19 16 4 5 6 4 5 6 →= 19 5 6 4 5 2 6 , 20 21 4 5 6 4 5 2 →= 20 17 6 4 5 2 2 , 20 21 4 5 6 4 5 6 →= 20 17 6 4 5 2 6 , 1 6 4 5 6 4 16 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 16 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 16 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 16 4 →= 8 2 6 4 5 6 4 , 12 6 4 5 6 4 16 4 →= 12 2 6 4 5 6 4 , 17 6 4 5 6 4 16 4 →= 17 2 6 4 5 6 4 , 18 6 4 5 6 4 16 4 →= 18 2 6 4 5 6 4 , 4 16 4 5 6 4 16 4 →= 4 5 6 4 5 6 4 , 19 16 4 5 6 4 16 4 →= 19 5 6 4 5 6 4 , 20 21 4 5 6 4 16 4 →= 20 17 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 1 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 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 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 ⎟ ⎜ 0 1 0 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 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 13 ↦ 12, 14 ↦ 13, 10 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 122-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 10 11 2 2 2 13 ⟶ 10 12 4 5 2 2 13 , 10 11 2 2 2 14 ⟶ 10 12 4 5 2 2 14 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 0 1 2 2 6 15 ⟶ 0 3 4 5 2 6 15 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 7 8 2 2 6 15 ⟶ 7 9 4 5 2 6 15 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 10 11 2 2 6 15 ⟶ 10 12 4 5 2 6 15 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 16 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 16 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 16 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 1 2 2 2 2 13 →= 1 6 4 5 2 2 13 , 1 2 2 2 2 14 →= 1 6 4 5 2 2 14 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 13 →= 2 6 4 5 2 2 13 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 13 →= 5 6 4 5 2 2 13 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 8 2 2 2 2 13 →= 8 6 4 5 2 2 13 , 8 2 2 2 2 14 →= 8 6 4 5 2 2 14 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 11 2 2 2 2 13 →= 11 6 4 5 2 2 13 , 11 2 2 2 2 14 →= 11 6 4 5 2 2 14 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 17 2 2 2 2 13 →= 17 6 4 5 2 2 13 , 17 2 2 2 2 14 →= 17 6 4 5 2 2 14 , 18 2 2 2 2 2 →= 18 6 4 5 2 2 2 , 18 2 2 2 2 6 →= 18 6 4 5 2 2 6 , 18 2 2 2 2 13 →= 18 6 4 5 2 2 13 , 18 2 2 2 2 14 →= 18 6 4 5 2 2 14 , 4 5 2 2 2 2 →= 4 16 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 16 4 5 2 2 6 , 4 5 2 2 2 13 →= 4 16 4 5 2 2 13 , 4 5 2 2 2 14 →= 4 16 4 5 2 2 14 , 19 5 2 2 2 2 →= 19 16 4 5 2 2 2 , 19 5 2 2 2 6 →= 19 16 4 5 2 2 6 , 19 5 2 2 2 13 →= 19 16 4 5 2 2 13 , 19 5 2 2 2 14 →= 19 16 4 5 2 2 14 , 20 17 2 2 2 2 →= 20 21 4 5 2 2 2 , 20 17 2 2 2 6 →= 20 21 4 5 2 2 6 , 20 17 2 2 2 13 →= 20 21 4 5 2 2 13 , 20 17 2 2 2 14 →= 20 21 4 5 2 2 14 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 1 2 2 2 6 15 →= 1 6 4 5 2 6 15 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 15 →= 2 6 4 5 2 6 15 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 15 →= 5 6 4 5 2 6 15 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 8 2 2 2 6 15 →= 8 6 4 5 2 6 15 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 11 2 2 2 6 15 →= 11 6 4 5 2 6 15 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 15 →= 17 6 4 5 2 6 15 , 18 2 2 2 6 4 →= 18 6 4 5 2 6 4 , 18 2 2 2 6 15 →= 18 6 4 5 2 6 15 , 4 5 2 2 6 4 →= 4 16 4 5 2 6 4 , 4 5 2 2 6 15 →= 4 16 4 5 2 6 15 , 19 5 2 2 6 4 →= 19 16 4 5 2 6 4 , 19 5 2 2 6 15 →= 19 16 4 5 2 6 15 , 20 17 2 2 6 4 →= 20 21 4 5 2 6 4 , 20 17 2 2 6 15 →= 20 21 4 5 2 6 15 , 1 2 2 2 13 22 →= 1 6 4 5 2 13 22 , 2 2 2 2 13 22 →= 2 6 4 5 2 13 22 , 5 2 2 2 13 22 →= 5 6 4 5 2 13 22 , 8 2 2 2 13 22 →= 8 6 4 5 2 13 22 , 11 2 2 2 13 22 →= 11 6 4 5 2 13 22 , 17 2 2 2 13 22 →= 17 6 4 5 2 13 22 , 18 2 2 2 13 22 →= 18 6 4 5 2 13 22 , 4 5 2 2 13 22 →= 4 16 4 5 2 13 22 , 19 5 2 2 13 22 →= 19 16 4 5 2 13 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 18 6 4 5 6 4 5 2 →= 18 2 6 4 5 2 2 , 18 6 4 5 6 4 5 6 →= 18 2 6 4 5 2 6 , 4 16 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 16 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 19 16 4 5 6 4 5 2 →= 19 5 6 4 5 2 2 , 19 16 4 5 6 4 5 6 →= 19 5 6 4 5 2 6 , 20 21 4 5 6 4 5 2 →= 20 17 6 4 5 2 2 , 20 21 4 5 6 4 5 6 →= 20 17 6 4 5 2 6 , 1 6 4 5 6 4 16 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 16 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 16 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 16 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 16 4 →= 11 2 6 4 5 6 4 , 17 6 4 5 6 4 16 4 →= 17 2 6 4 5 6 4 , 18 6 4 5 6 4 16 4 →= 18 2 6 4 5 6 4 , 4 16 4 5 6 4 16 4 →= 4 5 6 4 5 6 4 , 19 16 4 5 6 4 16 4 →= 19 5 6 4 5 6 4 , 20 21 4 5 6 4 16 4 →= 20 17 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 1 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 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 0 ⎟ ⎜ 0 0 0 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 0 ⎟ ⎜ 0 1 0 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 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 14 ↦ 13, 15 ↦ 14, 16 ↦ 15, 13 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 121-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 10 11 2 2 2 13 ⟶ 10 12 4 5 2 2 13 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 0 1 2 2 6 14 ⟶ 0 3 4 5 2 6 14 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 7 8 2 2 6 14 ⟶ 7 9 4 5 2 6 14 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 10 11 2 2 6 14 ⟶ 10 12 4 5 2 6 14 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 15 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 15 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 15 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 1 2 2 2 2 16 →= 1 6 4 5 2 2 16 , 1 2 2 2 2 13 →= 1 6 4 5 2 2 13 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 16 →= 2 6 4 5 2 2 16 , 2 2 2 2 2 13 →= 2 6 4 5 2 2 13 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 16 →= 5 6 4 5 2 2 16 , 5 2 2 2 2 13 →= 5 6 4 5 2 2 13 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 8 2 2 2 2 16 →= 8 6 4 5 2 2 16 , 8 2 2 2 2 13 →= 8 6 4 5 2 2 13 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 11 2 2 2 2 16 →= 11 6 4 5 2 2 16 , 11 2 2 2 2 13 →= 11 6 4 5 2 2 13 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 17 2 2 2 2 16 →= 17 6 4 5 2 2 16 , 17 2 2 2 2 13 →= 17 6 4 5 2 2 13 , 18 2 2 2 2 2 →= 18 6 4 5 2 2 2 , 18 2 2 2 2 6 →= 18 6 4 5 2 2 6 , 18 2 2 2 2 16 →= 18 6 4 5 2 2 16 , 18 2 2 2 2 13 →= 18 6 4 5 2 2 13 , 4 5 2 2 2 2 →= 4 15 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 15 4 5 2 2 6 , 4 5 2 2 2 16 →= 4 15 4 5 2 2 16 , 4 5 2 2 2 13 →= 4 15 4 5 2 2 13 , 19 5 2 2 2 2 →= 19 15 4 5 2 2 2 , 19 5 2 2 2 6 →= 19 15 4 5 2 2 6 , 19 5 2 2 2 16 →= 19 15 4 5 2 2 16 , 19 5 2 2 2 13 →= 19 15 4 5 2 2 13 , 20 17 2 2 2 2 →= 20 21 4 5 2 2 2 , 20 17 2 2 2 6 →= 20 21 4 5 2 2 6 , 20 17 2 2 2 16 →= 20 21 4 5 2 2 16 , 20 17 2 2 2 13 →= 20 21 4 5 2 2 13 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 1 2 2 2 6 14 →= 1 6 4 5 2 6 14 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 14 →= 2 6 4 5 2 6 14 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 14 →= 5 6 4 5 2 6 14 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 8 2 2 2 6 14 →= 8 6 4 5 2 6 14 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 11 2 2 2 6 14 →= 11 6 4 5 2 6 14 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 14 →= 17 6 4 5 2 6 14 , 18 2 2 2 6 4 →= 18 6 4 5 2 6 4 , 18 2 2 2 6 14 →= 18 6 4 5 2 6 14 , 4 5 2 2 6 4 →= 4 15 4 5 2 6 4 , 4 5 2 2 6 14 →= 4 15 4 5 2 6 14 , 19 5 2 2 6 4 →= 19 15 4 5 2 6 4 , 19 5 2 2 6 14 →= 19 15 4 5 2 6 14 , 20 17 2 2 6 4 →= 20 21 4 5 2 6 4 , 20 17 2 2 6 14 →= 20 21 4 5 2 6 14 , 1 2 2 2 16 22 →= 1 6 4 5 2 16 22 , 2 2 2 2 16 22 →= 2 6 4 5 2 16 22 , 5 2 2 2 16 22 →= 5 6 4 5 2 16 22 , 8 2 2 2 16 22 →= 8 6 4 5 2 16 22 , 11 2 2 2 16 22 →= 11 6 4 5 2 16 22 , 17 2 2 2 16 22 →= 17 6 4 5 2 16 22 , 18 2 2 2 16 22 →= 18 6 4 5 2 16 22 , 4 5 2 2 16 22 →= 4 15 4 5 2 16 22 , 19 5 2 2 16 22 →= 19 15 4 5 2 16 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 18 6 4 5 6 4 5 2 →= 18 2 6 4 5 2 2 , 18 6 4 5 6 4 5 6 →= 18 2 6 4 5 2 6 , 4 15 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 15 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 19 15 4 5 6 4 5 2 →= 19 5 6 4 5 2 2 , 19 15 4 5 6 4 5 6 →= 19 5 6 4 5 2 6 , 20 21 4 5 6 4 5 2 →= 20 17 6 4 5 2 2 , 20 21 4 5 6 4 5 6 →= 20 17 6 4 5 2 6 , 1 6 4 5 6 4 15 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 15 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 15 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 15 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 15 4 →= 11 2 6 4 5 6 4 , 17 6 4 5 6 4 15 4 →= 17 2 6 4 5 6 4 , 18 6 4 5 6 4 15 4 →= 18 2 6 4 5 6 4 , 4 15 4 5 6 4 15 4 →= 4 5 6 4 5 6 4 , 19 15 4 5 6 4 15 4 →= 19 5 6 4 5 6 4 , 20 21 4 5 6 4 15 4 →= 20 17 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 1 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 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 0 ⎟ ⎜ 0 0 0 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 0 ⎟ ⎜ 0 1 0 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 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 14 ↦ 13, 15 ↦ 14, 16 ↦ 15, 13 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 120-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 0 1 2 2 6 13 ⟶ 0 3 4 5 2 6 13 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 7 8 2 2 6 13 ⟶ 7 9 4 5 2 6 13 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 10 11 2 2 6 13 ⟶ 10 12 4 5 2 6 13 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 14 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 14 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 14 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 1 2 2 2 2 15 →= 1 6 4 5 2 2 15 , 1 2 2 2 2 16 →= 1 6 4 5 2 2 16 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 2 2 2 2 2 16 →= 2 6 4 5 2 2 16 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 5 2 2 2 2 16 →= 5 6 4 5 2 2 16 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 8 2 2 2 2 15 →= 8 6 4 5 2 2 15 , 8 2 2 2 2 16 →= 8 6 4 5 2 2 16 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 11 2 2 2 2 15 →= 11 6 4 5 2 2 15 , 11 2 2 2 2 16 →= 11 6 4 5 2 2 16 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 17 2 2 2 2 15 →= 17 6 4 5 2 2 15 , 17 2 2 2 2 16 →= 17 6 4 5 2 2 16 , 18 2 2 2 2 2 →= 18 6 4 5 2 2 2 , 18 2 2 2 2 6 →= 18 6 4 5 2 2 6 , 18 2 2 2 2 15 →= 18 6 4 5 2 2 15 , 18 2 2 2 2 16 →= 18 6 4 5 2 2 16 , 4 5 2 2 2 2 →= 4 14 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 14 4 5 2 2 6 , 4 5 2 2 2 15 →= 4 14 4 5 2 2 15 , 4 5 2 2 2 16 →= 4 14 4 5 2 2 16 , 19 5 2 2 2 2 →= 19 14 4 5 2 2 2 , 19 5 2 2 2 6 →= 19 14 4 5 2 2 6 , 19 5 2 2 2 15 →= 19 14 4 5 2 2 15 , 19 5 2 2 2 16 →= 19 14 4 5 2 2 16 , 20 17 2 2 2 2 →= 20 21 4 5 2 2 2 , 20 17 2 2 2 6 →= 20 21 4 5 2 2 6 , 20 17 2 2 2 15 →= 20 21 4 5 2 2 15 , 20 17 2 2 2 16 →= 20 21 4 5 2 2 16 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 1 2 2 2 6 13 →= 1 6 4 5 2 6 13 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 13 →= 2 6 4 5 2 6 13 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 13 →= 5 6 4 5 2 6 13 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 8 2 2 2 6 13 →= 8 6 4 5 2 6 13 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 11 2 2 2 6 13 →= 11 6 4 5 2 6 13 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 13 →= 17 6 4 5 2 6 13 , 18 2 2 2 6 4 →= 18 6 4 5 2 6 4 , 18 2 2 2 6 13 →= 18 6 4 5 2 6 13 , 4 5 2 2 6 4 →= 4 14 4 5 2 6 4 , 4 5 2 2 6 13 →= 4 14 4 5 2 6 13 , 19 5 2 2 6 4 →= 19 14 4 5 2 6 4 , 19 5 2 2 6 13 →= 19 14 4 5 2 6 13 , 20 17 2 2 6 4 →= 20 21 4 5 2 6 4 , 20 17 2 2 6 13 →= 20 21 4 5 2 6 13 , 1 2 2 2 15 22 →= 1 6 4 5 2 15 22 , 2 2 2 2 15 22 →= 2 6 4 5 2 15 22 , 5 2 2 2 15 22 →= 5 6 4 5 2 15 22 , 8 2 2 2 15 22 →= 8 6 4 5 2 15 22 , 11 2 2 2 15 22 →= 11 6 4 5 2 15 22 , 17 2 2 2 15 22 →= 17 6 4 5 2 15 22 , 18 2 2 2 15 22 →= 18 6 4 5 2 15 22 , 4 5 2 2 15 22 →= 4 14 4 5 2 15 22 , 19 5 2 2 15 22 →= 19 14 4 5 2 15 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 18 6 4 5 6 4 5 2 →= 18 2 6 4 5 2 2 , 18 6 4 5 6 4 5 6 →= 18 2 6 4 5 2 6 , 4 14 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 14 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 19 14 4 5 6 4 5 2 →= 19 5 6 4 5 2 2 , 19 14 4 5 6 4 5 6 →= 19 5 6 4 5 2 6 , 20 21 4 5 6 4 5 2 →= 20 17 6 4 5 2 2 , 20 21 4 5 6 4 5 6 →= 20 17 6 4 5 2 6 , 1 6 4 5 6 4 14 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 14 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 14 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 14 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 14 4 →= 11 2 6 4 5 6 4 , 17 6 4 5 6 4 14 4 →= 17 2 6 4 5 6 4 , 18 6 4 5 6 4 14 4 →= 18 2 6 4 5 6 4 , 4 14 4 5 6 4 14 4 →= 4 5 6 4 5 6 4 , 19 14 4 5 6 4 14 4 →= 19 5 6 4 5 6 4 , 20 21 4 5 6 4 14 4 →= 20 17 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 0 ↦ ⎛ ⎞ ⎜ 1 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 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 ⎟ ⎜ 0 1 0 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 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 1 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 119-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 7 8 2 2 6 13 ⟶ 7 9 4 5 2 6 13 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 10 11 2 2 6 13 ⟶ 10 12 4 5 2 6 13 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 14 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 14 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 14 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 1 2 2 2 2 15 →= 1 6 4 5 2 2 15 , 1 2 2 2 2 16 →= 1 6 4 5 2 2 16 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 2 2 2 2 2 16 →= 2 6 4 5 2 2 16 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 5 2 2 2 2 16 →= 5 6 4 5 2 2 16 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 8 2 2 2 2 15 →= 8 6 4 5 2 2 15 , 8 2 2 2 2 16 →= 8 6 4 5 2 2 16 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 11 2 2 2 2 15 →= 11 6 4 5 2 2 15 , 11 2 2 2 2 16 →= 11 6 4 5 2 2 16 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 17 2 2 2 2 15 →= 17 6 4 5 2 2 15 , 17 2 2 2 2 16 →= 17 6 4 5 2 2 16 , 18 2 2 2 2 2 →= 18 6 4 5 2 2 2 , 18 2 2 2 2 6 →= 18 6 4 5 2 2 6 , 18 2 2 2 2 15 →= 18 6 4 5 2 2 15 , 18 2 2 2 2 16 →= 18 6 4 5 2 2 16 , 4 5 2 2 2 2 →= 4 14 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 14 4 5 2 2 6 , 4 5 2 2 2 15 →= 4 14 4 5 2 2 15 , 4 5 2 2 2 16 →= 4 14 4 5 2 2 16 , 19 5 2 2 2 2 →= 19 14 4 5 2 2 2 , 19 5 2 2 2 6 →= 19 14 4 5 2 2 6 , 19 5 2 2 2 15 →= 19 14 4 5 2 2 15 , 19 5 2 2 2 16 →= 19 14 4 5 2 2 16 , 20 17 2 2 2 2 →= 20 21 4 5 2 2 2 , 20 17 2 2 2 6 →= 20 21 4 5 2 2 6 , 20 17 2 2 2 15 →= 20 21 4 5 2 2 15 , 20 17 2 2 2 16 →= 20 21 4 5 2 2 16 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 1 2 2 2 6 13 →= 1 6 4 5 2 6 13 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 13 →= 2 6 4 5 2 6 13 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 13 →= 5 6 4 5 2 6 13 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 8 2 2 2 6 13 →= 8 6 4 5 2 6 13 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 11 2 2 2 6 13 →= 11 6 4 5 2 6 13 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 13 →= 17 6 4 5 2 6 13 , 18 2 2 2 6 4 →= 18 6 4 5 2 6 4 , 18 2 2 2 6 13 →= 18 6 4 5 2 6 13 , 4 5 2 2 6 4 →= 4 14 4 5 2 6 4 , 4 5 2 2 6 13 →= 4 14 4 5 2 6 13 , 19 5 2 2 6 4 →= 19 14 4 5 2 6 4 , 19 5 2 2 6 13 →= 19 14 4 5 2 6 13 , 20 17 2 2 6 4 →= 20 21 4 5 2 6 4 , 20 17 2 2 6 13 →= 20 21 4 5 2 6 13 , 1 2 2 2 15 22 →= 1 6 4 5 2 15 22 , 2 2 2 2 15 22 →= 2 6 4 5 2 15 22 , 5 2 2 2 15 22 →= 5 6 4 5 2 15 22 , 8 2 2 2 15 22 →= 8 6 4 5 2 15 22 , 11 2 2 2 15 22 →= 11 6 4 5 2 15 22 , 17 2 2 2 15 22 →= 17 6 4 5 2 15 22 , 18 2 2 2 15 22 →= 18 6 4 5 2 15 22 , 4 5 2 2 15 22 →= 4 14 4 5 2 15 22 , 19 5 2 2 15 22 →= 19 14 4 5 2 15 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 18 6 4 5 6 4 5 2 →= 18 2 6 4 5 2 2 , 18 6 4 5 6 4 5 6 →= 18 2 6 4 5 2 6 , 4 14 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 14 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 19 14 4 5 6 4 5 2 →= 19 5 6 4 5 2 2 , 19 14 4 5 6 4 5 6 →= 19 5 6 4 5 2 6 , 20 21 4 5 6 4 5 2 →= 20 17 6 4 5 2 2 , 20 21 4 5 6 4 5 6 →= 20 17 6 4 5 2 6 , 1 6 4 5 6 4 14 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 14 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 14 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 14 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 14 4 →= 11 2 6 4 5 6 4 , 17 6 4 5 6 4 14 4 →= 17 2 6 4 5 6 4 , 18 6 4 5 6 4 14 4 →= 18 2 6 4 5 6 4 , 4 14 4 5 6 4 14 4 →= 4 5 6 4 5 6 4 , 19 14 4 5 6 4 14 4 →= 19 5 6 4 5 6 4 , 20 21 4 5 6 4 14 4 →= 20 17 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 1 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 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 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 ⎟ ⎜ 0 1 0 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 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 118-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 10 11 2 2 6 13 ⟶ 10 12 4 5 2 6 13 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 14 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 14 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 14 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 1 2 2 2 2 15 →= 1 6 4 5 2 2 15 , 1 2 2 2 2 16 →= 1 6 4 5 2 2 16 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 2 2 2 2 2 16 →= 2 6 4 5 2 2 16 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 5 2 2 2 2 16 →= 5 6 4 5 2 2 16 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 8 2 2 2 2 15 →= 8 6 4 5 2 2 15 , 8 2 2 2 2 16 →= 8 6 4 5 2 2 16 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 11 2 2 2 2 15 →= 11 6 4 5 2 2 15 , 11 2 2 2 2 16 →= 11 6 4 5 2 2 16 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 17 2 2 2 2 15 →= 17 6 4 5 2 2 15 , 17 2 2 2 2 16 →= 17 6 4 5 2 2 16 , 18 2 2 2 2 2 →= 18 6 4 5 2 2 2 , 18 2 2 2 2 6 →= 18 6 4 5 2 2 6 , 18 2 2 2 2 15 →= 18 6 4 5 2 2 15 , 18 2 2 2 2 16 →= 18 6 4 5 2 2 16 , 4 5 2 2 2 2 →= 4 14 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 14 4 5 2 2 6 , 4 5 2 2 2 15 →= 4 14 4 5 2 2 15 , 4 5 2 2 2 16 →= 4 14 4 5 2 2 16 , 19 5 2 2 2 2 →= 19 14 4 5 2 2 2 , 19 5 2 2 2 6 →= 19 14 4 5 2 2 6 , 19 5 2 2 2 15 →= 19 14 4 5 2 2 15 , 19 5 2 2 2 16 →= 19 14 4 5 2 2 16 , 20 17 2 2 2 2 →= 20 21 4 5 2 2 2 , 20 17 2 2 2 6 →= 20 21 4 5 2 2 6 , 20 17 2 2 2 15 →= 20 21 4 5 2 2 15 , 20 17 2 2 2 16 →= 20 21 4 5 2 2 16 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 1 2 2 2 6 13 →= 1 6 4 5 2 6 13 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 13 →= 2 6 4 5 2 6 13 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 13 →= 5 6 4 5 2 6 13 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 8 2 2 2 6 13 →= 8 6 4 5 2 6 13 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 11 2 2 2 6 13 →= 11 6 4 5 2 6 13 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 13 →= 17 6 4 5 2 6 13 , 18 2 2 2 6 4 →= 18 6 4 5 2 6 4 , 18 2 2 2 6 13 →= 18 6 4 5 2 6 13 , 4 5 2 2 6 4 →= 4 14 4 5 2 6 4 , 4 5 2 2 6 13 →= 4 14 4 5 2 6 13 , 19 5 2 2 6 4 →= 19 14 4 5 2 6 4 , 19 5 2 2 6 13 →= 19 14 4 5 2 6 13 , 20 17 2 2 6 4 →= 20 21 4 5 2 6 4 , 20 17 2 2 6 13 →= 20 21 4 5 2 6 13 , 1 2 2 2 15 22 →= 1 6 4 5 2 15 22 , 2 2 2 2 15 22 →= 2 6 4 5 2 15 22 , 5 2 2 2 15 22 →= 5 6 4 5 2 15 22 , 8 2 2 2 15 22 →= 8 6 4 5 2 15 22 , 11 2 2 2 15 22 →= 11 6 4 5 2 15 22 , 17 2 2 2 15 22 →= 17 6 4 5 2 15 22 , 18 2 2 2 15 22 →= 18 6 4 5 2 15 22 , 4 5 2 2 15 22 →= 4 14 4 5 2 15 22 , 19 5 2 2 15 22 →= 19 14 4 5 2 15 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 18 6 4 5 6 4 5 2 →= 18 2 6 4 5 2 2 , 18 6 4 5 6 4 5 6 →= 18 2 6 4 5 2 6 , 4 14 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 14 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 19 14 4 5 6 4 5 2 →= 19 5 6 4 5 2 2 , 19 14 4 5 6 4 5 6 →= 19 5 6 4 5 2 6 , 20 21 4 5 6 4 5 2 →= 20 17 6 4 5 2 2 , 20 21 4 5 6 4 5 6 →= 20 17 6 4 5 2 6 , 1 6 4 5 6 4 14 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 14 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 14 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 14 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 14 4 →= 11 2 6 4 5 6 4 , 17 6 4 5 6 4 14 4 →= 17 2 6 4 5 6 4 , 18 6 4 5 6 4 14 4 →= 18 2 6 4 5 6 4 , 4 14 4 5 6 4 14 4 →= 4 5 6 4 5 6 4 , 19 14 4 5 6 4 14 4 →= 19 5 6 4 5 6 4 , 20 21 4 5 6 4 14 4 →= 20 17 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 1 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 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 0 ⎟ ⎜ 0 0 0 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 0 ⎟ ⎜ 0 1 0 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 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 14 ↦ 13, 15 ↦ 14, 16 ↦ 15, 17 ↦ 16, 18 ↦ 17, 19 ↦ 18, 20 ↦ 19, 21 ↦ 20, 13 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 117-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 1 2 2 2 2 14 →= 1 6 4 5 2 2 14 , 1 2 2 2 2 15 →= 1 6 4 5 2 2 15 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 8 2 2 2 2 14 →= 8 6 4 5 2 2 14 , 8 2 2 2 2 15 →= 8 6 4 5 2 2 15 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 11 2 2 2 2 14 →= 11 6 4 5 2 2 14 , 11 2 2 2 2 15 →= 11 6 4 5 2 2 15 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 16 2 2 2 2 14 →= 16 6 4 5 2 2 14 , 16 2 2 2 2 15 →= 16 6 4 5 2 2 15 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 17 2 2 2 2 14 →= 17 6 4 5 2 2 14 , 17 2 2 2 2 15 →= 17 6 4 5 2 2 15 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 18 5 2 2 2 14 →= 18 13 4 5 2 2 14 , 18 5 2 2 2 15 →= 18 13 4 5 2 2 15 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 19 16 2 2 2 14 →= 19 20 4 5 2 2 14 , 19 16 2 2 2 15 →= 19 20 4 5 2 2 15 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 1 2 2 2 6 21 →= 1 6 4 5 2 6 21 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 8 2 2 2 6 21 →= 8 6 4 5 2 6 21 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 11 2 2 2 6 21 →= 11 6 4 5 2 6 21 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 16 2 2 2 6 21 →= 16 6 4 5 2 6 21 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 21 →= 17 6 4 5 2 6 21 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 18 5 2 2 6 21 →= 18 13 4 5 2 6 21 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 19 16 2 2 6 21 →= 19 20 4 5 2 6 21 , 1 2 2 2 14 22 →= 1 6 4 5 2 14 22 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 8 2 2 2 14 22 →= 8 6 4 5 2 14 22 , 11 2 2 2 14 22 →= 11 6 4 5 2 14 22 , 16 2 2 2 14 22 →= 16 6 4 5 2 14 22 , 17 2 2 2 14 22 →= 17 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 13 4 5 2 14 22 , 18 5 2 2 14 22 →= 18 13 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 0 0 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 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 13 ↦ 13, 15 ↦ 14, 14 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 116-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 1 2 2 2 2 14 →= 1 6 4 5 2 2 14 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 8 2 2 2 2 15 →= 8 6 4 5 2 2 15 , 8 2 2 2 2 14 →= 8 6 4 5 2 2 14 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 11 2 2 2 2 15 →= 11 6 4 5 2 2 15 , 11 2 2 2 2 14 →= 11 6 4 5 2 2 14 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 16 2 2 2 2 15 →= 16 6 4 5 2 2 15 , 16 2 2 2 2 14 →= 16 6 4 5 2 2 14 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 17 2 2 2 2 15 →= 17 6 4 5 2 2 15 , 17 2 2 2 2 14 →= 17 6 4 5 2 2 14 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 18 5 2 2 2 15 →= 18 13 4 5 2 2 15 , 18 5 2 2 2 14 →= 18 13 4 5 2 2 14 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 19 16 2 2 2 15 →= 19 20 4 5 2 2 15 , 19 16 2 2 2 14 →= 19 20 4 5 2 2 14 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 1 2 2 2 6 21 →= 1 6 4 5 2 6 21 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 8 2 2 2 6 21 →= 8 6 4 5 2 6 21 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 11 2 2 2 6 21 →= 11 6 4 5 2 6 21 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 16 2 2 2 6 21 →= 16 6 4 5 2 6 21 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 21 →= 17 6 4 5 2 6 21 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 18 5 2 2 6 21 →= 18 13 4 5 2 6 21 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 19 16 2 2 6 21 →= 19 20 4 5 2 6 21 , 1 2 2 2 15 22 →= 1 6 4 5 2 15 22 , 2 2 2 2 15 22 →= 2 6 4 5 2 15 22 , 5 2 2 2 15 22 →= 5 6 4 5 2 15 22 , 8 2 2 2 15 22 →= 8 6 4 5 2 15 22 , 11 2 2 2 15 22 →= 11 6 4 5 2 15 22 , 16 2 2 2 15 22 →= 16 6 4 5 2 15 22 , 17 2 2 2 15 22 →= 17 6 4 5 2 15 22 , 4 5 2 2 15 22 →= 4 13 4 5 2 15 22 , 18 5 2 2 15 22 →= 18 13 4 5 2 15 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 0 0 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 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 13 ↦ 13, 15 ↦ 14, 14 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 115-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 8 2 2 2 2 14 →= 8 6 4 5 2 2 14 , 8 2 2 2 2 15 →= 8 6 4 5 2 2 15 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 11 2 2 2 2 14 →= 11 6 4 5 2 2 14 , 11 2 2 2 2 15 →= 11 6 4 5 2 2 15 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 16 2 2 2 2 14 →= 16 6 4 5 2 2 14 , 16 2 2 2 2 15 →= 16 6 4 5 2 2 15 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 17 2 2 2 2 14 →= 17 6 4 5 2 2 14 , 17 2 2 2 2 15 →= 17 6 4 5 2 2 15 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 18 5 2 2 2 14 →= 18 13 4 5 2 2 14 , 18 5 2 2 2 15 →= 18 13 4 5 2 2 15 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 19 16 2 2 2 14 →= 19 20 4 5 2 2 14 , 19 16 2 2 2 15 →= 19 20 4 5 2 2 15 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 1 2 2 2 6 21 →= 1 6 4 5 2 6 21 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 8 2 2 2 6 21 →= 8 6 4 5 2 6 21 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 11 2 2 2 6 21 →= 11 6 4 5 2 6 21 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 16 2 2 2 6 21 →= 16 6 4 5 2 6 21 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 21 →= 17 6 4 5 2 6 21 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 18 5 2 2 6 21 →= 18 13 4 5 2 6 21 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 19 16 2 2 6 21 →= 19 20 4 5 2 6 21 , 1 2 2 2 14 22 →= 1 6 4 5 2 14 22 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 8 2 2 2 14 22 →= 8 6 4 5 2 14 22 , 11 2 2 2 14 22 →= 11 6 4 5 2 14 22 , 16 2 2 2 14 22 →= 16 6 4 5 2 14 22 , 17 2 2 2 14 22 →= 17 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 13 4 5 2 14 22 , 18 5 2 2 14 22 →= 18 13 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 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 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 114-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 8 2 2 2 2 15 →= 8 6 4 5 2 2 15 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 11 2 2 2 2 14 →= 11 6 4 5 2 2 14 , 11 2 2 2 2 15 →= 11 6 4 5 2 2 15 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 16 2 2 2 2 14 →= 16 6 4 5 2 2 14 , 16 2 2 2 2 15 →= 16 6 4 5 2 2 15 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 17 2 2 2 2 14 →= 17 6 4 5 2 2 14 , 17 2 2 2 2 15 →= 17 6 4 5 2 2 15 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 18 5 2 2 2 14 →= 18 13 4 5 2 2 14 , 18 5 2 2 2 15 →= 18 13 4 5 2 2 15 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 19 16 2 2 2 14 →= 19 20 4 5 2 2 14 , 19 16 2 2 2 15 →= 19 20 4 5 2 2 15 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 1 2 2 2 6 21 →= 1 6 4 5 2 6 21 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 8 2 2 2 6 21 →= 8 6 4 5 2 6 21 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 11 2 2 2 6 21 →= 11 6 4 5 2 6 21 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 16 2 2 2 6 21 →= 16 6 4 5 2 6 21 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 21 →= 17 6 4 5 2 6 21 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 18 5 2 2 6 21 →= 18 13 4 5 2 6 21 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 19 16 2 2 6 21 →= 19 20 4 5 2 6 21 , 1 2 2 2 14 22 →= 1 6 4 5 2 14 22 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 8 2 2 2 14 22 →= 8 6 4 5 2 14 22 , 11 2 2 2 14 22 →= 11 6 4 5 2 14 22 , 16 2 2 2 14 22 →= 16 6 4 5 2 14 22 , 17 2 2 2 14 22 →= 17 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 13 4 5 2 14 22 , 18 5 2 2 14 22 →= 18 13 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 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 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 113-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 11 2 2 2 2 14 →= 11 6 4 5 2 2 14 , 11 2 2 2 2 15 →= 11 6 4 5 2 2 15 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 16 2 2 2 2 14 →= 16 6 4 5 2 2 14 , 16 2 2 2 2 15 →= 16 6 4 5 2 2 15 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 17 2 2 2 2 14 →= 17 6 4 5 2 2 14 , 17 2 2 2 2 15 →= 17 6 4 5 2 2 15 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 18 5 2 2 2 14 →= 18 13 4 5 2 2 14 , 18 5 2 2 2 15 →= 18 13 4 5 2 2 15 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 19 16 2 2 2 14 →= 19 20 4 5 2 2 14 , 19 16 2 2 2 15 →= 19 20 4 5 2 2 15 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 1 2 2 2 6 21 →= 1 6 4 5 2 6 21 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 8 2 2 2 6 21 →= 8 6 4 5 2 6 21 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 11 2 2 2 6 21 →= 11 6 4 5 2 6 21 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 16 2 2 2 6 21 →= 16 6 4 5 2 6 21 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 21 →= 17 6 4 5 2 6 21 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 18 5 2 2 6 21 →= 18 13 4 5 2 6 21 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 19 16 2 2 6 21 →= 19 20 4 5 2 6 21 , 1 2 2 2 14 22 →= 1 6 4 5 2 14 22 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 8 2 2 2 14 22 →= 8 6 4 5 2 14 22 , 11 2 2 2 14 22 →= 11 6 4 5 2 14 22 , 16 2 2 2 14 22 →= 16 6 4 5 2 14 22 , 17 2 2 2 14 22 →= 17 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 13 4 5 2 14 22 , 18 5 2 2 14 22 →= 18 13 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 112-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 11 2 2 2 2 15 →= 11 6 4 5 2 2 15 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 16 2 2 2 2 14 →= 16 6 4 5 2 2 14 , 16 2 2 2 2 15 →= 16 6 4 5 2 2 15 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 17 2 2 2 2 14 →= 17 6 4 5 2 2 14 , 17 2 2 2 2 15 →= 17 6 4 5 2 2 15 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 18 5 2 2 2 14 →= 18 13 4 5 2 2 14 , 18 5 2 2 2 15 →= 18 13 4 5 2 2 15 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 19 16 2 2 2 14 →= 19 20 4 5 2 2 14 , 19 16 2 2 2 15 →= 19 20 4 5 2 2 15 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 1 2 2 2 6 21 →= 1 6 4 5 2 6 21 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 8 2 2 2 6 21 →= 8 6 4 5 2 6 21 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 11 2 2 2 6 21 →= 11 6 4 5 2 6 21 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 16 2 2 2 6 21 →= 16 6 4 5 2 6 21 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 21 →= 17 6 4 5 2 6 21 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 18 5 2 2 6 21 →= 18 13 4 5 2 6 21 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 19 16 2 2 6 21 →= 19 20 4 5 2 6 21 , 1 2 2 2 14 22 →= 1 6 4 5 2 14 22 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 8 2 2 2 14 22 →= 8 6 4 5 2 14 22 , 11 2 2 2 14 22 →= 11 6 4 5 2 14 22 , 16 2 2 2 14 22 →= 16 6 4 5 2 14 22 , 17 2 2 2 14 22 →= 17 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 13 4 5 2 14 22 , 18 5 2 2 14 22 →= 18 13 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 111-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 16 2 2 2 2 14 →= 16 6 4 5 2 2 14 , 16 2 2 2 2 15 →= 16 6 4 5 2 2 15 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 17 2 2 2 2 14 →= 17 6 4 5 2 2 14 , 17 2 2 2 2 15 →= 17 6 4 5 2 2 15 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 18 5 2 2 2 14 →= 18 13 4 5 2 2 14 , 18 5 2 2 2 15 →= 18 13 4 5 2 2 15 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 19 16 2 2 2 14 →= 19 20 4 5 2 2 14 , 19 16 2 2 2 15 →= 19 20 4 5 2 2 15 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 1 2 2 2 6 21 →= 1 6 4 5 2 6 21 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 8 2 2 2 6 21 →= 8 6 4 5 2 6 21 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 11 2 2 2 6 21 →= 11 6 4 5 2 6 21 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 16 2 2 2 6 21 →= 16 6 4 5 2 6 21 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 21 →= 17 6 4 5 2 6 21 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 18 5 2 2 6 21 →= 18 13 4 5 2 6 21 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 19 16 2 2 6 21 →= 19 20 4 5 2 6 21 , 1 2 2 2 14 22 →= 1 6 4 5 2 14 22 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 8 2 2 2 14 22 →= 8 6 4 5 2 14 22 , 11 2 2 2 14 22 →= 11 6 4 5 2 14 22 , 16 2 2 2 14 22 →= 16 6 4 5 2 14 22 , 17 2 2 2 14 22 →= 17 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 13 4 5 2 14 22 , 18 5 2 2 14 22 →= 18 13 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 110-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 16 2 2 2 2 15 →= 16 6 4 5 2 2 15 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 17 2 2 2 2 14 →= 17 6 4 5 2 2 14 , 17 2 2 2 2 15 →= 17 6 4 5 2 2 15 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 18 5 2 2 2 14 →= 18 13 4 5 2 2 14 , 18 5 2 2 2 15 →= 18 13 4 5 2 2 15 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 19 16 2 2 2 14 →= 19 20 4 5 2 2 14 , 19 16 2 2 2 15 →= 19 20 4 5 2 2 15 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 1 2 2 2 6 21 →= 1 6 4 5 2 6 21 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 8 2 2 2 6 21 →= 8 6 4 5 2 6 21 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 11 2 2 2 6 21 →= 11 6 4 5 2 6 21 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 16 2 2 2 6 21 →= 16 6 4 5 2 6 21 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 21 →= 17 6 4 5 2 6 21 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 18 5 2 2 6 21 →= 18 13 4 5 2 6 21 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 19 16 2 2 6 21 →= 19 20 4 5 2 6 21 , 1 2 2 2 14 22 →= 1 6 4 5 2 14 22 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 8 2 2 2 14 22 →= 8 6 4 5 2 14 22 , 11 2 2 2 14 22 →= 11 6 4 5 2 14 22 , 16 2 2 2 14 22 →= 16 6 4 5 2 14 22 , 17 2 2 2 14 22 →= 17 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 13 4 5 2 14 22 , 18 5 2 2 14 22 →= 18 13 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 109-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 17 2 2 2 2 14 →= 17 6 4 5 2 2 14 , 17 2 2 2 2 15 →= 17 6 4 5 2 2 15 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 18 5 2 2 2 14 →= 18 13 4 5 2 2 14 , 18 5 2 2 2 15 →= 18 13 4 5 2 2 15 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 19 16 2 2 2 14 →= 19 20 4 5 2 2 14 , 19 16 2 2 2 15 →= 19 20 4 5 2 2 15 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 1 2 2 2 6 21 →= 1 6 4 5 2 6 21 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 8 2 2 2 6 21 →= 8 6 4 5 2 6 21 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 11 2 2 2 6 21 →= 11 6 4 5 2 6 21 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 16 2 2 2 6 21 →= 16 6 4 5 2 6 21 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 21 →= 17 6 4 5 2 6 21 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 18 5 2 2 6 21 →= 18 13 4 5 2 6 21 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 19 16 2 2 6 21 →= 19 20 4 5 2 6 21 , 1 2 2 2 14 22 →= 1 6 4 5 2 14 22 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 8 2 2 2 14 22 →= 8 6 4 5 2 14 22 , 11 2 2 2 14 22 →= 11 6 4 5 2 14 22 , 16 2 2 2 14 22 →= 16 6 4 5 2 14 22 , 17 2 2 2 14 22 →= 17 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 13 4 5 2 14 22 , 18 5 2 2 14 22 →= 18 13 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 108-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 17 2 2 2 2 15 →= 17 6 4 5 2 2 15 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 18 5 2 2 2 14 →= 18 13 4 5 2 2 14 , 18 5 2 2 2 15 →= 18 13 4 5 2 2 15 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 19 16 2 2 2 14 →= 19 20 4 5 2 2 14 , 19 16 2 2 2 15 →= 19 20 4 5 2 2 15 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 1 2 2 2 6 21 →= 1 6 4 5 2 6 21 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 8 2 2 2 6 21 →= 8 6 4 5 2 6 21 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 11 2 2 2 6 21 →= 11 6 4 5 2 6 21 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 16 2 2 2 6 21 →= 16 6 4 5 2 6 21 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 21 →= 17 6 4 5 2 6 21 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 18 5 2 2 6 21 →= 18 13 4 5 2 6 21 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 19 16 2 2 6 21 →= 19 20 4 5 2 6 21 , 1 2 2 2 14 22 →= 1 6 4 5 2 14 22 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 8 2 2 2 14 22 →= 8 6 4 5 2 14 22 , 11 2 2 2 14 22 →= 11 6 4 5 2 14 22 , 16 2 2 2 14 22 →= 16 6 4 5 2 14 22 , 17 2 2 2 14 22 →= 17 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 13 4 5 2 14 22 , 18 5 2 2 14 22 →= 18 13 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 107-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 18 5 2 2 2 14 →= 18 13 4 5 2 2 14 , 18 5 2 2 2 15 →= 18 13 4 5 2 2 15 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 19 16 2 2 2 14 →= 19 20 4 5 2 2 14 , 19 16 2 2 2 15 →= 19 20 4 5 2 2 15 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 1 2 2 2 6 21 →= 1 6 4 5 2 6 21 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 8 2 2 2 6 21 →= 8 6 4 5 2 6 21 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 11 2 2 2 6 21 →= 11 6 4 5 2 6 21 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 16 2 2 2 6 21 →= 16 6 4 5 2 6 21 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 21 →= 17 6 4 5 2 6 21 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 18 5 2 2 6 21 →= 18 13 4 5 2 6 21 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 19 16 2 2 6 21 →= 19 20 4 5 2 6 21 , 1 2 2 2 14 22 →= 1 6 4 5 2 14 22 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 8 2 2 2 14 22 →= 8 6 4 5 2 14 22 , 11 2 2 2 14 22 →= 11 6 4 5 2 14 22 , 16 2 2 2 14 22 →= 16 6 4 5 2 14 22 , 17 2 2 2 14 22 →= 17 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 13 4 5 2 14 22 , 18 5 2 2 14 22 →= 18 13 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 1 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 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 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 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 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 106-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 18 5 2 2 2 15 →= 18 13 4 5 2 2 15 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 19 16 2 2 2 14 →= 19 20 4 5 2 2 14 , 19 16 2 2 2 15 →= 19 20 4 5 2 2 15 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 1 2 2 2 6 21 →= 1 6 4 5 2 6 21 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 8 2 2 2 6 21 →= 8 6 4 5 2 6 21 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 11 2 2 2 6 21 →= 11 6 4 5 2 6 21 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 16 2 2 2 6 21 →= 16 6 4 5 2 6 21 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 21 →= 17 6 4 5 2 6 21 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 18 5 2 2 6 21 →= 18 13 4 5 2 6 21 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 19 16 2 2 6 21 →= 19 20 4 5 2 6 21 , 1 2 2 2 14 22 →= 1 6 4 5 2 14 22 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 8 2 2 2 14 22 →= 8 6 4 5 2 14 22 , 11 2 2 2 14 22 →= 11 6 4 5 2 14 22 , 16 2 2 2 14 22 →= 16 6 4 5 2 14 22 , 17 2 2 2 14 22 →= 17 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 13 4 5 2 14 22 , 18 5 2 2 14 22 →= 18 13 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 1 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 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 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 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 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 105-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 19 16 2 2 2 14 →= 19 20 4 5 2 2 14 , 19 16 2 2 2 15 →= 19 20 4 5 2 2 15 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 1 2 2 2 6 21 →= 1 6 4 5 2 6 21 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 8 2 2 2 6 21 →= 8 6 4 5 2 6 21 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 11 2 2 2 6 21 →= 11 6 4 5 2 6 21 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 16 2 2 2 6 21 →= 16 6 4 5 2 6 21 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 21 →= 17 6 4 5 2 6 21 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 18 5 2 2 6 21 →= 18 13 4 5 2 6 21 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 19 16 2 2 6 21 →= 19 20 4 5 2 6 21 , 1 2 2 2 14 22 →= 1 6 4 5 2 14 22 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 8 2 2 2 14 22 →= 8 6 4 5 2 14 22 , 11 2 2 2 14 22 →= 11 6 4 5 2 14 22 , 16 2 2 2 14 22 →= 16 6 4 5 2 14 22 , 17 2 2 2 14 22 →= 17 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 13 4 5 2 14 22 , 18 5 2 2 14 22 →= 18 13 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 1 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 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 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 104-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 19 16 2 2 2 15 →= 19 20 4 5 2 2 15 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 1 2 2 2 6 21 →= 1 6 4 5 2 6 21 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 8 2 2 2 6 21 →= 8 6 4 5 2 6 21 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 11 2 2 2 6 21 →= 11 6 4 5 2 6 21 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 16 2 2 2 6 21 →= 16 6 4 5 2 6 21 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 21 →= 17 6 4 5 2 6 21 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 18 5 2 2 6 21 →= 18 13 4 5 2 6 21 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 19 16 2 2 6 21 →= 19 20 4 5 2 6 21 , 1 2 2 2 14 22 →= 1 6 4 5 2 14 22 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 8 2 2 2 14 22 →= 8 6 4 5 2 14 22 , 11 2 2 2 14 22 →= 11 6 4 5 2 14 22 , 16 2 2 2 14 22 →= 16 6 4 5 2 14 22 , 17 2 2 2 14 22 →= 17 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 13 4 5 2 14 22 , 18 5 2 2 14 22 →= 18 13 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 1 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 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 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 103-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 1 2 2 2 6 21 →= 1 6 4 5 2 6 21 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 8 2 2 2 6 21 →= 8 6 4 5 2 6 21 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 11 2 2 2 6 21 →= 11 6 4 5 2 6 21 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 16 2 2 2 6 21 →= 16 6 4 5 2 6 21 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 21 →= 17 6 4 5 2 6 21 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 18 5 2 2 6 21 →= 18 13 4 5 2 6 21 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 19 16 2 2 6 21 →= 19 20 4 5 2 6 21 , 1 2 2 2 14 22 →= 1 6 4 5 2 14 22 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 8 2 2 2 14 22 →= 8 6 4 5 2 14 22 , 11 2 2 2 14 22 →= 11 6 4 5 2 14 22 , 16 2 2 2 14 22 →= 16 6 4 5 2 14 22 , 17 2 2 2 14 22 →= 17 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 13 4 5 2 14 22 , 18 5 2 2 14 22 →= 18 13 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 0 0 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 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 102-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 8 2 2 2 6 21 →= 8 6 4 5 2 6 21 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 11 2 2 2 6 21 →= 11 6 4 5 2 6 21 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 16 2 2 2 6 21 →= 16 6 4 5 2 6 21 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 21 →= 17 6 4 5 2 6 21 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 18 5 2 2 6 21 →= 18 13 4 5 2 6 21 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 19 16 2 2 6 21 →= 19 20 4 5 2 6 21 , 1 2 2 2 14 22 →= 1 6 4 5 2 14 22 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 8 2 2 2 14 22 →= 8 6 4 5 2 14 22 , 11 2 2 2 14 22 →= 11 6 4 5 2 14 22 , 16 2 2 2 14 22 →= 16 6 4 5 2 14 22 , 17 2 2 2 14 22 →= 17 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 13 4 5 2 14 22 , 18 5 2 2 14 22 →= 18 13 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 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 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 101-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 11 2 2 2 6 21 →= 11 6 4 5 2 6 21 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 16 2 2 2 6 21 →= 16 6 4 5 2 6 21 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 21 →= 17 6 4 5 2 6 21 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 18 5 2 2 6 21 →= 18 13 4 5 2 6 21 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 19 16 2 2 6 21 →= 19 20 4 5 2 6 21 , 1 2 2 2 14 22 →= 1 6 4 5 2 14 22 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 8 2 2 2 14 22 →= 8 6 4 5 2 14 22 , 11 2 2 2 14 22 →= 11 6 4 5 2 14 22 , 16 2 2 2 14 22 →= 16 6 4 5 2 14 22 , 17 2 2 2 14 22 →= 17 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 13 4 5 2 14 22 , 18 5 2 2 14 22 →= 18 13 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 100-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 16 2 2 2 6 21 →= 16 6 4 5 2 6 21 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 21 →= 17 6 4 5 2 6 21 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 18 5 2 2 6 21 →= 18 13 4 5 2 6 21 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 19 16 2 2 6 21 →= 19 20 4 5 2 6 21 , 1 2 2 2 14 22 →= 1 6 4 5 2 14 22 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 8 2 2 2 14 22 →= 8 6 4 5 2 14 22 , 11 2 2 2 14 22 →= 11 6 4 5 2 14 22 , 16 2 2 2 14 22 →= 16 6 4 5 2 14 22 , 17 2 2 2 14 22 →= 17 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 13 4 5 2 14 22 , 18 5 2 2 14 22 →= 18 13 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 99-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 17 2 2 2 6 21 →= 17 6 4 5 2 6 21 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 18 5 2 2 6 21 →= 18 13 4 5 2 6 21 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 19 16 2 2 6 21 →= 19 20 4 5 2 6 21 , 1 2 2 2 14 22 →= 1 6 4 5 2 14 22 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 8 2 2 2 14 22 →= 8 6 4 5 2 14 22 , 11 2 2 2 14 22 →= 11 6 4 5 2 14 22 , 16 2 2 2 14 22 →= 16 6 4 5 2 14 22 , 17 2 2 2 14 22 →= 17 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 13 4 5 2 14 22 , 18 5 2 2 14 22 →= 18 13 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 98-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 18 5 2 2 6 21 →= 18 13 4 5 2 6 21 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 19 16 2 2 6 21 →= 19 20 4 5 2 6 21 , 1 2 2 2 14 22 →= 1 6 4 5 2 14 22 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 8 2 2 2 14 22 →= 8 6 4 5 2 14 22 , 11 2 2 2 14 22 →= 11 6 4 5 2 14 22 , 16 2 2 2 14 22 →= 16 6 4 5 2 14 22 , 17 2 2 2 14 22 →= 17 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 13 4 5 2 14 22 , 18 5 2 2 14 22 →= 18 13 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 1 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 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 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 97-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 19 16 2 2 6 21 →= 19 20 4 5 2 6 21 , 1 2 2 2 14 22 →= 1 6 4 5 2 14 22 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 8 2 2 2 14 22 →= 8 6 4 5 2 14 22 , 11 2 2 2 14 22 →= 11 6 4 5 2 14 22 , 16 2 2 2 14 22 →= 16 6 4 5 2 14 22 , 17 2 2 2 14 22 →= 17 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 13 4 5 2 14 22 , 18 5 2 2 14 22 →= 18 13 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 1 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 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 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 96-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 1 2 2 2 14 22 →= 1 6 4 5 2 14 22 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 8 2 2 2 14 22 →= 8 6 4 5 2 14 22 , 11 2 2 2 14 22 →= 11 6 4 5 2 14 22 , 16 2 2 2 14 22 →= 16 6 4 5 2 14 22 , 17 2 2 2 14 22 →= 17 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 13 4 5 2 14 22 , 18 5 2 2 14 22 →= 18 13 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 0 0 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 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 95-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 8 2 2 2 14 22 →= 8 6 4 5 2 14 22 , 11 2 2 2 14 22 →= 11 6 4 5 2 14 22 , 16 2 2 2 14 22 →= 16 6 4 5 2 14 22 , 17 2 2 2 14 22 →= 17 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 13 4 5 2 14 22 , 18 5 2 2 14 22 →= 18 13 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 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 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 94-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 11 2 2 2 14 22 →= 11 6 4 5 2 14 22 , 16 2 2 2 14 22 →= 16 6 4 5 2 14 22 , 17 2 2 2 14 22 →= 17 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 13 4 5 2 14 22 , 18 5 2 2 14 22 →= 18 13 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 93-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 16 2 2 2 14 22 →= 16 6 4 5 2 14 22 , 17 2 2 2 14 22 →= 17 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 13 4 5 2 14 22 , 18 5 2 2 14 22 →= 18 13 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 92-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 17 2 2 2 14 22 →= 17 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 13 4 5 2 14 22 , 18 5 2 2 14 22 →= 18 13 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 91-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 13 4 5 2 14 22 , 18 5 2 2 14 22 →= 18 13 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 7: 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 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 1 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 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 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 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 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 ⎟ ⎜ 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 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 0 0 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 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 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 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 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 90-rule system { 0 1 2 2 2 2 ⟶ 0 3 4 5 2 2 2 , 0 1 2 2 2 6 ⟶ 0 3 4 5 2 2 6 , 7 8 2 2 2 2 ⟶ 7 9 4 5 2 2 2 , 7 8 2 2 2 6 ⟶ 7 9 4 5 2 2 6 , 10 11 2 2 2 2 ⟶ 10 12 4 5 2 2 2 , 10 11 2 2 2 6 ⟶ 10 12 4 5 2 2 6 , 0 1 2 2 6 4 ⟶ 0 3 4 5 2 6 4 , 7 8 2 2 6 4 ⟶ 7 9 4 5 2 6 4 , 10 11 2 2 6 4 ⟶ 10 12 4 5 2 6 4 , 0 3 4 5 6 4 5 2 ⟶ 0 1 6 4 5 2 2 , 0 3 4 5 6 4 5 6 ⟶ 0 1 6 4 5 2 6 , 7 9 4 5 6 4 5 2 ⟶ 7 8 6 4 5 2 2 , 7 9 4 5 6 4 5 6 ⟶ 7 8 6 4 5 2 6 , 10 12 4 5 6 4 5 2 ⟶ 10 11 6 4 5 2 2 , 10 12 4 5 6 4 5 6 ⟶ 10 11 6 4 5 2 6 , 0 3 4 5 6 4 13 4 ⟶ 0 1 6 4 5 6 4 , 7 9 4 5 6 4 13 4 ⟶ 7 8 6 4 5 6 4 , 10 12 4 5 6 4 13 4 ⟶ 10 11 6 4 5 6 4 , 1 2 2 2 2 2 →= 1 6 4 5 2 2 2 , 1 2 2 2 2 6 →= 1 6 4 5 2 2 6 , 2 2 2 2 2 2 →= 2 6 4 5 2 2 2 , 2 2 2 2 2 6 →= 2 6 4 5 2 2 6 , 2 2 2 2 2 14 →= 2 6 4 5 2 2 14 , 2 2 2 2 2 15 →= 2 6 4 5 2 2 15 , 5 2 2 2 2 2 →= 5 6 4 5 2 2 2 , 5 2 2 2 2 6 →= 5 6 4 5 2 2 6 , 5 2 2 2 2 14 →= 5 6 4 5 2 2 14 , 5 2 2 2 2 15 →= 5 6 4 5 2 2 15 , 8 2 2 2 2 2 →= 8 6 4 5 2 2 2 , 8 2 2 2 2 6 →= 8 6 4 5 2 2 6 , 11 2 2 2 2 2 →= 11 6 4 5 2 2 2 , 11 2 2 2 2 6 →= 11 6 4 5 2 2 6 , 16 2 2 2 2 2 →= 16 6 4 5 2 2 2 , 16 2 2 2 2 6 →= 16 6 4 5 2 2 6 , 17 2 2 2 2 2 →= 17 6 4 5 2 2 2 , 17 2 2 2 2 6 →= 17 6 4 5 2 2 6 , 4 5 2 2 2 2 →= 4 13 4 5 2 2 2 , 4 5 2 2 2 6 →= 4 13 4 5 2 2 6 , 4 5 2 2 2 14 →= 4 13 4 5 2 2 14 , 4 5 2 2 2 15 →= 4 13 4 5 2 2 15 , 18 5 2 2 2 2 →= 18 13 4 5 2 2 2 , 18 5 2 2 2 6 →= 18 13 4 5 2 2 6 , 19 16 2 2 2 2 →= 19 20 4 5 2 2 2 , 19 16 2 2 2 6 →= 19 20 4 5 2 2 6 , 1 2 2 2 6 4 →= 1 6 4 5 2 6 4 , 2 2 2 2 6 4 →= 2 6 4 5 2 6 4 , 2 2 2 2 6 21 →= 2 6 4 5 2 6 21 , 5 2 2 2 6 4 →= 5 6 4 5 2 6 4 , 5 2 2 2 6 21 →= 5 6 4 5 2 6 21 , 8 2 2 2 6 4 →= 8 6 4 5 2 6 4 , 11 2 2 2 6 4 →= 11 6 4 5 2 6 4 , 16 2 2 2 6 4 →= 16 6 4 5 2 6 4 , 17 2 2 2 6 4 →= 17 6 4 5 2 6 4 , 4 5 2 2 6 4 →= 4 13 4 5 2 6 4 , 4 5 2 2 6 21 →= 4 13 4 5 2 6 21 , 18 5 2 2 6 4 →= 18 13 4 5 2 6 4 , 19 16 2 2 6 4 →= 19 20 4 5 2 6 4 , 2 2 2 2 14 22 →= 2 6 4 5 2 14 22 , 5 2 2 2 14 22 →= 5 6 4 5 2 14 22 , 4 5 2 2 14 22 →= 4 13 4 5 2 14 22 , 1 6 4 5 6 4 5 2 →= 1 2 6 4 5 2 2 , 1 6 4 5 6 4 5 6 →= 1 2 6 4 5 2 6 , 2 6 4 5 6 4 5 2 →= 2 2 6 4 5 2 2 , 2 6 4 5 6 4 5 6 →= 2 2 6 4 5 2 6 , 5 6 4 5 6 4 5 2 →= 5 2 6 4 5 2 2 , 5 6 4 5 6 4 5 6 →= 5 2 6 4 5 2 6 , 8 6 4 5 6 4 5 2 →= 8 2 6 4 5 2 2 , 8 6 4 5 6 4 5 6 →= 8 2 6 4 5 2 6 , 11 6 4 5 6 4 5 2 →= 11 2 6 4 5 2 2 , 11 6 4 5 6 4 5 6 →= 11 2 6 4 5 2 6 , 16 6 4 5 6 4 5 2 →= 16 2 6 4 5 2 2 , 16 6 4 5 6 4 5 6 →= 16 2 6 4 5 2 6 , 17 6 4 5 6 4 5 2 →= 17 2 6 4 5 2 2 , 17 6 4 5 6 4 5 6 →= 17 2 6 4 5 2 6 , 4 13 4 5 6 4 5 2 →= 4 5 6 4 5 2 2 , 4 13 4 5 6 4 5 6 →= 4 5 6 4 5 2 6 , 18 13 4 5 6 4 5 2 →= 18 5 6 4 5 2 2 , 18 13 4 5 6 4 5 6 →= 18 5 6 4 5 2 6 , 19 20 4 5 6 4 5 2 →= 19 16 6 4 5 2 2 , 19 20 4 5 6 4 5 6 →= 19 16 6 4 5 2 6 , 1 6 4 5 6 4 13 4 →= 1 2 6 4 5 6 4 , 2 6 4 5 6 4 13 4 →= 2 2 6 4 5 6 4 , 5 6 4 5 6 4 13 4 →= 5 2 6 4 5 6 4 , 8 6 4 5 6 4 13 4 →= 8 2 6 4 5 6 4 , 11 6 4 5 6 4 13 4 →= 11 2 6 4 5 6 4 , 16 6 4 5 6 4 13 4 →= 16 2 6 4 5 6 4 , 17 6 4 5 6 4 13 4 →= 17 2 6 4 5 6 4 , 4 13 4 5 6 4 13 4 →= 4 5 6 4 5 6 4 , 18 13 4 5 6 4 13 4 →= 18 5 6 4 5 6 4 , 19 20 4 5 6 4 13 4 →= 19 16 6 4 5 6 4 } Applying sparse tiling TROC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (23,0) ↦ 0, (0,1) ↦ 1, (1,2) ↦ 2, (2,2) ↦ 3, (0,3) ↦ 4, (3,4) ↦ 5, (4,5) ↦ 6, (5,2) ↦ 7, (2,6) ↦ 8, (2,24) ↦ 9, (2,14) ↦ 10, (2,15) ↦ 11, (6,4) ↦ 12, (6,21) ↦ 13, (6,24) ↦ 14, (23,7) ↦ 15, (7,8) ↦ 16, (8,2) ↦ 17, (7,9) ↦ 18, (9,4) ↦ 19, (23,10) ↦ 20, (10,11) ↦ 21, (11,2) ↦ 22, (10,12) ↦ 23, (12,4) ↦ 24, (4,24) ↦ 25, (4,13) ↦ 26, (5,6) ↦ 27, (1,6) ↦ 28, (8,6) ↦ 29, (11,6) ↦ 30, (13,4) ↦ 31, (23,1) ↦ 32, (16,2) ↦ 33, (17,2) ↦ 34, (23,2) ↦ 35, (14,22) ↦ 36, (14,24) ↦ 37, (15,24) ↦ 38, (18,5) ↦ 39, (23,5) ↦ 40, (23,8) ↦ 41, (23,11) ↦ 42, (19,16) ↦ 43, (16,6) ↦ 44, (23,16) ↦ 45, (23,17) ↦ 46, (17,6) ↦ 47, (20,4) ↦ 48, (23,4) ↦ 49, (23,18) ↦ 50, (18,13) ↦ 51, (23,19) ↦ 52, (19,20) ↦ 53, (21,24) ↦ 54, (22,24) ↦ 55 }, it remains to prove termination of the 794-rule system { 0 1 2 3 3 3 3 ⟶ 0 4 5 6 7 3 3 3 , 0 1 2 3 3 3 8 ⟶ 0 4 5 6 7 3 3 8 , 0 1 2 3 3 3 9 ⟶ 0 4 5 6 7 3 3 9 , 0 1 2 3 3 3 10 ⟶ 0 4 5 6 7 3 3 10 , 0 1 2 3 3 3 11 ⟶ 0 4 5 6 7 3 3 11 , 0 1 2 3 3 8 12 ⟶ 0 4 5 6 7 3 8 12 , 0 1 2 3 3 8 13 ⟶ 0 4 5 6 7 3 8 13 , 0 1 2 3 3 8 14 ⟶ 0 4 5 6 7 3 8 14 , 15 16 17 3 3 3 3 ⟶ 15 18 19 6 7 3 3 3 , 15 16 17 3 3 3 8 ⟶ 15 18 19 6 7 3 3 8 , 15 16 17 3 3 3 9 ⟶ 15 18 19 6 7 3 3 9 , 15 16 17 3 3 3 10 ⟶ 15 18 19 6 7 3 3 10 , 15 16 17 3 3 3 11 ⟶ 15 18 19 6 7 3 3 11 , 15 16 17 3 3 8 12 ⟶ 15 18 19 6 7 3 8 12 , 15 16 17 3 3 8 13 ⟶ 15 18 19 6 7 3 8 13 , 15 16 17 3 3 8 14 ⟶ 15 18 19 6 7 3 8 14 , 20 21 22 3 3 3 3 ⟶ 20 23 24 6 7 3 3 3 , 20 21 22 3 3 3 8 ⟶ 20 23 24 6 7 3 3 8 , 20 21 22 3 3 3 9 ⟶ 20 23 24 6 7 3 3 9 , 20 21 22 3 3 3 10 ⟶ 20 23 24 6 7 3 3 10 , 20 21 22 3 3 3 11 ⟶ 20 23 24 6 7 3 3 11 , 20 21 22 3 3 8 12 ⟶ 20 23 24 6 7 3 8 12 , 20 21 22 3 3 8 13 ⟶ 20 23 24 6 7 3 8 13 , 20 21 22 3 3 8 14 ⟶ 20 23 24 6 7 3 8 14 , 0 1 2 3 8 12 6 ⟶ 0 4 5 6 7 8 12 6 , 0 1 2 3 8 12 25 ⟶ 0 4 5 6 7 8 12 25 , 0 1 2 3 8 12 26 ⟶ 0 4 5 6 7 8 12 26 , 15 16 17 3 8 12 6 ⟶ 15 18 19 6 7 8 12 6 , 15 16 17 3 8 12 25 ⟶ 15 18 19 6 7 8 12 25 , 15 16 17 3 8 12 26 ⟶ 15 18 19 6 7 8 12 26 , 20 21 22 3 8 12 6 ⟶ 20 23 24 6 7 8 12 6 , 20 21 22 3 8 12 25 ⟶ 20 23 24 6 7 8 12 25 , 20 21 22 3 8 12 26 ⟶ 20 23 24 6 7 8 12 26 , 0 4 5 6 27 12 6 7 3 ⟶ 0 1 28 12 6 7 3 3 , 0 4 5 6 27 12 6 7 8 ⟶ 0 1 28 12 6 7 3 8 , 0 4 5 6 27 12 6 7 9 ⟶ 0 1 28 12 6 7 3 9 , 0 4 5 6 27 12 6 7 10 ⟶ 0 1 28 12 6 7 3 10 , 0 4 5 6 27 12 6 7 11 ⟶ 0 1 28 12 6 7 3 11 , 0 4 5 6 27 12 6 27 12 ⟶ 0 1 28 12 6 7 8 12 , 0 4 5 6 27 12 6 27 13 ⟶ 0 1 28 12 6 7 8 13 , 0 4 5 6 27 12 6 27 14 ⟶ 0 1 28 12 6 7 8 14 , 15 18 19 6 27 12 6 7 3 ⟶ 15 16 29 12 6 7 3 3 , 15 18 19 6 27 12 6 7 8 ⟶ 15 16 29 12 6 7 3 8 , 15 18 19 6 27 12 6 7 9 ⟶ 15 16 29 12 6 7 3 9 , 15 18 19 6 27 12 6 7 10 ⟶ 15 16 29 12 6 7 3 10 , 15 18 19 6 27 12 6 7 11 ⟶ 15 16 29 12 6 7 3 11 , 15 18 19 6 27 12 6 27 12 ⟶ 15 16 29 12 6 7 8 12 , 15 18 19 6 27 12 6 27 13 ⟶ 15 16 29 12 6 7 8 13 , 15 18 19 6 27 12 6 27 14 ⟶ 15 16 29 12 6 7 8 14 , 20 23 24 6 27 12 6 7 3 ⟶ 20 21 30 12 6 7 3 3 , 20 23 24 6 27 12 6 7 8 ⟶ 20 21 30 12 6 7 3 8 , 20 23 24 6 27 12 6 7 9 ⟶ 20 21 30 12 6 7 3 9 , 20 23 24 6 27 12 6 7 10 ⟶ 20 21 30 12 6 7 3 10 , 20 23 24 6 27 12 6 7 11 ⟶ 20 21 30 12 6 7 3 11 , 20 23 24 6 27 12 6 27 12 ⟶ 20 21 30 12 6 7 8 12 , 20 23 24 6 27 12 6 27 13 ⟶ 20 21 30 12 6 7 8 13 , 20 23 24 6 27 12 6 27 14 ⟶ 20 21 30 12 6 7 8 14 , 0 4 5 6 27 12 26 31 6 ⟶ 0 1 28 12 6 27 12 6 , 0 4 5 6 27 12 26 31 25 ⟶ 0 1 28 12 6 27 12 25 , 0 4 5 6 27 12 26 31 26 ⟶ 0 1 28 12 6 27 12 26 , 15 18 19 6 27 12 26 31 6 ⟶ 15 16 29 12 6 27 12 6 , 15 18 19 6 27 12 26 31 25 ⟶ 15 16 29 12 6 27 12 25 , 15 18 19 6 27 12 26 31 26 ⟶ 15 16 29 12 6 27 12 26 , 20 23 24 6 27 12 26 31 6 ⟶ 20 21 30 12 6 27 12 6 , 20 23 24 6 27 12 26 31 25 ⟶ 20 21 30 12 6 27 12 25 , 20 23 24 6 27 12 26 31 26 ⟶ 20 21 30 12 6 27 12 26 , 1 2 3 3 3 3 3 →= 1 28 12 6 7 3 3 3 , 1 2 3 3 3 3 8 →= 1 28 12 6 7 3 3 8 , 1 2 3 3 3 3 9 →= 1 28 12 6 7 3 3 9 , 1 2 3 3 3 3 10 →= 1 28 12 6 7 3 3 10 , 1 2 3 3 3 3 11 →= 1 28 12 6 7 3 3 11 , 32 2 3 3 3 3 3 →= 32 28 12 6 7 3 3 3 , 32 2 3 3 3 3 8 →= 32 28 12 6 7 3 3 8 , 32 2 3 3 3 3 9 →= 32 28 12 6 7 3 3 9 , 32 2 3 3 3 3 10 →= 32 28 12 6 7 3 3 10 , 32 2 3 3 3 3 11 →= 32 28 12 6 7 3 3 11 , 1 2 3 3 3 8 12 →= 1 28 12 6 7 3 8 12 , 1 2 3 3 3 8 13 →= 1 28 12 6 7 3 8 13 , 1 2 3 3 3 8 14 →= 1 28 12 6 7 3 8 14 , 32 2 3 3 3 8 12 →= 32 28 12 6 7 3 8 12 , 32 2 3 3 3 8 13 →= 32 28 12 6 7 3 8 13 , 32 2 3 3 3 8 14 →= 32 28 12 6 7 3 8 14 , 33 3 3 3 3 3 3 →= 33 8 12 6 7 3 3 3 , 33 3 3 3 3 3 8 →= 33 8 12 6 7 3 3 8 , 33 3 3 3 3 3 9 →= 33 8 12 6 7 3 3 9 , 33 3 3 3 3 3 10 →= 33 8 12 6 7 3 3 10 , 33 3 3 3 3 3 11 →= 33 8 12 6 7 3 3 11 , 2 3 3 3 3 3 3 →= 2 8 12 6 7 3 3 3 , 2 3 3 3 3 3 8 →= 2 8 12 6 7 3 3 8 , 2 3 3 3 3 3 9 →= 2 8 12 6 7 3 3 9 , 2 3 3 3 3 3 10 →= 2 8 12 6 7 3 3 10 , 2 3 3 3 3 3 11 →= 2 8 12 6 7 3 3 11 , 34 3 3 3 3 3 3 →= 34 8 12 6 7 3 3 3 , 34 3 3 3 3 3 8 →= 34 8 12 6 7 3 3 8 , 34 3 3 3 3 3 9 →= 34 8 12 6 7 3 3 9 , 34 3 3 3 3 3 10 →= 34 8 12 6 7 3 3 10 , 34 3 3 3 3 3 11 →= 34 8 12 6 7 3 3 11 , 3 3 3 3 3 3 3 →= 3 8 12 6 7 3 3 3 , 3 3 3 3 3 3 8 →= 3 8 12 6 7 3 3 8 , 3 3 3 3 3 3 9 →= 3 8 12 6 7 3 3 9 , 3 3 3 3 3 3 10 →= 3 8 12 6 7 3 3 10 , 3 3 3 3 3 3 11 →= 3 8 12 6 7 3 3 11 , 7 3 3 3 3 3 3 →= 7 8 12 6 7 3 3 3 , 7 3 3 3 3 3 8 →= 7 8 12 6 7 3 3 8 , 7 3 3 3 3 3 9 →= 7 8 12 6 7 3 3 9 , 7 3 3 3 3 3 10 →= 7 8 12 6 7 3 3 10 , 7 3 3 3 3 3 11 →= 7 8 12 6 7 3 3 11 , 35 3 3 3 3 3 3 →= 35 8 12 6 7 3 3 3 , 35 3 3 3 3 3 8 →= 35 8 12 6 7 3 3 8 , 35 3 3 3 3 3 9 →= 35 8 12 6 7 3 3 9 , 35 3 3 3 3 3 10 →= 35 8 12 6 7 3 3 10 , 35 3 3 3 3 3 11 →= 35 8 12 6 7 3 3 11 , 17 3 3 3 3 3 3 →= 17 8 12 6 7 3 3 3 , 17 3 3 3 3 3 8 →= 17 8 12 6 7 3 3 8 , 17 3 3 3 3 3 9 →= 17 8 12 6 7 3 3 9 , 17 3 3 3 3 3 10 →= 17 8 12 6 7 3 3 10 , 17 3 3 3 3 3 11 →= 17 8 12 6 7 3 3 11 , 22 3 3 3 3 3 3 →= 22 8 12 6 7 3 3 3 , 22 3 3 3 3 3 8 →= 22 8 12 6 7 3 3 8 , 22 3 3 3 3 3 9 →= 22 8 12 6 7 3 3 9 , 22 3 3 3 3 3 10 →= 22 8 12 6 7 3 3 10 , 22 3 3 3 3 3 11 →= 22 8 12 6 7 3 3 11 , 33 3 3 3 3 8 12 →= 33 8 12 6 7 3 8 12 , 33 3 3 3 3 8 13 →= 33 8 12 6 7 3 8 13 , 33 3 3 3 3 8 14 →= 33 8 12 6 7 3 8 14 , 2 3 3 3 3 8 12 →= 2 8 12 6 7 3 8 12 , 2 3 3 3 3 8 13 →= 2 8 12 6 7 3 8 13 , 2 3 3 3 3 8 14 →= 2 8 12 6 7 3 8 14 , 34 3 3 3 3 8 12 →= 34 8 12 6 7 3 8 12 , 34 3 3 3 3 8 13 →= 34 8 12 6 7 3 8 13 , 34 3 3 3 3 8 14 →= 34 8 12 6 7 3 8 14 , 3 3 3 3 3 8 12 →= 3 8 12 6 7 3 8 12 , 3 3 3 3 3 8 13 →= 3 8 12 6 7 3 8 13 , 3 3 3 3 3 8 14 →= 3 8 12 6 7 3 8 14 , 7 3 3 3 3 8 12 →= 7 8 12 6 7 3 8 12 , 7 3 3 3 3 8 13 →= 7 8 12 6 7 3 8 13 , 7 3 3 3 3 8 14 →= 7 8 12 6 7 3 8 14 , 35 3 3 3 3 8 12 →= 35 8 12 6 7 3 8 12 , 35 3 3 3 3 8 13 →= 35 8 12 6 7 3 8 13 , 35 3 3 3 3 8 14 →= 35 8 12 6 7 3 8 14 , 17 3 3 3 3 8 12 →= 17 8 12 6 7 3 8 12 , 17 3 3 3 3 8 13 →= 17 8 12 6 7 3 8 13 , 17 3 3 3 3 8 14 →= 17 8 12 6 7 3 8 14 , 22 3 3 3 3 8 12 →= 22 8 12 6 7 3 8 12 , 22 3 3 3 3 8 13 →= 22 8 12 6 7 3 8 13 , 22 3 3 3 3 8 14 →= 22 8 12 6 7 3 8 14 , 33 3 3 3 3 10 36 →= 33 8 12 6 7 3 10 36 , 33 3 3 3 3 10 37 →= 33 8 12 6 7 3 10 37 , 2 3 3 3 3 10 36 →= 2 8 12 6 7 3 10 36 , 2 3 3 3 3 10 37 →= 2 8 12 6 7 3 10 37 , 34 3 3 3 3 10 36 →= 34 8 12 6 7 3 10 36 , 34 3 3 3 3 10 37 →= 34 8 12 6 7 3 10 37 , 3 3 3 3 3 10 36 →= 3 8 12 6 7 3 10 36 , 3 3 3 3 3 10 37 →= 3 8 12 6 7 3 10 37 , 7 3 3 3 3 10 36 →= 7 8 12 6 7 3 10 36 , 7 3 3 3 3 10 37 →= 7 8 12 6 7 3 10 37 , 35 3 3 3 3 10 36 →= 35 8 12 6 7 3 10 36 , 35 3 3 3 3 10 37 →= 35 8 12 6 7 3 10 37 , 17 3 3 3 3 10 36 →= 17 8 12 6 7 3 10 36 , 17 3 3 3 3 10 37 →= 17 8 12 6 7 3 10 37 , 22 3 3 3 3 10 36 →= 22 8 12 6 7 3 10 36 , 22 3 3 3 3 10 37 →= 22 8 12 6 7 3 10 37 , 33 3 3 3 3 11 38 →= 33 8 12 6 7 3 11 38 , 2 3 3 3 3 11 38 →= 2 8 12 6 7 3 11 38 , 34 3 3 3 3 11 38 →= 34 8 12 6 7 3 11 38 , 3 3 3 3 3 11 38 →= 3 8 12 6 7 3 11 38 , 7 3 3 3 3 11 38 →= 7 8 12 6 7 3 11 38 , 35 3 3 3 3 11 38 →= 35 8 12 6 7 3 11 38 , 17 3 3 3 3 11 38 →= 17 8 12 6 7 3 11 38 , 22 3 3 3 3 11 38 →= 22 8 12 6 7 3 11 38 , 39 7 3 3 3 3 3 →= 39 27 12 6 7 3 3 3 , 39 7 3 3 3 3 8 →= 39 27 12 6 7 3 3 8 , 39 7 3 3 3 3 9 →= 39 27 12 6 7 3 3 9 , 39 7 3 3 3 3 10 →= 39 27 12 6 7 3 3 10 , 39 7 3 3 3 3 11 →= 39 27 12 6 7 3 3 11 , 6 7 3 3 3 3 3 →= 6 27 12 6 7 3 3 3 , 6 7 3 3 3 3 8 →= 6 27 12 6 7 3 3 8 , 6 7 3 3 3 3 9 →= 6 27 12 6 7 3 3 9 , 6 7 3 3 3 3 10 →= 6 27 12 6 7 3 3 10 , 6 7 3 3 3 3 11 →= 6 27 12 6 7 3 3 11 , 40 7 3 3 3 3 3 →= 40 27 12 6 7 3 3 3 , 40 7 3 3 3 3 8 →= 40 27 12 6 7 3 3 8 , 40 7 3 3 3 3 9 →= 40 27 12 6 7 3 3 9 , 40 7 3 3 3 3 10 →= 40 27 12 6 7 3 3 10 , 40 7 3 3 3 3 11 →= 40 27 12 6 7 3 3 11 , 39 7 3 3 3 8 12 →= 39 27 12 6 7 3 8 12 , 39 7 3 3 3 8 13 →= 39 27 12 6 7 3 8 13 , 39 7 3 3 3 8 14 →= 39 27 12 6 7 3 8 14 , 6 7 3 3 3 8 12 →= 6 27 12 6 7 3 8 12 , 6 7 3 3 3 8 13 →= 6 27 12 6 7 3 8 13 , 6 7 3 3 3 8 14 →= 6 27 12 6 7 3 8 14 , 40 7 3 3 3 8 12 →= 40 27 12 6 7 3 8 12 , 40 7 3 3 3 8 13 →= 40 27 12 6 7 3 8 13 , 40 7 3 3 3 8 14 →= 40 27 12 6 7 3 8 14 , 39 7 3 3 3 10 36 →= 39 27 12 6 7 3 10 36 , 39 7 3 3 3 10 37 →= 39 27 12 6 7 3 10 37 , 6 7 3 3 3 10 36 →= 6 27 12 6 7 3 10 36 , 6 7 3 3 3 10 37 →= 6 27 12 6 7 3 10 37 , 40 7 3 3 3 10 36 →= 40 27 12 6 7 3 10 36 , 40 7 3 3 3 10 37 →= 40 27 12 6 7 3 10 37 , 39 7 3 3 3 11 38 →= 39 27 12 6 7 3 11 38 , 6 7 3 3 3 11 38 →= 6 27 12 6 7 3 11 38 , 40 7 3 3 3 11 38 →= 40 27 12 6 7 3 11 38 , 16 17 3 3 3 3 3 →= 16 29 12 6 7 3 3 3 , 16 17 3 3 3 3 8 →= 16 29 12 6 7 3 3 8 , 16 17 3 3 3 3 9 →= 16 29 12 6 7 3 3 9 , 16 17 3 3 3 3 10 →= 16 29 12 6 7 3 3 10 , 16 17 3 3 3 3 11 →= 16 29 12 6 7 3 3 11 , 41 17 3 3 3 3 3 →= 41 29 12 6 7 3 3 3 , 41 17 3 3 3 3 8 →= 41 29 12 6 7 3 3 8 , 41 17 3 3 3 3 9 →= 41 29 12 6 7 3 3 9 , 41 17 3 3 3 3 10 →= 41 29 12 6 7 3 3 10 , 41 17 3 3 3 3 11 →= 41 29 12 6 7 3 3 11 , 16 17 3 3 3 8 12 →= 16 29 12 6 7 3 8 12 , 16 17 3 3 3 8 13 →= 16 29 12 6 7 3 8 13 , 16 17 3 3 3 8 14 →= 16 29 12 6 7 3 8 14 , 41 17 3 3 3 8 12 →= 41 29 12 6 7 3 8 12 , 41 17 3 3 3 8 13 →= 41 29 12 6 7 3 8 13 , 41 17 3 3 3 8 14 →= 41 29 12 6 7 3 8 14 , 42 22 3 3 3 3 3 →= 42 30 12 6 7 3 3 3 , 42 22 3 3 3 3 8 →= 42 30 12 6 7 3 3 8 , 42 22 3 3 3 3 9 →= 42 30 12 6 7 3 3 9 , 42 22 3 3 3 3 10 →= 42 30 12 6 7 3 3 10 , 42 22 3 3 3 3 11 →= 42 30 12 6 7 3 3 11 , 21 22 3 3 3 3 3 →= 21 30 12 6 7 3 3 3 , 21 22 3 3 3 3 8 →= 21 30 12 6 7 3 3 8 , 21 22 3 3 3 3 9 →= 21 30 12 6 7 3 3 9 , 21 22 3 3 3 3 10 →= 21 30 12 6 7 3 3 10 , 21 22 3 3 3 3 11 →= 21 30 12 6 7 3 3 11 , 42 22 3 3 3 8 12 →= 42 30 12 6 7 3 8 12 , 42 22 3 3 3 8 13 →= 42 30 12 6 7 3 8 13 , 42 22 3 3 3 8 14 →= 42 30 12 6 7 3 8 14 , 21 22 3 3 3 8 12 →= 21 30 12 6 7 3 8 12 , 21 22 3 3 3 8 13 →= 21 30 12 6 7 3 8 13 , 21 22 3 3 3 8 14 →= 21 30 12 6 7 3 8 14 , 43 33 3 3 3 3 3 →= 43 44 12 6 7 3 3 3 , 43 33 3 3 3 3 8 →= 43 44 12 6 7 3 3 8 , 43 33 3 3 3 3 9 →= 43 44 12 6 7 3 3 9 , 43 33 3 3 3 3 10 →= 43 44 12 6 7 3 3 10 , 43 33 3 3 3 3 11 →= 43 44 12 6 7 3 3 11 , 45 33 3 3 3 3 3 →= 45 44 12 6 7 3 3 3 , 45 33 3 3 3 3 8 →= 45 44 12 6 7 3 3 8 , 45 33 3 3 3 3 9 →= 45 44 12 6 7 3 3 9 , 45 33 3 3 3 3 10 →= 45 44 12 6 7 3 3 10 , 45 33 3 3 3 3 11 →= 45 44 12 6 7 3 3 11 , 43 33 3 3 3 8 12 →= 43 44 12 6 7 3 8 12 , 43 33 3 3 3 8 13 →= 43 44 12 6 7 3 8 13 , 43 33 3 3 3 8 14 →= 43 44 12 6 7 3 8 14 , 45 33 3 3 3 8 12 →= 45 44 12 6 7 3 8 12 , 45 33 3 3 3 8 13 →= 45 44 12 6 7 3 8 13 , 45 33 3 3 3 8 14 →= 45 44 12 6 7 3 8 14 , 46 34 3 3 3 3 3 →= 46 47 12 6 7 3 3 3 , 46 34 3 3 3 3 8 →= 46 47 12 6 7 3 3 8 , 46 34 3 3 3 3 9 →= 46 47 12 6 7 3 3 9 , 46 34 3 3 3 3 10 →= 46 47 12 6 7 3 3 10 , 46 34 3 3 3 3 11 →= 46 47 12 6 7 3 3 11 , 46 34 3 3 3 8 12 →= 46 47 12 6 7 3 8 12 , 46 34 3 3 3 8 13 →= 46 47 12 6 7 3 8 13 , 46 34 3 3 3 8 14 →= 46 47 12 6 7 3 8 14 , 5 6 7 3 3 3 3 →= 5 26 31 6 7 3 3 3 , 5 6 7 3 3 3 8 →= 5 26 31 6 7 3 3 8 , 5 6 7 3 3 3 9 →= 5 26 31 6 7 3 3 9 , 5 6 7 3 3 3 10 →= 5 26 31 6 7 3 3 10 , 5 6 7 3 3 3 11 →= 5 26 31 6 7 3 3 11 , 48 6 7 3 3 3 3 →= 48 26 31 6 7 3 3 3 , 48 6 7 3 3 3 8 →= 48 26 31 6 7 3 3 8 , 48 6 7 3 3 3 9 →= 48 26 31 6 7 3 3 9 , 48 6 7 3 3 3 10 →= 48 26 31 6 7 3 3 10 , 48 6 7 3 3 3 11 →= 48 26 31 6 7 3 3 11 , 12 6 7 3 3 3 3 →= 12 26 31 6 7 3 3 3 , 12 6 7 3 3 3 8 →= 12 26 31 6 7 3 3 8 , 12 6 7 3 3 3 9 →= 12 26 31 6 7 3 3 9 , 12 6 7 3 3 3 10 →= 12 26 31 6 7 3 3 10 , 12 6 7 3 3 3 11 →= 12 26 31 6 7 3 3 11 , 49 6 7 3 3 3 3 →= 49 26 31 6 7 3 3 3 , 49 6 7 3 3 3 8 →= 49 26 31 6 7 3 3 8 , 49 6 7 3 3 3 9 →= 49 26 31 6 7 3 3 9 , 49 6 7 3 3 3 10 →= 49 26 31 6 7 3 3 10 , 49 6 7 3 3 3 11 →= 49 26 31 6 7 3 3 11 , 19 6 7 3 3 3 3 →= 19 26 31 6 7 3 3 3 , 19 6 7 3 3 3 8 →= 19 26 31 6 7 3 3 8 , 19 6 7 3 3 3 9 →= 19 26 31 6 7 3 3 9 , 19 6 7 3 3 3 10 →= 19 26 31 6 7 3 3 10 , 19 6 7 3 3 3 11 →= 19 26 31 6 7 3 3 11 , 24 6 7 3 3 3 3 →= 24 26 31 6 7 3 3 3 , 24 6 7 3 3 3 8 →= 24 26 31 6 7 3 3 8 , 24 6 7 3 3 3 9 →= 24 26 31 6 7 3 3 9 , 24 6 7 3 3 3 10 →= 24 26 31 6 7 3 3 10 , 24 6 7 3 3 3 11 →= 24 26 31 6 7 3 3 11 , 31 6 7 3 3 3 3 →= 31 26 31 6 7 3 3 3 , 31 6 7 3 3 3 8 →= 31 26 31 6 7 3 3 8 , 31 6 7 3 3 3 9 →= 31 26 31 6 7 3 3 9 , 31 6 7 3 3 3 10 →= 31 26 31 6 7 3 3 10 , 31 6 7 3 3 3 11 →= 31 26 31 6 7 3 3 11 , 5 6 7 3 3 8 12 →= 5 26 31 6 7 3 8 12 , 5 6 7 3 3 8 13 →= 5 26 31 6 7 3 8 13 , 5 6 7 3 3 8 14 →= 5 26 31 6 7 3 8 14 , 48 6 7 3 3 8 12 →= 48 26 31 6 7 3 8 12 , 48 6 7 3 3 8 13 →= 48 26 31 6 7 3 8 13 , 48 6 7 3 3 8 14 →= 48 26 31 6 7 3 8 14 , 12 6 7 3 3 8 12 →= 12 26 31 6 7 3 8 12 , 12 6 7 3 3 8 13 →= 12 26 31 6 7 3 8 13 , 12 6 7 3 3 8 14 →= 12 26 31 6 7 3 8 14 , 49 6 7 3 3 8 12 →= 49 26 31 6 7 3 8 12 , 49 6 7 3 3 8 13 →= 49 26 31 6 7 3 8 13 , 49 6 7 3 3 8 14 →= 49 26 31 6 7 3 8 14 , 19 6 7 3 3 8 12 →= 19 26 31 6 7 3 8 12 , 19 6 7 3 3 8 13 →= 19 26 31 6 7 3 8 13 , 19 6 7 3 3 8 14 →= 19 26 31 6 7 3 8 14 , 24 6 7 3 3 8 12 →= 24 26 31 6 7 3 8 12 , 24 6 7 3 3 8 13 →= 24 26 31 6 7 3 8 13 , 24 6 7 3 3 8 14 →= 24 26 31 6 7 3 8 14 , 31 6 7 3 3 8 12 →= 31 26 31 6 7 3 8 12 , 31 6 7 3 3 8 13 →= 31 26 31 6 7 3 8 13 , 31 6 7 3 3 8 14 →= 31 26 31 6 7 3 8 14 , 5 6 7 3 3 10 36 →= 5 26 31 6 7 3 10 36 , 5 6 7 3 3 10 37 →= 5 26 31 6 7 3 10 37 , 48 6 7 3 3 10 36 →= 48 26 31 6 7 3 10 36 , 48 6 7 3 3 10 37 →= 48 26 31 6 7 3 10 37 , 12 6 7 3 3 10 36 →= 12 26 31 6 7 3 10 36 , 12 6 7 3 3 10 37 →= 12 26 31 6 7 3 10 37 , 49 6 7 3 3 10 36 →= 49 26 31 6 7 3 10 36 , 49 6 7 3 3 10 37 →= 49 26 31 6 7 3 10 37 , 19 6 7 3 3 10 36 →= 19 26 31 6 7 3 10 36 , 19 6 7 3 3 10 37 →= 19 26 31 6 7 3 10 37 , 24 6 7 3 3 10 36 →= 24 26 31 6 7 3 10 36 , 24 6 7 3 3 10 37 →= 24 26 31 6 7 3 10 37 , 31 6 7 3 3 10 36 →= 31 26 31 6 7 3 10 36 , 31 6 7 3 3 10 37 →= 31 26 31 6 7 3 10 37 , 5 6 7 3 3 11 38 →= 5 26 31 6 7 3 11 38 , 48 6 7 3 3 11 38 →= 48 26 31 6 7 3 11 38 , 12 6 7 3 3 11 38 →= 12 26 31 6 7 3 11 38 , 49 6 7 3 3 11 38 →= 49 26 31 6 7 3 11 38 , 19 6 7 3 3 11 38 →= 19 26 31 6 7 3 11 38 , 24 6 7 3 3 11 38 →= 24 26 31 6 7 3 11 38 , 31 6 7 3 3 11 38 →= 31 26 31 6 7 3 11 38 , 50 39 7 3 3 3 3 →= 50 51 31 6 7 3 3 3 , 50 39 7 3 3 3 8 →= 50 51 31 6 7 3 3 8 , 50 39 7 3 3 3 9 →= 50 51 31 6 7 3 3 9 , 50 39 7 3 3 3 10 →= 50 51 31 6 7 3 3 10 , 50 39 7 3 3 3 11 →= 50 51 31 6 7 3 3 11 , 50 39 7 3 3 8 12 →= 50 51 31 6 7 3 8 12 , 50 39 7 3 3 8 13 →= 50 51 31 6 7 3 8 13 , 50 39 7 3 3 8 14 →= 50 51 31 6 7 3 8 14 , 52 43 33 3 3 3 3 →= 52 53 48 6 7 3 3 3 , 52 43 33 3 3 3 8 →= 52 53 48 6 7 3 3 8 , 52 43 33 3 3 3 9 →= 52 53 48 6 7 3 3 9 , 52 43 33 3 3 3 10 →= 52 53 48 6 7 3 3 10 , 52 43 33 3 3 3 11 →= 52 53 48 6 7 3 3 11 , 52 43 33 3 3 8 12 →= 52 53 48 6 7 3 8 12 , 52 43 33 3 3 8 13 →= 52 53 48 6 7 3 8 13 , 52 43 33 3 3 8 14 →= 52 53 48 6 7 3 8 14 , 1 2 3 3 8 12 6 →= 1 28 12 6 7 8 12 6 , 1 2 3 3 8 12 25 →= 1 28 12 6 7 8 12 25 , 1 2 3 3 8 12 26 →= 1 28 12 6 7 8 12 26 , 32 2 3 3 8 12 6 →= 32 28 12 6 7 8 12 6 , 32 2 3 3 8 12 25 →= 32 28 12 6 7 8 12 25 , 32 2 3 3 8 12 26 →= 32 28 12 6 7 8 12 26 , 33 3 3 3 8 12 6 →= 33 8 12 6 7 8 12 6 , 33 3 3 3 8 12 25 →= 33 8 12 6 7 8 12 25 , 33 3 3 3 8 12 26 →= 33 8 12 6 7 8 12 26 , 2 3 3 3 8 12 6 →= 2 8 12 6 7 8 12 6 , 2 3 3 3 8 12 25 →= 2 8 12 6 7 8 12 25 , 2 3 3 3 8 12 26 →= 2 8 12 6 7 8 12 26 , 34 3 3 3 8 12 6 →= 34 8 12 6 7 8 12 6 , 34 3 3 3 8 12 25 →= 34 8 12 6 7 8 12 25 , 34 3 3 3 8 12 26 →= 34 8 12 6 7 8 12 26 , 3 3 3 3 8 12 6 →= 3 8 12 6 7 8 12 6 , 3 3 3 3 8 12 25 →= 3 8 12 6 7 8 12 25 , 3 3 3 3 8 12 26 →= 3 8 12 6 7 8 12 26 , 7 3 3 3 8 12 6 →= 7 8 12 6 7 8 12 6 , 7 3 3 3 8 12 25 →= 7 8 12 6 7 8 12 25 , 7 3 3 3 8 12 26 →= 7 8 12 6 7 8 12 26 , 35 3 3 3 8 12 6 →= 35 8 12 6 7 8 12 6 , 35 3 3 3 8 12 25 →= 35 8 12 6 7 8 12 25 , 35 3 3 3 8 12 26 →= 35 8 12 6 7 8 12 26 , 17 3 3 3 8 12 6 →= 17 8 12 6 7 8 12 6 , 17 3 3 3 8 12 25 →= 17 8 12 6 7 8 12 25 , 17 3 3 3 8 12 26 →= 17 8 12 6 7 8 12 26 , 22 3 3 3 8 12 6 →= 22 8 12 6 7 8 12 6 , 22 3 3 3 8 12 25 →= 22 8 12 6 7 8 12 25 , 22 3 3 3 8 12 26 →= 22 8 12 6 7 8 12 26 , 33 3 3 3 8 13 54 →= 33 8 12 6 7 8 13 54 , 2 3 3 3 8 13 54 →= 2 8 12 6 7 8 13 54 , 34 3 3 3 8 13 54 →= 34 8 12 6 7 8 13 54 , 3 3 3 3 8 13 54 →= 3 8 12 6 7 8 13 54 , 7 3 3 3 8 13 54 →= 7 8 12 6 7 8 13 54 , 35 3 3 3 8 13 54 →= 35 8 12 6 7 8 13 54 , 17 3 3 3 8 13 54 →= 17 8 12 6 7 8 13 54 , 22 3 3 3 8 13 54 →= 22 8 12 6 7 8 13 54 , 39 7 3 3 8 12 6 →= 39 27 12 6 7 8 12 6 , 39 7 3 3 8 12 25 →= 39 27 12 6 7 8 12 25 , 39 7 3 3 8 12 26 →= 39 27 12 6 7 8 12 26 , 6 7 3 3 8 12 6 →= 6 27 12 6 7 8 12 6 , 6 7 3 3 8 12 25 →= 6 27 12 6 7 8 12 25 , 6 7 3 3 8 12 26 →= 6 27 12 6 7 8 12 26 , 40 7 3 3 8 12 6 →= 40 27 12 6 7 8 12 6 , 40 7 3 3 8 12 25 →= 40 27 12 6 7 8 12 25 , 40 7 3 3 8 12 26 →= 40 27 12 6 7 8 12 26 , 39 7 3 3 8 13 54 →= 39 27 12 6 7 8 13 54 , 6 7 3 3 8 13 54 →= 6 27 12 6 7 8 13 54 , 40 7 3 3 8 13 54 →= 40 27 12 6 7 8 13 54 , 16 17 3 3 8 12 6 →= 16 29 12 6 7 8 12 6 , 16 17 3 3 8 12 25 →= 16 29 12 6 7 8 12 25 , 16 17 3 3 8 12 26 →= 16 29 12 6 7 8 12 26 , 41 17 3 3 8 12 6 →= 41 29 12 6 7 8 12 6 , 41 17 3 3 8 12 25 →= 41 29 12 6 7 8 12 25 , 41 17 3 3 8 12 26 →= 41 29 12 6 7 8 12 26 , 42 22 3 3 8 12 6 →= 42 30 12 6 7 8 12 6 , 42 22 3 3 8 12 25 →= 42 30 12 6 7 8 12 25 , 42 22 3 3 8 12 26 →= 42 30 12 6 7 8 12 26 , 21 22 3 3 8 12 6 →= 21 30 12 6 7 8 12 6 , 21 22 3 3 8 12 25 →= 21 30 12 6 7 8 12 25 , 21 22 3 3 8 12 26 →= 21 30 12 6 7 8 12 26 , 43 33 3 3 8 12 6 →= 43 44 12 6 7 8 12 6 , 43 33 3 3 8 12 25 →= 43 44 12 6 7 8 12 25 , 43 33 3 3 8 12 26 →= 43 44 12 6 7 8 12 26 , 45 33 3 3 8 12 6 →= 45 44 12 6 7 8 12 6 , 45 33 3 3 8 12 25 →= 45 44 12 6 7 8 12 25 , 45 33 3 3 8 12 26 →= 45 44 12 6 7 8 12 26 , 46 34 3 3 8 12 6 →= 46 47 12 6 7 8 12 6 , 46 34 3 3 8 12 25 →= 46 47 12 6 7 8 12 25 , 46 34 3 3 8 12 26 →= 46 47 12 6 7 8 12 26 , 5 6 7 3 8 12 6 →= 5 26 31 6 7 8 12 6 , 5 6 7 3 8 12 25 →= 5 26 31 6 7 8 12 25 , 5 6 7 3 8 12 26 →= 5 26 31 6 7 8 12 26 , 48 6 7 3 8 12 6 →= 48 26 31 6 7 8 12 6 , 48 6 7 3 8 12 25 →= 48 26 31 6 7 8 12 25 , 48 6 7 3 8 12 26 →= 48 26 31 6 7 8 12 26 , 12 6 7 3 8 12 6 →= 12 26 31 6 7 8 12 6 , 12 6 7 3 8 12 25 →= 12 26 31 6 7 8 12 25 , 12 6 7 3 8 12 26 →= 12 26 31 6 7 8 12 26 , 49 6 7 3 8 12 6 →= 49 26 31 6 7 8 12 6 , 49 6 7 3 8 12 25 →= 49 26 31 6 7 8 12 25 , 49 6 7 3 8 12 26 →= 49 26 31 6 7 8 12 26 , 19 6 7 3 8 12 6 →= 19 26 31 6 7 8 12 6 , 19 6 7 3 8 12 25 →= 19 26 31 6 7 8 12 25 , 19 6 7 3 8 12 26 →= 19 26 31 6 7 8 12 26 , 24 6 7 3 8 12 6 →= 24 26 31 6 7 8 12 6 , 24 6 7 3 8 12 25 →= 24 26 31 6 7 8 12 25 , 24 6 7 3 8 12 26 →= 24 26 31 6 7 8 12 26 , 31 6 7 3 8 12 6 →= 31 26 31 6 7 8 12 6 , 31 6 7 3 8 12 25 →= 31 26 31 6 7 8 12 25 , 31 6 7 3 8 12 26 →= 31 26 31 6 7 8 12 26 , 5 6 7 3 8 13 54 →= 5 26 31 6 7 8 13 54 , 48 6 7 3 8 13 54 →= 48 26 31 6 7 8 13 54 , 12 6 7 3 8 13 54 →= 12 26 31 6 7 8 13 54 , 49 6 7 3 8 13 54 →= 49 26 31 6 7 8 13 54 , 19 6 7 3 8 13 54 →= 19 26 31 6 7 8 13 54 , 24 6 7 3 8 13 54 →= 24 26 31 6 7 8 13 54 , 31 6 7 3 8 13 54 →= 31 26 31 6 7 8 13 54 , 50 39 7 3 8 12 6 →= 50 51 31 6 7 8 12 6 , 50 39 7 3 8 12 25 →= 50 51 31 6 7 8 12 25 , 50 39 7 3 8 12 26 →= 50 51 31 6 7 8 12 26 , 52 43 33 3 8 12 6 →= 52 53 48 6 7 8 12 6 , 52 43 33 3 8 12 25 →= 52 53 48 6 7 8 12 25 , 52 43 33 3 8 12 26 →= 52 53 48 6 7 8 12 26 , 33 3 3 3 10 36 55 →= 33 8 12 6 7 10 36 55 , 2 3 3 3 10 36 55 →= 2 8 12 6 7 10 36 55 , 34 3 3 3 10 36 55 →= 34 8 12 6 7 10 36 55 , 3 3 3 3 10 36 55 →= 3 8 12 6 7 10 36 55 , 7 3 3 3 10 36 55 →= 7 8 12 6 7 10 36 55 , 35 3 3 3 10 36 55 →= 35 8 12 6 7 10 36 55 , 17 3 3 3 10 36 55 →= 17 8 12 6 7 10 36 55 , 22 3 3 3 10 36 55 →= 22 8 12 6 7 10 36 55 , 39 7 3 3 10 36 55 →= 39 27 12 6 7 10 36 55 , 6 7 3 3 10 36 55 →= 6 27 12 6 7 10 36 55 , 40 7 3 3 10 36 55 →= 40 27 12 6 7 10 36 55 , 5 6 7 3 10 36 55 →= 5 26 31 6 7 10 36 55 , 48 6 7 3 10 36 55 →= 48 26 31 6 7 10 36 55 , 12 6 7 3 10 36 55 →= 12 26 31 6 7 10 36 55 , 49 6 7 3 10 36 55 →= 49 26 31 6 7 10 36 55 , 19 6 7 3 10 36 55 →= 19 26 31 6 7 10 36 55 , 24 6 7 3 10 36 55 →= 24 26 31 6 7 10 36 55 , 31 6 7 3 10 36 55 →= 31 26 31 6 7 10 36 55 , 1 28 12 6 27 12 6 7 3 →= 1 2 8 12 6 7 3 3 , 1 28 12 6 27 12 6 7 8 →= 1 2 8 12 6 7 3 8 , 1 28 12 6 27 12 6 7 9 →= 1 2 8 12 6 7 3 9 , 1 28 12 6 27 12 6 7 10 →= 1 2 8 12 6 7 3 10 , 1 28 12 6 27 12 6 7 11 →= 1 2 8 12 6 7 3 11 , 32 28 12 6 27 12 6 7 3 →= 32 2 8 12 6 7 3 3 , 32 28 12 6 27 12 6 7 8 →= 32 2 8 12 6 7 3 8 , 32 28 12 6 27 12 6 7 9 →= 32 2 8 12 6 7 3 9 , 32 28 12 6 27 12 6 7 10 →= 32 2 8 12 6 7 3 10 , 32 28 12 6 27 12 6 7 11 →= 32 2 8 12 6 7 3 11 , 1 28 12 6 27 12 6 27 12 →= 1 2 8 12 6 7 8 12 , 1 28 12 6 27 12 6 27 13 →= 1 2 8 12 6 7 8 13 , 1 28 12 6 27 12 6 27 14 →= 1 2 8 12 6 7 8 14 , 32 28 12 6 27 12 6 27 12 →= 32 2 8 12 6 7 8 12 , 32 28 12 6 27 12 6 27 13 →= 32 2 8 12 6 7 8 13 , 32 28 12 6 27 12 6 27 14 →= 32 2 8 12 6 7 8 14 , 33 8 12 6 27 12 6 7 3 →= 33 3 8 12 6 7 3 3 , 33 8 12 6 27 12 6 7 8 →= 33 3 8 12 6 7 3 8 , 33 8 12 6 27 12 6 7 9 →= 33 3 8 12 6 7 3 9 , 33 8 12 6 27 12 6 7 10 →= 33 3 8 12 6 7 3 10 , 33 8 12 6 27 12 6 7 11 →= 33 3 8 12 6 7 3 11 , 2 8 12 6 27 12 6 7 3 →= 2 3 8 12 6 7 3 3 , 2 8 12 6 27 12 6 7 8 →= 2 3 8 12 6 7 3 8 , 2 8 12 6 27 12 6 7 9 →= 2 3 8 12 6 7 3 9 , 2 8 12 6 27 12 6 7 10 →= 2 3 8 12 6 7 3 10 , 2 8 12 6 27 12 6 7 11 →= 2 3 8 12 6 7 3 11 , 34 8 12 6 27 12 6 7 3 →= 34 3 8 12 6 7 3 3 , 34 8 12 6 27 12 6 7 8 →= 34 3 8 12 6 7 3 8 , 34 8 12 6 27 12 6 7 9 →= 34 3 8 12 6 7 3 9 , 34 8 12 6 27 12 6 7 10 →= 34 3 8 12 6 7 3 10 , 34 8 12 6 27 12 6 7 11 →= 34 3 8 12 6 7 3 11 , 3 8 12 6 27 12 6 7 3 →= 3 3 8 12 6 7 3 3 , 3 8 12 6 27 12 6 7 8 →= 3 3 8 12 6 7 3 8 , 3 8 12 6 27 12 6 7 9 →= 3 3 8 12 6 7 3 9 , 3 8 12 6 27 12 6 7 10 →= 3 3 8 12 6 7 3 10 , 3 8 12 6 27 12 6 7 11 →= 3 3 8 12 6 7 3 11 , 7 8 12 6 27 12 6 7 3 →= 7 3 8 12 6 7 3 3 , 7 8 12 6 27 12 6 7 8 →= 7 3 8 12 6 7 3 8 , 7 8 12 6 27 12 6 7 9 →= 7 3 8 12 6 7 3 9 , 7 8 12 6 27 12 6 7 10 →= 7 3 8 12 6 7 3 10 , 7 8 12 6 27 12 6 7 11 →= 7 3 8 12 6 7 3 11 , 35 8 12 6 27 12 6 7 3 →= 35 3 8 12 6 7 3 3 , 35 8 12 6 27 12 6 7 8 →= 35 3 8 12 6 7 3 8 , 35 8 12 6 27 12 6 7 9 →= 35 3 8 12 6 7 3 9 , 35 8 12 6 27 12 6 7 10 →= 35 3 8 12 6 7 3 10 , 35 8 12 6 27 12 6 7 11 →= 35 3 8 12 6 7 3 11 , 17 8 12 6 27 12 6 7 3 →= 17 3 8 12 6 7 3 3 , 17 8 12 6 27 12 6 7 8 →= 17 3 8 12 6 7 3 8 , 17 8 12 6 27 12 6 7 9 →= 17 3 8 12 6 7 3 9 , 17 8 12 6 27 12 6 7 10 →= 17 3 8 12 6 7 3 10 , 17 8 12 6 27 12 6 7 11 →= 17 3 8 12 6 7 3 11 , 22 8 12 6 27 12 6 7 3 →= 22 3 8 12 6 7 3 3 , 22 8 12 6 27 12 6 7 8 →= 22 3 8 12 6 7 3 8 , 22 8 12 6 27 12 6 7 9 →= 22 3 8 12 6 7 3 9 , 22 8 12 6 27 12 6 7 10 →= 22 3 8 12 6 7 3 10 , 22 8 12 6 27 12 6 7 11 →= 22 3 8 12 6 7 3 11 , 33 8 12 6 27 12 6 27 12 →= 33 3 8 12 6 7 8 12 , 33 8 12 6 27 12 6 27 13 →= 33 3 8 12 6 7 8 13 , 33 8 12 6 27 12 6 27 14 →= 33 3 8 12 6 7 8 14 , 2 8 12 6 27 12 6 27 12 →= 2 3 8 12 6 7 8 12 , 2 8 12 6 27 12 6 27 13 →= 2 3 8 12 6 7 8 13 , 2 8 12 6 27 12 6 27 14 →= 2 3 8 12 6 7 8 14 , 34 8 12 6 27 12 6 27 12 →= 34 3 8 12 6 7 8 12 , 34 8 12 6 27 12 6 27 13 →= 34 3 8 12 6 7 8 13 , 34 8 12 6 27 12 6 27 14 →= 34 3 8 12 6 7 8 14 , 3 8 12 6 27 12 6 27 12 →= 3 3 8 12 6 7 8 12 , 3 8 12 6 27 12 6 27 13 →= 3 3 8 12 6 7 8 13 , 3 8 12 6 27 12 6 27 14 →= 3 3 8 12 6 7 8 14 , 7 8 12 6 27 12 6 27 12 →= 7 3 8 12 6 7 8 12 , 7 8 12 6 27 12 6 27 13 →= 7 3 8 12 6 7 8 13 , 7 8 12 6 27 12 6 27 14 →= 7 3 8 12 6 7 8 14 , 35 8 12 6 27 12 6 27 12 →= 35 3 8 12 6 7 8 12 , 35 8 12 6 27 12 6 27 13 →= 35 3 8 12 6 7 8 13 , 35 8 12 6 27 12 6 27 14 →= 35 3 8 12 6 7 8 14 , 17 8 12 6 27 12 6 27 12 →= 17 3 8 12 6 7 8 12 , 17 8 12 6 27 12 6 27 13 →= 17 3 8 12 6 7 8 13 , 17 8 12 6 27 12 6 27 14 →= 17 3 8 12 6 7 8 14 , 22 8 12 6 27 12 6 27 12 →= 22 3 8 12 6 7 8 12 , 22 8 12 6 27 12 6 27 13 →= 22 3 8 12 6 7 8 13 , 22 8 12 6 27 12 6 27 14 →= 22 3 8 12 6 7 8 14 , 39 27 12 6 27 12 6 7 3 →= 39 7 8 12 6 7 3 3 , 39 27 12 6 27 12 6 7 8 →= 39 7 8 12 6 7 3 8 , 39 27 12 6 27 12 6 7 9 →= 39 7 8 12 6 7 3 9 , 39 27 12 6 27 12 6 7 10 →= 39 7 8 12 6 7 3 10 , 39 27 12 6 27 12 6 7 11 →= 39 7 8 12 6 7 3 11 , 6 27 12 6 27 12 6 7 3 →= 6 7 8 12 6 7 3 3 , 6 27 12 6 27 12 6 7 8 →= 6 7 8 12 6 7 3 8 , 6 27 12 6 27 12 6 7 9 →= 6 7 8 12 6 7 3 9 , 6 27 12 6 27 12 6 7 10 →= 6 7 8 12 6 7 3 10 , 6 27 12 6 27 12 6 7 11 →= 6 7 8 12 6 7 3 11 , 40 27 12 6 27 12 6 7 3 →= 40 7 8 12 6 7 3 3 , 40 27 12 6 27 12 6 7 8 →= 40 7 8 12 6 7 3 8 , 40 27 12 6 27 12 6 7 9 →= 40 7 8 12 6 7 3 9 , 40 27 12 6 27 12 6 7 10 →= 40 7 8 12 6 7 3 10 , 40 27 12 6 27 12 6 7 11 →= 40 7 8 12 6 7 3 11 , 39 27 12 6 27 12 6 27 12 →= 39 7 8 12 6 7 8 12 , 39 27 12 6 27 12 6 27 13 →= 39 7 8 12 6 7 8 13 , 39 27 12 6 27 12 6 27 14 →= 39 7 8 12 6 7 8 14 , 6 27 12 6 27 12 6 27 12 →= 6 7 8 12 6 7 8 12 , 6 27 12 6 27 12 6 27 13 →= 6 7 8 12 6 7 8 13 , 6 27 12 6 27 12 6 27 14 →= 6 7 8 12 6 7 8 14 , 40 27 12 6 27 12 6 27 12 →= 40 7 8 12 6 7 8 12 , 40 27 12 6 27 12 6 27 13 →= 40 7 8 12 6 7 8 13 , 40 27 12 6 27 12 6 27 14 →= 40 7 8 12 6 7 8 14 , 16 29 12 6 27 12 6 7 3 →= 16 17 8 12 6 7 3 3 , 16 29 12 6 27 12 6 7 8 →= 16 17 8 12 6 7 3 8 , 16 29 12 6 27 12 6 7 9 →= 16 17 8 12 6 7 3 9 , 16 29 12 6 27 12 6 7 10 →= 16 17 8 12 6 7 3 10 , 16 29 12 6 27 12 6 7 11 →= 16 17 8 12 6 7 3 11 , 41 29 12 6 27 12 6 7 3 →= 41 17 8 12 6 7 3 3 , 41 29 12 6 27 12 6 7 8 →= 41 17 8 12 6 7 3 8 , 41 29 12 6 27 12 6 7 9 →= 41 17 8 12 6 7 3 9 , 41 29 12 6 27 12 6 7 10 →= 41 17 8 12 6 7 3 10 , 41 29 12 6 27 12 6 7 11 →= 41 17 8 12 6 7 3 11 , 16 29 12 6 27 12 6 27 12 →= 16 17 8 12 6 7 8 12 , 16 29 12 6 27 12 6 27 13 →= 16 17 8 12 6 7 8 13 , 16 29 12 6 27 12 6 27 14 →= 16 17 8 12 6 7 8 14 , 41 29 12 6 27 12 6 27 12 →= 41 17 8 12 6 7 8 12 , 41 29 12 6 27 12 6 27 13 →= 41 17 8 12 6 7 8 13 , 41 29 12 6 27 12 6 27 14 →= 41 17 8 12 6 7 8 14 , 42 30 12 6 27 12 6 7 3 →= 42 22 8 12 6 7 3 3 , 42 30 12 6 27 12 6 7 8 →= 42 22 8 12 6 7 3 8 , 42 30 12 6 27 12 6 7 9 →= 42 22 8 12 6 7 3 9 , 42 30 12 6 27 12 6 7 10 →= 42 22 8 12 6 7 3 10 , 42 30 12 6 27 12 6 7 11 →= 42 22 8 12 6 7 3 11 , 21 30 12 6 27 12 6 7 3 →= 21 22 8 12 6 7 3 3 , 21 30 12 6 27 12 6 7 8 →= 21 22 8 12 6 7 3 8 , 21 30 12 6 27 12 6 7 9 →= 21 22 8 12 6 7 3 9 , 21 30 12 6 27 12 6 7 10 →= 21 22 8 12 6 7 3 10 , 21 30 12 6 27 12 6 7 11 →= 21 22 8 12 6 7 3 11 , 42 30 12 6 27 12 6 27 12 →= 42 22 8 12 6 7 8 12 , 42 30 12 6 27 12 6 27 13 →= 42 22 8 12 6 7 8 13 , 42 30 12 6 27 12 6 27 14 →= 42 22 8 12 6 7 8 14 , 21 30 12 6 27 12 6 27 12 →= 21 22 8 12 6 7 8 12 , 21 30 12 6 27 12 6 27 13 →= 21 22 8 12 6 7 8 13 , 21 30 12 6 27 12 6 27 14 →= 21 22 8 12 6 7 8 14 , 43 44 12 6 27 12 6 7 3 →= 43 33 8 12 6 7 3 3 , 43 44 12 6 27 12 6 7 8 →= 43 33 8 12 6 7 3 8 , 43 44 12 6 27 12 6 7 9 →= 43 33 8 12 6 7 3 9 , 43 44 12 6 27 12 6 7 10 →= 43 33 8 12 6 7 3 10 , 43 44 12 6 27 12 6 7 11 →= 43 33 8 12 6 7 3 11 , 45 44 12 6 27 12 6 7 3 →= 45 33 8 12 6 7 3 3 , 45 44 12 6 27 12 6 7 8 →= 45 33 8 12 6 7 3 8 , 45 44 12 6 27 12 6 7 9 →= 45 33 8 12 6 7 3 9 , 45 44 12 6 27 12 6 7 10 →= 45 33 8 12 6 7 3 10 , 45 44 12 6 27 12 6 7 11 →= 45 33 8 12 6 7 3 11 , 43 44 12 6 27 12 6 27 12 →= 43 33 8 12 6 7 8 12 , 43 44 12 6 27 12 6 27 13 →= 43 33 8 12 6 7 8 13 , 43 44 12 6 27 12 6 27 14 →= 43 33 8 12 6 7 8 14 , 45 44 12 6 27 12 6 27 12 →= 45 33 8 12 6 7 8 12 , 45 44 12 6 27 12 6 27 13 →= 45 33 8 12 6 7 8 13 , 45 44 12 6 27 12 6 27 14 →= 45 33 8 12 6 7 8 14 , 46 47 12 6 27 12 6 7 3 →= 46 34 8 12 6 7 3 3 , 46 47 12 6 27 12 6 7 8 →= 46 34 8 12 6 7 3 8 , 46 47 12 6 27 12 6 7 9 →= 46 34 8 12 6 7 3 9 , 46 47 12 6 27 12 6 7 10 →= 46 34 8 12 6 7 3 10 , 46 47 12 6 27 12 6 7 11 →= 46 34 8 12 6 7 3 11 , 46 47 12 6 27 12 6 27 12 →= 46 34 8 12 6 7 8 12 , 46 47 12 6 27 12 6 27 13 →= 46 34 8 12 6 7 8 13 , 46 47 12 6 27 12 6 27 14 →= 46 34 8 12 6 7 8 14 , 5 26 31 6 27 12 6 7 3 →= 5 6 27 12 6 7 3 3 , 5 26 31 6 27 12 6 7 8 →= 5 6 27 12 6 7 3 8 , 5 26 31 6 27 12 6 7 9 →= 5 6 27 12 6 7 3 9 , 5 26 31 6 27 12 6 7 10 →= 5 6 27 12 6 7 3 10 , 5 26 31 6 27 12 6 7 11 →= 5 6 27 12 6 7 3 11 , 48 26 31 6 27 12 6 7 3 →= 48 6 27 12 6 7 3 3 , 48 26 31 6 27 12 6 7 8 →= 48 6 27 12 6 7 3 8 , 48 26 31 6 27 12 6 7 9 →= 48 6 27 12 6 7 3 9 , 48 26 31 6 27 12 6 7 10 →= 48 6 27 12 6 7 3 10 , 48 26 31 6 27 12 6 7 11 →= 48 6 27 12 6 7 3 11 , 12 26 31 6 27 12 6 7 3 →= 12 6 27 12 6 7 3 3 , 12 26 31 6 27 12 6 7 8 →= 12 6 27 12 6 7 3 8 , 12 26 31 6 27 12 6 7 9 →= 12 6 27 12 6 7 3 9 , 12 26 31 6 27 12 6 7 10 →= 12 6 27 12 6 7 3 10 , 12 26 31 6 27 12 6 7 11 →= 12 6 27 12 6 7 3 11 , 49 26 31 6 27 12 6 7 3 →= 49 6 27 12 6 7 3 3 , 49 26 31 6 27 12 6 7 8 →= 49 6 27 12 6 7 3 8 , 49 26 31 6 27 12 6 7 9 →= 49 6 27 12 6 7 3 9 , 49 26 31 6 27 12 6 7 10 →= 49 6 27 12 6 7 3 10 , 49 26 31 6 27 12 6 7 11 →= 49 6 27 12 6 7 3 11 , 19 26 31 6 27 12 6 7 3 →= 19 6 27 12 6 7 3 3 , 19 26 31 6 27 12 6 7 8 →= 19 6 27 12 6 7 3 8 , 19 26 31 6 27 12 6 7 9 →= 19 6 27 12 6 7 3 9 , 19 26 31 6 27 12 6 7 10 →= 19 6 27 12 6 7 3 10 , 19 26 31 6 27 12 6 7 11 →= 19 6 27 12 6 7 3 11 , 24 26 31 6 27 12 6 7 3 →= 24 6 27 12 6 7 3 3 , 24 26 31 6 27 12 6 7 8 →= 24 6 27 12 6 7 3 8 , 24 26 31 6 27 12 6 7 9 →= 24 6 27 12 6 7 3 9 , 24 26 31 6 27 12 6 7 10 →= 24 6 27 12 6 7 3 10 , 24 26 31 6 27 12 6 7 11 →= 24 6 27 12 6 7 3 11 , 31 26 31 6 27 12 6 7 3 →= 31 6 27 12 6 7 3 3 , 31 26 31 6 27 12 6 7 8 →= 31 6 27 12 6 7 3 8 , 31 26 31 6 27 12 6 7 9 →= 31 6 27 12 6 7 3 9 , 31 26 31 6 27 12 6 7 10 →= 31 6 27 12 6 7 3 10 , 31 26 31 6 27 12 6 7 11 →= 31 6 27 12 6 7 3 11 , 5 26 31 6 27 12 6 27 12 →= 5 6 27 12 6 7 8 12 , 5 26 31 6 27 12 6 27 13 →= 5 6 27 12 6 7 8 13 , 5 26 31 6 27 12 6 27 14 →= 5 6 27 12 6 7 8 14 , 48 26 31 6 27 12 6 27 12 →= 48 6 27 12 6 7 8 12 , 48 26 31 6 27 12 6 27 13 →= 48 6 27 12 6 7 8 13 , 48 26 31 6 27 12 6 27 14 →= 48 6 27 12 6 7 8 14 , 12 26 31 6 27 12 6 27 12 →= 12 6 27 12 6 7 8 12 , 12 26 31 6 27 12 6 27 13 →= 12 6 27 12 6 7 8 13 , 12 26 31 6 27 12 6 27 14 →= 12 6 27 12 6 7 8 14 , 49 26 31 6 27 12 6 27 12 →= 49 6 27 12 6 7 8 12 , 49 26 31 6 27 12 6 27 13 →= 49 6 27 12 6 7 8 13 , 49 26 31 6 27 12 6 27 14 →= 49 6 27 12 6 7 8 14 , 19 26 31 6 27 12 6 27 12 →= 19 6 27 12 6 7 8 12 , 19 26 31 6 27 12 6 27 13 →= 19 6 27 12 6 7 8 13 , 19 26 31 6 27 12 6 27 14 →= 19 6 27 12 6 7 8 14 , 24 26 31 6 27 12 6 27 12 →= 24 6 27 12 6 7 8 12 , 24 26 31 6 27 12 6 27 13 →= 24 6 27 12 6 7 8 13 , 24 26 31 6 27 12 6 27 14 →= 24 6 27 12 6 7 8 14 , 31 26 31 6 27 12 6 27 12 →= 31 6 27 12 6 7 8 12 , 31 26 31 6 27 12 6 27 13 →= 31 6 27 12 6 7 8 13 , 31 26 31 6 27 12 6 27 14 →= 31 6 27 12 6 7 8 14 , 50 51 31 6 27 12 6 7 3 →= 50 39 27 12 6 7 3 3 , 50 51 31 6 27 12 6 7 8 →= 50 39 27 12 6 7 3 8 , 50 51 31 6 27 12 6 7 9 →= 50 39 27 12 6 7 3 9 , 50 51 31 6 27 12 6 7 10 →= 50 39 27 12 6 7 3 10 , 50 51 31 6 27 12 6 7 11 →= 50 39 27 12 6 7 3 11 , 50 51 31 6 27 12 6 27 12 →= 50 39 27 12 6 7 8 12 , 50 51 31 6 27 12 6 27 13 →= 50 39 27 12 6 7 8 13 , 50 51 31 6 27 12 6 27 14 →= 50 39 27 12 6 7 8 14 , 52 53 48 6 27 12 6 7 3 →= 52 43 44 12 6 7 3 3 , 52 53 48 6 27 12 6 7 8 →= 52 43 44 12 6 7 3 8 , 52 53 48 6 27 12 6 7 9 →= 52 43 44 12 6 7 3 9 , 52 53 48 6 27 12 6 7 10 →= 52 43 44 12 6 7 3 10 , 52 53 48 6 27 12 6 7 11 →= 52 43 44 12 6 7 3 11 , 52 53 48 6 27 12 6 27 12 →= 52 43 44 12 6 7 8 12 , 52 53 48 6 27 12 6 27 13 →= 52 43 44 12 6 7 8 13 , 52 53 48 6 27 12 6 27 14 →= 52 43 44 12 6 7 8 14 , 1 28 12 6 27 12 26 31 6 →= 1 2 8 12 6 27 12 6 , 1 28 12 6 27 12 26 31 25 →= 1 2 8 12 6 27 12 25 , 1 28 12 6 27 12 26 31 26 →= 1 2 8 12 6 27 12 26 , 32 28 12 6 27 12 26 31 6 →= 32 2 8 12 6 27 12 6 , 32 28 12 6 27 12 26 31 25 →= 32 2 8 12 6 27 12 25 , 32 28 12 6 27 12 26 31 26 →= 32 2 8 12 6 27 12 26 , 33 8 12 6 27 12 26 31 6 →= 33 3 8 12 6 27 12 6 , 33 8 12 6 27 12 26 31 25 →= 33 3 8 12 6 27 12 25 , 33 8 12 6 27 12 26 31 26 →= 33 3 8 12 6 27 12 26 , 2 8 12 6 27 12 26 31 6 →= 2 3 8 12 6 27 12 6 , 2 8 12 6 27 12 26 31 25 →= 2 3 8 12 6 27 12 25 , 2 8 12 6 27 12 26 31 26 →= 2 3 8 12 6 27 12 26 , 34 8 12 6 27 12 26 31 6 →= 34 3 8 12 6 27 12 6 , 34 8 12 6 27 12 26 31 25 →= 34 3 8 12 6 27 12 25 , 34 8 12 6 27 12 26 31 26 →= 34 3 8 12 6 27 12 26 , 3 8 12 6 27 12 26 31 6 →= 3 3 8 12 6 27 12 6 , 3 8 12 6 27 12 26 31 25 →= 3 3 8 12 6 27 12 25 , 3 8 12 6 27 12 26 31 26 →= 3 3 8 12 6 27 12 26 , 7 8 12 6 27 12 26 31 6 →= 7 3 8 12 6 27 12 6 , 7 8 12 6 27 12 26 31 25 →= 7 3 8 12 6 27 12 25 , 7 8 12 6 27 12 26 31 26 →= 7 3 8 12 6 27 12 26 , 35 8 12 6 27 12 26 31 6 →= 35 3 8 12 6 27 12 6 , 35 8 12 6 27 12 26 31 25 →= 35 3 8 12 6 27 12 25 , 35 8 12 6 27 12 26 31 26 →= 35 3 8 12 6 27 12 26 , 17 8 12 6 27 12 26 31 6 →= 17 3 8 12 6 27 12 6 , 17 8 12 6 27 12 26 31 25 →= 17 3 8 12 6 27 12 25 , 17 8 12 6 27 12 26 31 26 →= 17 3 8 12 6 27 12 26 , 22 8 12 6 27 12 26 31 6 →= 22 3 8 12 6 27 12 6 , 22 8 12 6 27 12 26 31 25 →= 22 3 8 12 6 27 12 25 , 22 8 12 6 27 12 26 31 26 →= 22 3 8 12 6 27 12 26 , 39 27 12 6 27 12 26 31 6 →= 39 7 8 12 6 27 12 6 , 39 27 12 6 27 12 26 31 25 →= 39 7 8 12 6 27 12 25 , 39 27 12 6 27 12 26 31 26 →= 39 7 8 12 6 27 12 26 , 6 27 12 6 27 12 26 31 6 →= 6 7 8 12 6 27 12 6 , 6 27 12 6 27 12 26 31 25 →= 6 7 8 12 6 27 12 25 , 6 27 12 6 27 12 26 31 26 →= 6 7 8 12 6 27 12 26 , 40 27 12 6 27 12 26 31 6 →= 40 7 8 12 6 27 12 6 , 40 27 12 6 27 12 26 31 25 →= 40 7 8 12 6 27 12 25 , 40 27 12 6 27 12 26 31 26 →= 40 7 8 12 6 27 12 26 , 16 29 12 6 27 12 26 31 6 →= 16 17 8 12 6 27 12 6 , 16 29 12 6 27 12 26 31 25 →= 16 17 8 12 6 27 12 25 , 16 29 12 6 27 12 26 31 26 →= 16 17 8 12 6 27 12 26 , 41 29 12 6 27 12 26 31 6 →= 41 17 8 12 6 27 12 6 , 41 29 12 6 27 12 26 31 25 →= 41 17 8 12 6 27 12 25 , 41 29 12 6 27 12 26 31 26 →= 41 17 8 12 6 27 12 26 , 42 30 12 6 27 12 26 31 6 →= 42 22 8 12 6 27 12 6 , 42 30 12 6 27 12 26 31 25 →= 42 22 8 12 6 27 12 25 , 42 30 12 6 27 12 26 31 26 →= 42 22 8 12 6 27 12 26 , 21 30 12 6 27 12 26 31 6 →= 21 22 8 12 6 27 12 6 , 21 30 12 6 27 12 26 31 25 →= 21 22 8 12 6 27 12 25 , 21 30 12 6 27 12 26 31 26 →= 21 22 8 12 6 27 12 26 , 43 44 12 6 27 12 26 31 6 →= 43 33 8 12 6 27 12 6 , 43 44 12 6 27 12 26 31 25 →= 43 33 8 12 6 27 12 25 , 43 44 12 6 27 12 26 31 26 →= 43 33 8 12 6 27 12 26 , 45 44 12 6 27 12 26 31 6 →= 45 33 8 12 6 27 12 6 , 45 44 12 6 27 12 26 31 25 →= 45 33 8 12 6 27 12 25 , 45 44 12 6 27 12 26 31 26 →= 45 33 8 12 6 27 12 26 , 46 47 12 6 27 12 26 31 6 →= 46 34 8 12 6 27 12 6 , 46 47 12 6 27 12 26 31 25 →= 46 34 8 12 6 27 12 25 , 46 47 12 6 27 12 26 31 26 →= 46 34 8 12 6 27 12 26 , 5 26 31 6 27 12 26 31 6 →= 5 6 27 12 6 27 12 6 , 5 26 31 6 27 12 26 31 25 →= 5 6 27 12 6 27 12 25 , 5 26 31 6 27 12 26 31 26 →= 5 6 27 12 6 27 12 26 , 48 26 31 6 27 12 26 31 6 →= 48 6 27 12 6 27 12 6 , 48 26 31 6 27 12 26 31 25 →= 48 6 27 12 6 27 12 25 , 48 26 31 6 27 12 26 31 26 →= 48 6 27 12 6 27 12 26 , 12 26 31 6 27 12 26 31 6 →= 12 6 27 12 6 27 12 6 , 12 26 31 6 27 12 26 31 25 →= 12 6 27 12 6 27 12 25 , 12 26 31 6 27 12 26 31 26 →= 12 6 27 12 6 27 12 26 , 49 26 31 6 27 12 26 31 6 →= 49 6 27 12 6 27 12 6 , 49 26 31 6 27 12 26 31 25 →= 49 6 27 12 6 27 12 25 , 49 26 31 6 27 12 26 31 26 →= 49 6 27 12 6 27 12 26 , 19 26 31 6 27 12 26 31 6 →= 19 6 27 12 6 27 12 6 , 19 26 31 6 27 12 26 31 25 →= 19 6 27 12 6 27 12 25 , 19 26 31 6 27 12 26 31 26 →= 19 6 27 12 6 27 12 26 , 24 26 31 6 27 12 26 31 6 →= 24 6 27 12 6 27 12 6 , 24 26 31 6 27 12 26 31 25 →= 24 6 27 12 6 27 12 25 , 24 26 31 6 27 12 26 31 26 →= 24 6 27 12 6 27 12 26 , 31 26 31 6 27 12 26 31 6 →= 31 6 27 12 6 27 12 6 , 31 26 31 6 27 12 26 31 25 →= 31 6 27 12 6 27 12 25 , 31 26 31 6 27 12 26 31 26 →= 31 6 27 12 6 27 12 26 , 50 51 31 6 27 12 26 31 6 →= 50 39 27 12 6 27 12 6 , 50 51 31 6 27 12 26 31 25 →= 50 39 27 12 6 27 12 25 , 50 51 31 6 27 12 26 31 26 →= 50 39 27 12 6 27 12 26 , 52 53 48 6 27 12 26 31 6 →= 52 43 44 12 6 27 12 6 , 52 53 48 6 27 12 26 31 25 →= 52 43 44 12 6 27 12 25 , 52 53 48 6 27 12 26 31 26 →= 52 43 44 12 6 27 12 26 } Applying sparse untiling TROCU(2) after reversal [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { 3 ↦ 0, 2 ↦ 1, 1 ↦ 2, 0 ↦ 3, 7 ↦ 4, 6 ↦ 5, 5 ↦ 6, 4 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13, 14 ↦ 14, 17 ↦ 15, 16 ↦ 16, 15 ↦ 17, 19 ↦ 18, 18 ↦ 19, 22 ↦ 20, 21 ↦ 21, 20 ↦ 22, 24 ↦ 23, 23 ↦ 24, 25 ↦ 25, 26 ↦ 26, 27 ↦ 27, 28 ↦ 28, 29 ↦ 29, 30 ↦ 30, 31 ↦ 31, 32 ↦ 32, 33 ↦ 33, 34 ↦ 34, 35 ↦ 35, 36 ↦ 36, 37 ↦ 37, 38 ↦ 38, 39 ↦ 39, 40 ↦ 40, 41 ↦ 41, 42 ↦ 42, 43 ↦ 43, 44 ↦ 44, 45 ↦ 45, 46 ↦ 46, 47 ↦ 47, 48 ↦ 48, 49 ↦ 49, 50 ↦ 50, 51 ↦ 51, 52 ↦ 52, 53 ↦ 53, 54 ↦ 54, 55 ↦ 55 }, it remains to prove termination of the 634-rule system { 0 1 2 3 3 3 3 ⟶ 0 4 5 6 7 3 3 3 , 0 1 2 3 3 3 8 ⟶ 0 4 5 6 7 3 3 8 , 0 1 2 3 3 3 9 ⟶ 0 4 5 6 7 3 3 9 , 0 1 2 3 3 3 10 ⟶ 0 4 5 6 7 3 3 10 , 0 1 2 3 3 3 11 ⟶ 0 4 5 6 7 3 3 11 , 0 1 2 3 3 8 12 ⟶ 0 4 5 6 7 3 8 12 , 0 1 2 3 3 8 13 ⟶ 0 4 5 6 7 3 8 13 , 0 1 2 3 3 8 14 ⟶ 0 4 5 6 7 3 8 14 , 15 16 17 3 3 3 3 ⟶ 15 18 19 6 7 3 3 3 , 15 16 17 3 3 3 8 ⟶ 15 18 19 6 7 3 3 8 , 15 16 17 3 3 3 9 ⟶ 15 18 19 6 7 3 3 9 , 15 16 17 3 3 3 10 ⟶ 15 18 19 6 7 3 3 10 , 15 16 17 3 3 3 11 ⟶ 15 18 19 6 7 3 3 11 , 15 16 17 3 3 8 12 ⟶ 15 18 19 6 7 3 8 12 , 15 16 17 3 3 8 13 ⟶ 15 18 19 6 7 3 8 13 , 15 16 17 3 3 8 14 ⟶ 15 18 19 6 7 3 8 14 , 20 21 22 3 3 3 3 ⟶ 20 23 24 6 7 3 3 3 , 20 21 22 3 3 3 8 ⟶ 20 23 24 6 7 3 3 8 , 20 21 22 3 3 3 9 ⟶ 20 23 24 6 7 3 3 9 , 20 21 22 3 3 3 10 ⟶ 20 23 24 6 7 3 3 10 , 20 21 22 3 3 3 11 ⟶ 20 23 24 6 7 3 3 11 , 20 21 22 3 3 8 12 ⟶ 20 23 24 6 7 3 8 12 , 20 21 22 3 3 8 13 ⟶ 20 23 24 6 7 3 8 13 , 20 21 22 3 3 8 14 ⟶ 20 23 24 6 7 3 8 14 , 0 1 2 3 8 12 6 ⟶ 0 4 5 6 7 8 12 6 , 0 1 2 3 8 12 25 ⟶ 0 4 5 6 7 8 12 25 , 0 1 2 3 8 12 26 ⟶ 0 4 5 6 7 8 12 26 , 15 16 17 3 8 12 6 ⟶ 15 18 19 6 7 8 12 6 , 15 16 17 3 8 12 25 ⟶ 15 18 19 6 7 8 12 25 , 15 16 17 3 8 12 26 ⟶ 15 18 19 6 7 8 12 26 , 20 21 22 3 8 12 6 ⟶ 20 23 24 6 7 8 12 6 , 20 21 22 3 8 12 25 ⟶ 20 23 24 6 7 8 12 25 , 20 21 22 3 8 12 26 ⟶ 20 23 24 6 7 8 12 26 , 0 4 5 6 27 12 6 7 3 ⟶ 0 1 28 12 6 7 3 3 , 0 4 5 6 27 12 6 7 8 ⟶ 0 1 28 12 6 7 3 8 , 0 4 5 6 27 12 6 7 10 ⟶ 0 1 28 12 6 7 3 10 , 0 4 5 6 27 12 6 27 12 ⟶ 0 1 28 12 6 7 8 12 , 15 18 19 6 27 12 6 7 3 ⟶ 15 16 29 12 6 7 3 3 , 15 18 19 6 27 12 6 7 8 ⟶ 15 16 29 12 6 7 3 8 , 15 18 19 6 27 12 6 7 10 ⟶ 15 16 29 12 6 7 3 10 , 15 18 19 6 27 12 6 27 12 ⟶ 15 16 29 12 6 7 8 12 , 20 23 24 6 27 12 6 7 3 ⟶ 20 21 30 12 6 7 3 3 , 20 23 24 6 27 12 6 7 8 ⟶ 20 21 30 12 6 7 3 8 , 20 23 24 6 27 12 6 7 10 ⟶ 20 21 30 12 6 7 3 10 , 20 23 24 6 27 12 6 27 12 ⟶ 20 21 30 12 6 7 8 12 , 0 4 5 6 27 12 26 31 6 ⟶ 0 1 28 12 6 27 12 6 , 0 4 5 6 27 12 26 31 26 ⟶ 0 1 28 12 6 27 12 26 , 15 18 19 6 27 12 26 31 6 ⟶ 15 16 29 12 6 27 12 6 , 15 18 19 6 27 12 26 31 26 ⟶ 15 16 29 12 6 27 12 26 , 20 23 24 6 27 12 26 31 6 ⟶ 20 21 30 12 6 27 12 6 , 20 23 24 6 27 12 26 31 26 ⟶ 20 21 30 12 6 27 12 26 , 1 2 3 3 3 3 3 →= 1 28 12 6 7 3 3 3 , 1 2 3 3 3 3 8 →= 1 28 12 6 7 3 3 8 , 1 2 3 3 3 3 9 →= 1 28 12 6 7 3 3 9 , 1 2 3 3 3 3 10 →= 1 28 12 6 7 3 3 10 , 1 2 3 3 3 3 11 →= 1 28 12 6 7 3 3 11 , 32 2 3 3 3 3 3 →= 32 28 12 6 7 3 3 3 , 32 2 3 3 3 3 8 →= 32 28 12 6 7 3 3 8 , 32 2 3 3 3 3 9 →= 32 28 12 6 7 3 3 9 , 32 2 3 3 3 3 10 →= 32 28 12 6 7 3 3 10 , 32 2 3 3 3 3 11 →= 32 28 12 6 7 3 3 11 , 1 2 3 3 3 8 12 →= 1 28 12 6 7 3 8 12 , 1 2 3 3 3 8 13 →= 1 28 12 6 7 3 8 13 , 1 2 3 3 3 8 14 →= 1 28 12 6 7 3 8 14 , 32 2 3 3 3 8 12 →= 32 28 12 6 7 3 8 12 , 32 2 3 3 3 8 13 →= 32 28 12 6 7 3 8 13 , 32 2 3 3 3 8 14 →= 32 28 12 6 7 3 8 14 , 33 3 3 3 3 3 3 →= 33 8 12 6 7 3 3 3 , 33 3 3 3 3 3 8 →= 33 8 12 6 7 3 3 8 , 33 3 3 3 3 3 9 →= 33 8 12 6 7 3 3 9 , 33 3 3 3 3 3 10 →= 33 8 12 6 7 3 3 10 , 33 3 3 3 3 3 11 →= 33 8 12 6 7 3 3 11 , 2 3 3 3 3 3 3 →= 2 8 12 6 7 3 3 3 , 2 3 3 3 3 3 8 →= 2 8 12 6 7 3 3 8 , 2 3 3 3 3 3 9 →= 2 8 12 6 7 3 3 9 , 2 3 3 3 3 3 10 →= 2 8 12 6 7 3 3 10 , 2 3 3 3 3 3 11 →= 2 8 12 6 7 3 3 11 , 34 3 3 3 3 3 3 →= 34 8 12 6 7 3 3 3 , 34 3 3 3 3 3 8 →= 34 8 12 6 7 3 3 8 , 34 3 3 3 3 3 9 →= 34 8 12 6 7 3 3 9 , 34 3 3 3 3 3 10 →= 34 8 12 6 7 3 3 10 , 34 3 3 3 3 3 11 →= 34 8 12 6 7 3 3 11 , 3 3 3 3 3 3 3 →= 3 8 12 6 7 3 3 3 , 3 3 3 3 3 3 8 →= 3 8 12 6 7 3 3 8 , 3 3 3 3 3 3 9 →= 3 8 12 6 7 3 3 9 , 3 3 3 3 3 3 10 →= 3 8 12 6 7 3 3 10 , 3 3 3 3 3 3 11 →= 3 8 12 6 7 3 3 11 , 7 3 3 3 3 3 3 →= 7 8 12 6 7 3 3 3 , 7 3 3 3 3 3 8 →= 7 8 12 6 7 3 3 8 , 7 3 3 3 3 3 9 →= 7 8 12 6 7 3 3 9 , 7 3 3 3 3 3 10 →= 7 8 12 6 7 3 3 10 , 7 3 3 3 3 3 11 →= 7 8 12 6 7 3 3 11 , 35 3 3 3 3 3 3 →= 35 8 12 6 7 3 3 3 , 35 3 3 3 3 3 8 →= 35 8 12 6 7 3 3 8 , 35 3 3 3 3 3 9 →= 35 8 12 6 7 3 3 9 , 35 3 3 3 3 3 10 →= 35 8 12 6 7 3 3 10 , 35 3 3 3 3 3 11 →= 35 8 12 6 7 3 3 11 , 17 3 3 3 3 3 3 →= 17 8 12 6 7 3 3 3 , 17 3 3 3 3 3 8 →= 17 8 12 6 7 3 3 8 , 17 3 3 3 3 3 9 →= 17 8 12 6 7 3 3 9 , 17 3 3 3 3 3 10 →= 17 8 12 6 7 3 3 10 , 17 3 3 3 3 3 11 →= 17 8 12 6 7 3 3 11 , 22 3 3 3 3 3 3 →= 22 8 12 6 7 3 3 3 , 22 3 3 3 3 3 8 →= 22 8 12 6 7 3 3 8 , 22 3 3 3 3 3 9 →= 22 8 12 6 7 3 3 9 , 22 3 3 3 3 3 10 →= 22 8 12 6 7 3 3 10 , 22 3 3 3 3 3 11 →= 22 8 12 6 7 3 3 11 , 33 3 3 3 3 8 12 →= 33 8 12 6 7 3 8 12 , 33 3 3 3 3 8 13 →= 33 8 12 6 7 3 8 13 , 33 3 3 3 3 8 14 →= 33 8 12 6 7 3 8 14 , 2 3 3 3 3 8 12 →= 2 8 12 6 7 3 8 12 , 2 3 3 3 3 8 13 →= 2 8 12 6 7 3 8 13 , 2 3 3 3 3 8 14 →= 2 8 12 6 7 3 8 14 , 34 3 3 3 3 8 12 →= 34 8 12 6 7 3 8 12 , 34 3 3 3 3 8 13 →= 34 8 12 6 7 3 8 13 , 34 3 3 3 3 8 14 →= 34 8 12 6 7 3 8 14 , 3 3 3 3 3 8 12 →= 3 8 12 6 7 3 8 12 , 3 3 3 3 3 8 13 →= 3 8 12 6 7 3 8 13 , 3 3 3 3 3 8 14 →= 3 8 12 6 7 3 8 14 , 7 3 3 3 3 8 12 →= 7 8 12 6 7 3 8 12 , 7 3 3 3 3 8 13 →= 7 8 12 6 7 3 8 13 , 7 3 3 3 3 8 14 →= 7 8 12 6 7 3 8 14 , 35 3 3 3 3 8 12 →= 35 8 12 6 7 3 8 12 , 35 3 3 3 3 8 13 →= 35 8 12 6 7 3 8 13 , 35 3 3 3 3 8 14 →= 35 8 12 6 7 3 8 14 , 17 3 3 3 3 8 12 →= 17 8 12 6 7 3 8 12 , 17 3 3 3 3 8 13 →= 17 8 12 6 7 3 8 13 , 17 3 3 3 3 8 14 →= 17 8 12 6 7 3 8 14 , 22 3 3 3 3 8 12 →= 22 8 12 6 7 3 8 12 , 22 3 3 3 3 8 13 →= 22 8 12 6 7 3 8 13 , 22 3 3 3 3 8 14 →= 22 8 12 6 7 3 8 14 , 33 3 3 3 3 10 36 →= 33 8 12 6 7 3 10 36 , 33 3 3 3 3 10 37 →= 33 8 12 6 7 3 10 37 , 2 3 3 3 3 10 36 →= 2 8 12 6 7 3 10 36 , 2 3 3 3 3 10 37 →= 2 8 12 6 7 3 10 37 , 34 3 3 3 3 10 36 →= 34 8 12 6 7 3 10 36 , 34 3 3 3 3 10 37 →= 34 8 12 6 7 3 10 37 , 3 3 3 3 3 10 36 →= 3 8 12 6 7 3 10 36 , 3 3 3 3 3 10 37 →= 3 8 12 6 7 3 10 37 , 7 3 3 3 3 10 36 →= 7 8 12 6 7 3 10 36 , 7 3 3 3 3 10 37 →= 7 8 12 6 7 3 10 37 , 35 3 3 3 3 10 36 →= 35 8 12 6 7 3 10 36 , 35 3 3 3 3 10 37 →= 35 8 12 6 7 3 10 37 , 17 3 3 3 3 10 36 →= 17 8 12 6 7 3 10 36 , 17 3 3 3 3 10 37 →= 17 8 12 6 7 3 10 37 , 22 3 3 3 3 10 36 →= 22 8 12 6 7 3 10 36 , 22 3 3 3 3 10 37 →= 22 8 12 6 7 3 10 37 , 33 3 3 3 3 11 38 →= 33 8 12 6 7 3 11 38 , 2 3 3 3 3 11 38 →= 2 8 12 6 7 3 11 38 , 34 3 3 3 3 11 38 →= 34 8 12 6 7 3 11 38 , 3 3 3 3 3 11 38 →= 3 8 12 6 7 3 11 38 , 7 3 3 3 3 11 38 →= 7 8 12 6 7 3 11 38 , 35 3 3 3 3 11 38 →= 35 8 12 6 7 3 11 38 , 17 3 3 3 3 11 38 →= 17 8 12 6 7 3 11 38 , 22 3 3 3 3 11 38 →= 22 8 12 6 7 3 11 38 , 39 7 3 3 3 3 3 →= 39 27 12 6 7 3 3 3 , 39 7 3 3 3 3 8 →= 39 27 12 6 7 3 3 8 , 39 7 3 3 3 3 9 →= 39 27 12 6 7 3 3 9 , 39 7 3 3 3 3 10 →= 39 27 12 6 7 3 3 10 , 39 7 3 3 3 3 11 →= 39 27 12 6 7 3 3 11 , 6 7 3 3 3 3 3 →= 6 27 12 6 7 3 3 3 , 6 7 3 3 3 3 8 →= 6 27 12 6 7 3 3 8 , 6 7 3 3 3 3 9 →= 6 27 12 6 7 3 3 9 , 6 7 3 3 3 3 10 →= 6 27 12 6 7 3 3 10 , 6 7 3 3 3 3 11 →= 6 27 12 6 7 3 3 11 , 40 7 3 3 3 3 3 →= 40 27 12 6 7 3 3 3 , 40 7 3 3 3 3 8 →= 40 27 12 6 7 3 3 8 , 40 7 3 3 3 3 9 →= 40 27 12 6 7 3 3 9 , 40 7 3 3 3 3 10 →= 40 27 12 6 7 3 3 10 , 40 7 3 3 3 3 11 →= 40 27 12 6 7 3 3 11 , 39 7 3 3 3 8 12 →= 39 27 12 6 7 3 8 12 , 39 7 3 3 3 8 13 →= 39 27 12 6 7 3 8 13 , 39 7 3 3 3 8 14 →= 39 27 12 6 7 3 8 14 , 6 7 3 3 3 8 12 →= 6 27 12 6 7 3 8 12 , 6 7 3 3 3 8 13 →= 6 27 12 6 7 3 8 13 , 6 7 3 3 3 8 14 →= 6 27 12 6 7 3 8 14 , 40 7 3 3 3 8 12 →= 40 27 12 6 7 3 8 12 , 40 7 3 3 3 8 13 →= 40 27 12 6 7 3 8 13 , 40 7 3 3 3 8 14 →= 40 27 12 6 7 3 8 14 , 39 7 3 3 3 10 36 →= 39 27 12 6 7 3 10 36 , 39 7 3 3 3 10 37 →= 39 27 12 6 7 3 10 37 , 6 7 3 3 3 10 36 →= 6 27 12 6 7 3 10 36 , 6 7 3 3 3 10 37 →= 6 27 12 6 7 3 10 37 , 40 7 3 3 3 10 36 →= 40 27 12 6 7 3 10 36 , 40 7 3 3 3 10 37 →= 40 27 12 6 7 3 10 37 , 39 7 3 3 3 11 38 →= 39 27 12 6 7 3 11 38 , 6 7 3 3 3 11 38 →= 6 27 12 6 7 3 11 38 , 40 7 3 3 3 11 38 →= 40 27 12 6 7 3 11 38 , 16 17 3 3 3 3 3 →= 16 29 12 6 7 3 3 3 , 16 17 3 3 3 3 8 →= 16 29 12 6 7 3 3 8 , 16 17 3 3 3 3 9 →= 16 29 12 6 7 3 3 9 , 16 17 3 3 3 3 10 →= 16 29 12 6 7 3 3 10 , 16 17 3 3 3 3 11 →= 16 29 12 6 7 3 3 11 , 41 17 3 3 3 3 3 →= 41 29 12 6 7 3 3 3 , 41 17 3 3 3 3 8 →= 41 29 12 6 7 3 3 8 , 41 17 3 3 3 3 9 →= 41 29 12 6 7 3 3 9 , 41 17 3 3 3 3 10 →= 41 29 12 6 7 3 3 10 , 41 17 3 3 3 3 11 →= 41 29 12 6 7 3 3 11 , 16 17 3 3 3 8 12 →= 16 29 12 6 7 3 8 12 , 16 17 3 3 3 8 13 →= 16 29 12 6 7 3 8 13 , 16 17 3 3 3 8 14 →= 16 29 12 6 7 3 8 14 , 41 17 3 3 3 8 12 →= 41 29 12 6 7 3 8 12 , 41 17 3 3 3 8 13 →= 41 29 12 6 7 3 8 13 , 41 17 3 3 3 8 14 →= 41 29 12 6 7 3 8 14 , 42 22 3 3 3 3 3 →= 42 30 12 6 7 3 3 3 , 42 22 3 3 3 3 8 →= 42 30 12 6 7 3 3 8 , 42 22 3 3 3 3 9 →= 42 30 12 6 7 3 3 9 , 42 22 3 3 3 3 10 →= 42 30 12 6 7 3 3 10 , 42 22 3 3 3 3 11 →= 42 30 12 6 7 3 3 11 , 21 22 3 3 3 3 3 →= 21 30 12 6 7 3 3 3 , 21 22 3 3 3 3 8 →= 21 30 12 6 7 3 3 8 , 21 22 3 3 3 3 9 →= 21 30 12 6 7 3 3 9 , 21 22 3 3 3 3 10 →= 21 30 12 6 7 3 3 10 , 21 22 3 3 3 3 11 →= 21 30 12 6 7 3 3 11 , 42 22 3 3 3 8 12 →= 42 30 12 6 7 3 8 12 , 42 22 3 3 3 8 13 →= 42 30 12 6 7 3 8 13 , 42 22 3 3 3 8 14 →= 42 30 12 6 7 3 8 14 , 21 22 3 3 3 8 12 →= 21 30 12 6 7 3 8 12 , 21 22 3 3 3 8 13 →= 21 30 12 6 7 3 8 13 , 21 22 3 3 3 8 14 →= 21 30 12 6 7 3 8 14 , 43 33 3 3 3 3 3 →= 43 44 12 6 7 3 3 3 , 43 33 3 3 3 3 8 →= 43 44 12 6 7 3 3 8 , 43 33 3 3 3 3 9 →= 43 44 12 6 7 3 3 9 , 43 33 3 3 3 3 10 →= 43 44 12 6 7 3 3 10 , 43 33 3 3 3 3 11 →= 43 44 12 6 7 3 3 11 , 45 33 3 3 3 3 3 →= 45 44 12 6 7 3 3 3 , 45 33 3 3 3 3 8 →= 45 44 12 6 7 3 3 8 , 45 33 3 3 3 3 9 →= 45 44 12 6 7 3 3 9 , 45 33 3 3 3 3 10 →= 45 44 12 6 7 3 3 10 , 45 33 3 3 3 3 11 →= 45 44 12 6 7 3 3 11 , 43 33 3 3 3 8 12 →= 43 44 12 6 7 3 8 12 , 43 33 3 3 3 8 13 →= 43 44 12 6 7 3 8 13 , 43 33 3 3 3 8 14 →= 43 44 12 6 7 3 8 14 , 45 33 3 3 3 8 12 →= 45 44 12 6 7 3 8 12 , 45 33 3 3 3 8 13 →= 45 44 12 6 7 3 8 13 , 45 33 3 3 3 8 14 →= 45 44 12 6 7 3 8 14 , 46 34 3 3 3 3 3 →= 46 47 12 6 7 3 3 3 , 46 34 3 3 3 3 8 →= 46 47 12 6 7 3 3 8 , 46 34 3 3 3 3 9 →= 46 47 12 6 7 3 3 9 , 46 34 3 3 3 3 10 →= 46 47 12 6 7 3 3 10 , 46 34 3 3 3 3 11 →= 46 47 12 6 7 3 3 11 , 46 34 3 3 3 8 12 →= 46 47 12 6 7 3 8 12 , 46 34 3 3 3 8 13 →= 46 47 12 6 7 3 8 13 , 46 34 3 3 3 8 14 →= 46 47 12 6 7 3 8 14 , 5 6 7 3 3 3 3 →= 5 26 31 6 7 3 3 3 , 5 6 7 3 3 3 8 →= 5 26 31 6 7 3 3 8 , 5 6 7 3 3 3 9 →= 5 26 31 6 7 3 3 9 , 5 6 7 3 3 3 10 →= 5 26 31 6 7 3 3 10 , 5 6 7 3 3 3 11 →= 5 26 31 6 7 3 3 11 , 48 6 7 3 3 3 3 →= 48 26 31 6 7 3 3 3 , 48 6 7 3 3 3 8 →= 48 26 31 6 7 3 3 8 , 48 6 7 3 3 3 9 →= 48 26 31 6 7 3 3 9 , 48 6 7 3 3 3 10 →= 48 26 31 6 7 3 3 10 , 48 6 7 3 3 3 11 →= 48 26 31 6 7 3 3 11 , 12 6 7 3 3 3 3 →= 12 26 31 6 7 3 3 3 , 12 6 7 3 3 3 8 →= 12 26 31 6 7 3 3 8 , 12 6 7 3 3 3 9 →= 12 26 31 6 7 3 3 9 , 12 6 7 3 3 3 10 →= 12 26 31 6 7 3 3 10 , 12 6 7 3 3 3 11 →= 12 26 31 6 7 3 3 11 , 49 6 7 3 3 3 3 →= 49 26 31 6 7 3 3 3 , 49 6 7 3 3 3 8 →= 49 26 31 6 7 3 3 8 , 49 6 7 3 3 3 9 →= 49 26 31 6 7 3 3 9 , 49 6 7 3 3 3 10 →= 49 26 31 6 7 3 3 10 , 49 6 7 3 3 3 11 →= 49 26 31 6 7 3 3 11 , 19 6 7 3 3 3 3 →= 19 26 31 6 7 3 3 3 , 19 6 7 3 3 3 8 →= 19 26 31 6 7 3 3 8 , 19 6 7 3 3 3 9 →= 19 26 31 6 7 3 3 9 , 19 6 7 3 3 3 10 →= 19 26 31 6 7 3 3 10 , 19 6 7 3 3 3 11 →= 19 26 31 6 7 3 3 11 , 24 6 7 3 3 3 3 →= 24 26 31 6 7 3 3 3 , 24 6 7 3 3 3 8 →= 24 26 31 6 7 3 3 8 , 24 6 7 3 3 3 9 →= 24 26 31 6 7 3 3 9 , 24 6 7 3 3 3 10 →= 24 26 31 6 7 3 3 10 , 24 6 7 3 3 3 11 →= 24 26 31 6 7 3 3 11 , 31 6 7 3 3 3 3 →= 31 26 31 6 7 3 3 3 , 31 6 7 3 3 3 8 →= 31 26 31 6 7 3 3 8 , 31 6 7 3 3 3 9 →= 31 26 31 6 7 3 3 9 , 31 6 7 3 3 3 10 →= 31 26 31 6 7 3 3 10 , 31 6 7 3 3 3 11 →= 31 26 31 6 7 3 3 11 , 5 6 7 3 3 8 12 →= 5 26 31 6 7 3 8 12 , 5 6 7 3 3 8 13 →= 5 26 31 6 7 3 8 13 , 5 6 7 3 3 8 14 →= 5 26 31 6 7 3 8 14 , 48 6 7 3 3 8 12 →= 48 26 31 6 7 3 8 12 , 48 6 7 3 3 8 13 →= 48 26 31 6 7 3 8 13 , 48 6 7 3 3 8 14 →= 48 26 31 6 7 3 8 14 , 12 6 7 3 3 8 12 →= 12 26 31 6 7 3 8 12 , 12 6 7 3 3 8 13 →= 12 26 31 6 7 3 8 13 , 12 6 7 3 3 8 14 →= 12 26 31 6 7 3 8 14 , 49 6 7 3 3 8 12 →= 49 26 31 6 7 3 8 12 , 49 6 7 3 3 8 13 →= 49 26 31 6 7 3 8 13 , 49 6 7 3 3 8 14 →= 49 26 31 6 7 3 8 14 , 19 6 7 3 3 8 12 →= 19 26 31 6 7 3 8 12 , 19 6 7 3 3 8 13 →= 19 26 31 6 7 3 8 13 , 19 6 7 3 3 8 14 →= 19 26 31 6 7 3 8 14 , 24 6 7 3 3 8 12 →= 24 26 31 6 7 3 8 12 , 24 6 7 3 3 8 13 →= 24 26 31 6 7 3 8 13 , 24 6 7 3 3 8 14 →= 24 26 31 6 7 3 8 14 , 31 6 7 3 3 8 12 →= 31 26 31 6 7 3 8 12 , 31 6 7 3 3 8 13 →= 31 26 31 6 7 3 8 13 , 31 6 7 3 3 8 14 →= 31 26 31 6 7 3 8 14 , 5 6 7 3 3 10 36 →= 5 26 31 6 7 3 10 36 , 5 6 7 3 3 10 37 →= 5 26 31 6 7 3 10 37 , 48 6 7 3 3 10 36 →= 48 26 31 6 7 3 10 36 , 48 6 7 3 3 10 37 →= 48 26 31 6 7 3 10 37 , 12 6 7 3 3 10 36 →= 12 26 31 6 7 3 10 36 , 12 6 7 3 3 10 37 →= 12 26 31 6 7 3 10 37 , 49 6 7 3 3 10 36 →= 49 26 31 6 7 3 10 36 , 49 6 7 3 3 10 37 →= 49 26 31 6 7 3 10 37 , 19 6 7 3 3 10 36 →= 19 26 31 6 7 3 10 36 , 19 6 7 3 3 10 37 →= 19 26 31 6 7 3 10 37 , 24 6 7 3 3 10 36 →= 24 26 31 6 7 3 10 36 , 24 6 7 3 3 10 37 →= 24 26 31 6 7 3 10 37 , 31 6 7 3 3 10 36 →= 31 26 31 6 7 3 10 36 , 31 6 7 3 3 10 37 →= 31 26 31 6 7 3 10 37 , 5 6 7 3 3 11 38 →= 5 26 31 6 7 3 11 38 , 48 6 7 3 3 11 38 →= 48 26 31 6 7 3 11 38 , 12 6 7 3 3 11 38 →= 12 26 31 6 7 3 11 38 , 49 6 7 3 3 11 38 →= 49 26 31 6 7 3 11 38 , 19 6 7 3 3 11 38 →= 19 26 31 6 7 3 11 38 , 24 6 7 3 3 11 38 →= 24 26 31 6 7 3 11 38 , 31 6 7 3 3 11 38 →= 31 26 31 6 7 3 11 38 , 50 39 7 3 3 3 3 →= 50 51 31 6 7 3 3 3 , 50 39 7 3 3 3 8 →= 50 51 31 6 7 3 3 8 , 50 39 7 3 3 3 9 →= 50 51 31 6 7 3 3 9 , 50 39 7 3 3 3 10 →= 50 51 31 6 7 3 3 10 , 50 39 7 3 3 3 11 →= 50 51 31 6 7 3 3 11 , 50 39 7 3 3 8 12 →= 50 51 31 6 7 3 8 12 , 50 39 7 3 3 8 13 →= 50 51 31 6 7 3 8 13 , 50 39 7 3 3 8 14 →= 50 51 31 6 7 3 8 14 , 52 43 33 3 3 3 3 →= 52 53 48 6 7 3 3 3 , 52 43 33 3 3 3 8 →= 52 53 48 6 7 3 3 8 , 52 43 33 3 3 3 9 →= 52 53 48 6 7 3 3 9 , 52 43 33 3 3 3 10 →= 52 53 48 6 7 3 3 10 , 52 43 33 3 3 3 11 →= 52 53 48 6 7 3 3 11 , 52 43 33 3 3 8 12 →= 52 53 48 6 7 3 8 12 , 52 43 33 3 3 8 13 →= 52 53 48 6 7 3 8 13 , 52 43 33 3 3 8 14 →= 52 53 48 6 7 3 8 14 , 1 2 3 3 8 12 6 →= 1 28 12 6 7 8 12 6 , 1 2 3 3 8 12 25 →= 1 28 12 6 7 8 12 25 , 1 2 3 3 8 12 26 →= 1 28 12 6 7 8 12 26 , 32 2 3 3 8 12 6 →= 32 28 12 6 7 8 12 6 , 32 2 3 3 8 12 25 →= 32 28 12 6 7 8 12 25 , 32 2 3 3 8 12 26 →= 32 28 12 6 7 8 12 26 , 33 3 3 3 8 12 6 →= 33 8 12 6 7 8 12 6 , 33 3 3 3 8 12 25 →= 33 8 12 6 7 8 12 25 , 33 3 3 3 8 12 26 →= 33 8 12 6 7 8 12 26 , 2 3 3 3 8 12 6 →= 2 8 12 6 7 8 12 6 , 2 3 3 3 8 12 25 →= 2 8 12 6 7 8 12 25 , 2 3 3 3 8 12 26 →= 2 8 12 6 7 8 12 26 , 34 3 3 3 8 12 6 →= 34 8 12 6 7 8 12 6 , 34 3 3 3 8 12 25 →= 34 8 12 6 7 8 12 25 , 34 3 3 3 8 12 26 →= 34 8 12 6 7 8 12 26 , 3 3 3 3 8 12 6 →= 3 8 12 6 7 8 12 6 , 3 3 3 3 8 12 25 →= 3 8 12 6 7 8 12 25 , 3 3 3 3 8 12 26 →= 3 8 12 6 7 8 12 26 , 7 3 3 3 8 12 6 →= 7 8 12 6 7 8 12 6 , 7 3 3 3 8 12 25 →= 7 8 12 6 7 8 12 25 , 7 3 3 3 8 12 26 →= 7 8 12 6 7 8 12 26 , 35 3 3 3 8 12 6 →= 35 8 12 6 7 8 12 6 , 35 3 3 3 8 12 25 →= 35 8 12 6 7 8 12 25 , 35 3 3 3 8 12 26 →= 35 8 12 6 7 8 12 26 , 17 3 3 3 8 12 6 →= 17 8 12 6 7 8 12 6 , 17 3 3 3 8 12 25 →= 17 8 12 6 7 8 12 25 , 17 3 3 3 8 12 26 →= 17 8 12 6 7 8 12 26 , 22 3 3 3 8 12 6 →= 22 8 12 6 7 8 12 6 , 22 3 3 3 8 12 25 →= 22 8 12 6 7 8 12 25 , 22 3 3 3 8 12 26 →= 22 8 12 6 7 8 12 26 , 33 3 3 3 8 13 54 →= 33 8 12 6 7 8 13 54 , 2 3 3 3 8 13 54 →= 2 8 12 6 7 8 13 54 , 34 3 3 3 8 13 54 →= 34 8 12 6 7 8 13 54 , 3 3 3 3 8 13 54 →= 3 8 12 6 7 8 13 54 , 7 3 3 3 8 13 54 →= 7 8 12 6 7 8 13 54 , 35 3 3 3 8 13 54 →= 35 8 12 6 7 8 13 54 , 17 3 3 3 8 13 54 →= 17 8 12 6 7 8 13 54 , 22 3 3 3 8 13 54 →= 22 8 12 6 7 8 13 54 , 39 7 3 3 8 12 6 →= 39 27 12 6 7 8 12 6 , 39 7 3 3 8 12 25 →= 39 27 12 6 7 8 12 25 , 39 7 3 3 8 12 26 →= 39 27 12 6 7 8 12 26 , 6 7 3 3 8 12 6 →= 6 27 12 6 7 8 12 6 , 6 7 3 3 8 12 25 →= 6 27 12 6 7 8 12 25 , 6 7 3 3 8 12 26 →= 6 27 12 6 7 8 12 26 , 40 7 3 3 8 12 6 →= 40 27 12 6 7 8 12 6 , 40 7 3 3 8 12 25 →= 40 27 12 6 7 8 12 25 , 40 7 3 3 8 12 26 →= 40 27 12 6 7 8 12 26 , 39 7 3 3 8 13 54 →= 39 27 12 6 7 8 13 54 , 6 7 3 3 8 13 54 →= 6 27 12 6 7 8 13 54 , 40 7 3 3 8 13 54 →= 40 27 12 6 7 8 13 54 , 16 17 3 3 8 12 6 →= 16 29 12 6 7 8 12 6 , 16 17 3 3 8 12 25 →= 16 29 12 6 7 8 12 25 , 16 17 3 3 8 12 26 →= 16 29 12 6 7 8 12 26 , 41 17 3 3 8 12 6 →= 41 29 12 6 7 8 12 6 , 41 17 3 3 8 12 25 →= 41 29 12 6 7 8 12 25 , 41 17 3 3 8 12 26 →= 41 29 12 6 7 8 12 26 , 42 22 3 3 8 12 6 →= 42 30 12 6 7 8 12 6 , 42 22 3 3 8 12 25 →= 42 30 12 6 7 8 12 25 , 42 22 3 3 8 12 26 →= 42 30 12 6 7 8 12 26 , 21 22 3 3 8 12 6 →= 21 30 12 6 7 8 12 6 , 21 22 3 3 8 12 25 →= 21 30 12 6 7 8 12 25 , 21 22 3 3 8 12 26 →= 21 30 12 6 7 8 12 26 , 43 33 3 3 8 12 6 →= 43 44 12 6 7 8 12 6 , 43 33 3 3 8 12 25 →= 43 44 12 6 7 8 12 25 , 43 33 3 3 8 12 26 →= 43 44 12 6 7 8 12 26 , 45 33 3 3 8 12 6 →= 45 44 12 6 7 8 12 6 , 45 33 3 3 8 12 25 →= 45 44 12 6 7 8 12 25 , 45 33 3 3 8 12 26 →= 45 44 12 6 7 8 12 26 , 46 34 3 3 8 12 6 →= 46 47 12 6 7 8 12 6 , 46 34 3 3 8 12 25 →= 46 47 12 6 7 8 12 25 , 46 34 3 3 8 12 26 →= 46 47 12 6 7 8 12 26 , 5 6 7 3 8 12 6 →= 5 26 31 6 7 8 12 6 , 5 6 7 3 8 12 25 →= 5 26 31 6 7 8 12 25 , 5 6 7 3 8 12 26 →= 5 26 31 6 7 8 12 26 , 48 6 7 3 8 12 6 →= 48 26 31 6 7 8 12 6 , 48 6 7 3 8 12 25 →= 48 26 31 6 7 8 12 25 , 48 6 7 3 8 12 26 →= 48 26 31 6 7 8 12 26 , 12 6 7 3 8 12 6 →= 12 26 31 6 7 8 12 6 , 12 6 7 3 8 12 25 →= 12 26 31 6 7 8 12 25 , 12 6 7 3 8 12 26 →= 12 26 31 6 7 8 12 26 , 49 6 7 3 8 12 6 →= 49 26 31 6 7 8 12 6 , 49 6 7 3 8 12 25 →= 49 26 31 6 7 8 12 25 , 49 6 7 3 8 12 26 →= 49 26 31 6 7 8 12 26 , 19 6 7 3 8 12 6 →= 19 26 31 6 7 8 12 6 , 19 6 7 3 8 12 25 →= 19 26 31 6 7 8 12 25 , 19 6 7 3 8 12 26 →= 19 26 31 6 7 8 12 26 , 24 6 7 3 8 12 6 →= 24 26 31 6 7 8 12 6 , 24 6 7 3 8 12 25 →= 24 26 31 6 7 8 12 25 , 24 6 7 3 8 12 26 →= 24 26 31 6 7 8 12 26 , 31 6 7 3 8 12 6 →= 31 26 31 6 7 8 12 6 , 31 6 7 3 8 12 25 →= 31 26 31 6 7 8 12 25 , 31 6 7 3 8 12 26 →= 31 26 31 6 7 8 12 26 , 5 6 7 3 8 13 54 →= 5 26 31 6 7 8 13 54 , 48 6 7 3 8 13 54 →= 48 26 31 6 7 8 13 54 , 12 6 7 3 8 13 54 →= 12 26 31 6 7 8 13 54 , 49 6 7 3 8 13 54 →= 49 26 31 6 7 8 13 54 , 19 6 7 3 8 13 54 →= 19 26 31 6 7 8 13 54 , 24 6 7 3 8 13 54 →= 24 26 31 6 7 8 13 54 , 31 6 7 3 8 13 54 →= 31 26 31 6 7 8 13 54 , 50 39 7 3 8 12 6 →= 50 51 31 6 7 8 12 6 , 50 39 7 3 8 12 25 →= 50 51 31 6 7 8 12 25 , 50 39 7 3 8 12 26 →= 50 51 31 6 7 8 12 26 , 52 43 33 3 8 12 6 →= 52 53 48 6 7 8 12 6 , 52 43 33 3 8 12 25 →= 52 53 48 6 7 8 12 25 , 52 43 33 3 8 12 26 →= 52 53 48 6 7 8 12 26 , 33 3 3 3 10 36 55 →= 33 8 12 6 7 10 36 55 , 2 3 3 3 10 36 55 →= 2 8 12 6 7 10 36 55 , 34 3 3 3 10 36 55 →= 34 8 12 6 7 10 36 55 , 3 3 3 3 10 36 55 →= 3 8 12 6 7 10 36 55 , 7 3 3 3 10 36 55 →= 7 8 12 6 7 10 36 55 , 35 3 3 3 10 36 55 →= 35 8 12 6 7 10 36 55 , 17 3 3 3 10 36 55 →= 17 8 12 6 7 10 36 55 , 22 3 3 3 10 36 55 →= 22 8 12 6 7 10 36 55 , 39 7 3 3 10 36 55 →= 39 27 12 6 7 10 36 55 , 6 7 3 3 10 36 55 →= 6 27 12 6 7 10 36 55 , 40 7 3 3 10 36 55 →= 40 27 12 6 7 10 36 55 , 5 6 7 3 10 36 55 →= 5 26 31 6 7 10 36 55 , 48 6 7 3 10 36 55 →= 48 26 31 6 7 10 36 55 , 12 6 7 3 10 36 55 →= 12 26 31 6 7 10 36 55 , 49 6 7 3 10 36 55 →= 49 26 31 6 7 10 36 55 , 19 6 7 3 10 36 55 →= 19 26 31 6 7 10 36 55 , 24 6 7 3 10 36 55 →= 24 26 31 6 7 10 36 55 , 31 6 7 3 10 36 55 →= 31 26 31 6 7 10 36 55 , 1 28 12 6 27 12 6 7 3 →= 1 2 8 12 6 7 3 3 , 1 28 12 6 27 12 6 7 8 →= 1 2 8 12 6 7 3 8 , 1 28 12 6 27 12 6 7 10 →= 1 2 8 12 6 7 3 10 , 32 28 12 6 27 12 6 7 3 →= 32 2 8 12 6 7 3 3 , 32 28 12 6 27 12 6 7 8 →= 32 2 8 12 6 7 3 8 , 32 28 12 6 27 12 6 7 10 →= 32 2 8 12 6 7 3 10 , 1 28 12 6 27 12 6 27 12 →= 1 2 8 12 6 7 8 12 , 32 28 12 6 27 12 6 27 12 →= 32 2 8 12 6 7 8 12 , 33 8 12 6 27 12 6 7 3 →= 33 3 8 12 6 7 3 3 , 33 8 12 6 27 12 6 7 8 →= 33 3 8 12 6 7 3 8 , 33 8 12 6 27 12 6 7 10 →= 33 3 8 12 6 7 3 10 , 2 8 12 6 27 12 6 7 3 →= 2 3 8 12 6 7 3 3 , 2 8 12 6 27 12 6 7 8 →= 2 3 8 12 6 7 3 8 , 2 8 12 6 27 12 6 7 10 →= 2 3 8 12 6 7 3 10 , 34 8 12 6 27 12 6 7 3 →= 34 3 8 12 6 7 3 3 , 34 8 12 6 27 12 6 7 8 →= 34 3 8 12 6 7 3 8 , 34 8 12 6 27 12 6 7 10 →= 34 3 8 12 6 7 3 10 , 3 8 12 6 27 12 6 7 3 →= 3 3 8 12 6 7 3 3 , 3 8 12 6 27 12 6 7 8 →= 3 3 8 12 6 7 3 8 , 3 8 12 6 27 12 6 7 10 →= 3 3 8 12 6 7 3 10 , 7 8 12 6 27 12 6 7 3 →= 7 3 8 12 6 7 3 3 , 7 8 12 6 27 12 6 7 8 →= 7 3 8 12 6 7 3 8 , 7 8 12 6 27 12 6 7 10 →= 7 3 8 12 6 7 3 10 , 35 8 12 6 27 12 6 7 3 →= 35 3 8 12 6 7 3 3 , 35 8 12 6 27 12 6 7 8 →= 35 3 8 12 6 7 3 8 , 35 8 12 6 27 12 6 7 10 →= 35 3 8 12 6 7 3 10 , 17 8 12 6 27 12 6 7 3 →= 17 3 8 12 6 7 3 3 , 17 8 12 6 27 12 6 7 8 →= 17 3 8 12 6 7 3 8 , 17 8 12 6 27 12 6 7 10 →= 17 3 8 12 6 7 3 10 , 22 8 12 6 27 12 6 7 3 →= 22 3 8 12 6 7 3 3 , 22 8 12 6 27 12 6 7 8 →= 22 3 8 12 6 7 3 8 , 22 8 12 6 27 12 6 7 10 →= 22 3 8 12 6 7 3 10 , 33 8 12 6 27 12 6 27 12 →= 33 3 8 12 6 7 8 12 , 2 8 12 6 27 12 6 27 12 →= 2 3 8 12 6 7 8 12 , 34 8 12 6 27 12 6 27 12 →= 34 3 8 12 6 7 8 12 , 3 8 12 6 27 12 6 27 12 →= 3 3 8 12 6 7 8 12 , 7 8 12 6 27 12 6 27 12 →= 7 3 8 12 6 7 8 12 , 35 8 12 6 27 12 6 27 12 →= 35 3 8 12 6 7 8 12 , 17 8 12 6 27 12 6 27 12 →= 17 3 8 12 6 7 8 12 , 22 8 12 6 27 12 6 27 12 →= 22 3 8 12 6 7 8 12 , 39 27 12 6 27 12 6 7 3 →= 39 7 8 12 6 7 3 3 , 39 27 12 6 27 12 6 7 8 →= 39 7 8 12 6 7 3 8 , 39 27 12 6 27 12 6 7 10 →= 39 7 8 12 6 7 3 10 , 6 27 12 6 27 12 6 7 3 →= 6 7 8 12 6 7 3 3 , 6 27 12 6 27 12 6 7 8 →= 6 7 8 12 6 7 3 8 , 6 27 12 6 27 12 6 7 10 →= 6 7 8 12 6 7 3 10 , 40 27 12 6 27 12 6 7 3 →= 40 7 8 12 6 7 3 3 , 40 27 12 6 27 12 6 7 8 →= 40 7 8 12 6 7 3 8 , 40 27 12 6 27 12 6 7 10 →= 40 7 8 12 6 7 3 10 , 39 27 12 6 27 12 6 27 12 →= 39 7 8 12 6 7 8 12 , 6 27 12 6 27 12 6 27 12 →= 6 7 8 12 6 7 8 12 , 40 27 12 6 27 12 6 27 12 →= 40 7 8 12 6 7 8 12 , 16 29 12 6 27 12 6 7 3 →= 16 17 8 12 6 7 3 3 , 16 29 12 6 27 12 6 7 8 →= 16 17 8 12 6 7 3 8 , 16 29 12 6 27 12 6 7 10 →= 16 17 8 12 6 7 3 10 , 41 29 12 6 27 12 6 7 3 →= 41 17 8 12 6 7 3 3 , 41 29 12 6 27 12 6 7 8 →= 41 17 8 12 6 7 3 8 , 41 29 12 6 27 12 6 7 10 →= 41 17 8 12 6 7 3 10 , 16 29 12 6 27 12 6 27 12 →= 16 17 8 12 6 7 8 12 , 41 29 12 6 27 12 6 27 12 →= 41 17 8 12 6 7 8 12 , 42 30 12 6 27 12 6 7 3 →= 42 22 8 12 6 7 3 3 , 42 30 12 6 27 12 6 7 8 →= 42 22 8 12 6 7 3 8 , 42 30 12 6 27 12 6 7 10 →= 42 22 8 12 6 7 3 10 , 21 30 12 6 27 12 6 7 3 →= 21 22 8 12 6 7 3 3 , 21 30 12 6 27 12 6 7 8 →= 21 22 8 12 6 7 3 8 , 21 30 12 6 27 12 6 7 10 →= 21 22 8 12 6 7 3 10 , 42 30 12 6 27 12 6 27 12 →= 42 22 8 12 6 7 8 12 , 21 30 12 6 27 12 6 27 12 →= 21 22 8 12 6 7 8 12 , 43 44 12 6 27 12 6 7 3 →= 43 33 8 12 6 7 3 3 , 43 44 12 6 27 12 6 7 8 →= 43 33 8 12 6 7 3 8 , 43 44 12 6 27 12 6 7 10 →= 43 33 8 12 6 7 3 10 , 45 44 12 6 27 12 6 7 3 →= 45 33 8 12 6 7 3 3 , 45 44 12 6 27 12 6 7 8 →= 45 33 8 12 6 7 3 8 , 45 44 12 6 27 12 6 7 10 →= 45 33 8 12 6 7 3 10 , 43 44 12 6 27 12 6 27 12 →= 43 33 8 12 6 7 8 12 , 45 44 12 6 27 12 6 27 12 →= 45 33 8 12 6 7 8 12 , 46 47 12 6 27 12 6 7 3 →= 46 34 8 12 6 7 3 3 , 46 47 12 6 27 12 6 7 8 →= 46 34 8 12 6 7 3 8 , 46 47 12 6 27 12 6 7 10 →= 46 34 8 12 6 7 3 10 , 46 47 12 6 27 12 6 27 12 →= 46 34 8 12 6 7 8 12 , 5 26 31 6 27 12 6 7 3 →= 5 6 27 12 6 7 3 3 , 5 26 31 6 27 12 6 7 8 →= 5 6 27 12 6 7 3 8 , 5 26 31 6 27 12 6 7 10 →= 5 6 27 12 6 7 3 10 , 48 26 31 6 27 12 6 7 3 →= 48 6 27 12 6 7 3 3 , 48 26 31 6 27 12 6 7 8 →= 48 6 27 12 6 7 3 8 , 48 26 31 6 27 12 6 7 10 →= 48 6 27 12 6 7 3 10 , 12 26 31 6 27 12 6 7 3 →= 12 6 27 12 6 7 3 3 , 12 26 31 6 27 12 6 7 8 →= 12 6 27 12 6 7 3 8 , 12 26 31 6 27 12 6 7 10 →= 12 6 27 12 6 7 3 10 , 49 26 31 6 27 12 6 7 3 →= 49 6 27 12 6 7 3 3 , 49 26 31 6 27 12 6 7 8 →= 49 6 27 12 6 7 3 8 , 49 26 31 6 27 12 6 7 10 →= 49 6 27 12 6 7 3 10 , 19 26 31 6 27 12 6 7 3 →= 19 6 27 12 6 7 3 3 , 19 26 31 6 27 12 6 7 8 →= 19 6 27 12 6 7 3 8 , 19 26 31 6 27 12 6 7 10 →= 19 6 27 12 6 7 3 10 , 24 26 31 6 27 12 6 7 3 →= 24 6 27 12 6 7 3 3 , 24 26 31 6 27 12 6 7 8 →= 24 6 27 12 6 7 3 8 , 24 26 31 6 27 12 6 7 10 →= 24 6 27 12 6 7 3 10 , 31 26 31 6 27 12 6 7 3 →= 31 6 27 12 6 7 3 3 , 31 26 31 6 27 12 6 7 8 →= 31 6 27 12 6 7 3 8 , 31 26 31 6 27 12 6 7 10 →= 31 6 27 12 6 7 3 10 , 5 26 31 6 27 12 6 27 12 →= 5 6 27 12 6 7 8 12 , 48 26 31 6 27 12 6 27 12 →= 48 6 27 12 6 7 8 12 , 12 26 31 6 27 12 6 27 12 →= 12 6 27 12 6 7 8 12 , 49 26 31 6 27 12 6 27 12 →= 49 6 27 12 6 7 8 12 , 19 26 31 6 27 12 6 27 12 →= 19 6 27 12 6 7 8 12 , 24 26 31 6 27 12 6 27 12 →= 24 6 27 12 6 7 8 12 , 31 26 31 6 27 12 6 27 12 →= 31 6 27 12 6 7 8 12 , 50 51 31 6 27 12 6 7 3 →= 50 39 27 12 6 7 3 3 , 50 51 31 6 27 12 6 7 8 →= 50 39 27 12 6 7 3 8 , 50 51 31 6 27 12 6 7 10 →= 50 39 27 12 6 7 3 10 , 50 51 31 6 27 12 6 27 12 →= 50 39 27 12 6 7 8 12 , 52 53 48 6 27 12 6 7 3 →= 52 43 44 12 6 7 3 3 , 52 53 48 6 27 12 6 7 8 →= 52 43 44 12 6 7 3 8 , 52 53 48 6 27 12 6 7 10 →= 52 43 44 12 6 7 3 10 , 52 53 48 6 27 12 6 27 12 →= 52 43 44 12 6 7 8 12 , 1 28 12 6 27 12 26 31 6 →= 1 2 8 12 6 27 12 6 , 1 28 12 6 27 12 26 31 26 →= 1 2 8 12 6 27 12 26 , 32 28 12 6 27 12 26 31 6 →= 32 2 8 12 6 27 12 6 , 32 28 12 6 27 12 26 31 26 →= 32 2 8 12 6 27 12 26 , 33 8 12 6 27 12 26 31 6 →= 33 3 8 12 6 27 12 6 , 33 8 12 6 27 12 26 31 26 →= 33 3 8 12 6 27 12 26 , 2 8 12 6 27 12 26 31 6 →= 2 3 8 12 6 27 12 6 , 2 8 12 6 27 12 26 31 26 →= 2 3 8 12 6 27 12 26 , 34 8 12 6 27 12 26 31 6 →= 34 3 8 12 6 27 12 6 , 34 8 12 6 27 12 26 31 26 →= 34 3 8 12 6 27 12 26 , 3 8 12 6 27 12 26 31 6 →= 3 3 8 12 6 27 12 6 , 3 8 12 6 27 12 26 31 26 →= 3 3 8 12 6 27 12 26 , 7 8 12 6 27 12 26 31 6 →= 7 3 8 12 6 27 12 6 , 7 8 12 6 27 12 26 31 26 →= 7 3 8 12 6 27 12 26 , 35 8 12 6 27 12 26 31 6 →= 35 3 8 12 6 27 12 6 , 35 8 12 6 27 12 26 31 26 →= 35 3 8 12 6 27 12 26 , 17 8 12 6 27 12 26 31 6 →= 17 3 8 12 6 27 12 6 , 17 8 12 6 27 12 26 31 26 →= 17 3 8 12 6 27 12 26 , 22 8 12 6 27 12 26 31 6 →= 22 3 8 12 6 27 12 6 , 22 8 12 6 27 12 26 31 26 →= 22 3 8 12 6 27 12 26 , 39 27 12 6 27 12 26 31 6 →= 39 7 8 12 6 27 12 6 , 39 27 12 6 27 12 26 31 26 →= 39 7 8 12 6 27 12 26 , 6 27 12 6 27 12 26 31 6 →= 6 7 8 12 6 27 12 6 , 6 27 12 6 27 12 26 31 26 →= 6 7 8 12 6 27 12 26 , 40 27 12 6 27 12 26 31 6 →= 40 7 8 12 6 27 12 6 , 40 27 12 6 27 12 26 31 26 →= 40 7 8 12 6 27 12 26 , 16 29 12 6 27 12 26 31 6 →= 16 17 8 12 6 27 12 6 , 16 29 12 6 27 12 26 31 26 →= 16 17 8 12 6 27 12 26 , 41 29 12 6 27 12 26 31 6 →= 41 17 8 12 6 27 12 6 , 41 29 12 6 27 12 26 31 26 →= 41 17 8 12 6 27 12 26 , 42 30 12 6 27 12 26 31 6 →= 42 22 8 12 6 27 12 6 , 42 30 12 6 27 12 26 31 26 →= 42 22 8 12 6 27 12 26 , 21 30 12 6 27 12 26 31 6 →= 21 22 8 12 6 27 12 6 , 21 30 12 6 27 12 26 31 26 →= 21 22 8 12 6 27 12 26 , 43 44 12 6 27 12 26 31 6 →= 43 33 8 12 6 27 12 6 , 43 44 12 6 27 12 26 31 26 →= 43 33 8 12 6 27 12 26 , 45 44 12 6 27 12 26 31 6 →= 45 33 8 12 6 27 12 6 , 45 44 12 6 27 12 26 31 26 →= 45 33 8 12 6 27 12 26 , 46 47 12 6 27 12 26 31 6 →= 46 34 8 12 6 27 12 6 , 46 47 12 6 27 12 26 31 26 →= 46 34 8 12 6 27 12 26 , 5 26 31 6 27 12 26 31 6 →= 5 6 27 12 6 27 12 6 , 5 26 31 6 27 12 26 31 26 →= 5 6 27 12 6 27 12 26 , 48 26 31 6 27 12 26 31 6 →= 48 6 27 12 6 27 12 6 , 48 26 31 6 27 12 26 31 26 →= 48 6 27 12 6 27 12 26 , 12 26 31 6 27 12 26 31 6 →= 12 6 27 12 6 27 12 6 , 12 26 31 6 27 12 26 31 26 →= 12 6 27 12 6 27 12 26 , 49 26 31 6 27 12 26 31 6 →= 49 6 27 12 6 27 12 6 , 49 26 31 6 27 12 26 31 26 →= 49 6 27 12 6 27 12 26 , 19 26 31 6 27 12 26 31 6 →= 19 6 27 12 6 27 12 6 , 19 26 31 6 27 12 26 31 26 →= 19 6 27 12 6 27 12 26 , 24 26 31 6 27 12 26 31 6 →= 24 6 27 12 6 27 12 6 , 24 26 31 6 27 12 26 31 26 →= 24 6 27 12 6 27 12 26 , 31 26 31 6 27 12 26 31 6 →= 31 6 27 12 6 27 12 6 , 31 26 31 6 27 12 26 31 26 →= 31 6 27 12 6 27 12 26 , 50 51 31 6 27 12 26 31 6 →= 50 39 27 12 6 27 12 6 , 50 51 31 6 27 12 26 31 26 →= 50 39 27 12 6 27 12 26 , 52 53 48 6 27 12 26 31 6 →= 52 43 44 12 6 27 12 6 , 52 53 48 6 27 12 26 31 26 →= 52 43 44 12 6 27 12 26 } 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 0 ⎟ ⎜ 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 23 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 24 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 25 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 26 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 27 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 28 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 29 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 30 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 31 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 32 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 33 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 40 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 41 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 42 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 43 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 44 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 45 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 46 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 47 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 48 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 49 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 50 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 51 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 52 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 53 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 54 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 55 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 39 ↦ 0, 7 ↦ 1, 3 ↦ 2, 27 ↦ 3, 12 ↦ 4, 6 ↦ 5, 8 ↦ 6, 9 ↦ 7, 10 ↦ 8, 11 ↦ 9, 40 ↦ 10, 13 ↦ 11, 14 ↦ 12, 36 ↦ 13, 37 ↦ 14, 38 ↦ 15, 5 ↦ 16, 26 ↦ 17, 31 ↦ 18, 48 ↦ 19, 49 ↦ 20, 19 ↦ 21, 24 ↦ 22, 50 ↦ 23, 51 ↦ 24, 25 ↦ 25, 54 ↦ 26, 55 ↦ 27, 33 ↦ 28, 2 ↦ 29, 34 ↦ 30, 35 ↦ 31, 17 ↦ 32, 22 ↦ 33 }, it remains to prove termination of the 251-rule system { 0 1 2 2 2 2 2 →= 0 3 4 5 1 2 2 2 , 0 1 2 2 2 2 6 →= 0 3 4 5 1 2 2 6 , 0 1 2 2 2 2 7 →= 0 3 4 5 1 2 2 7 , 0 1 2 2 2 2 8 →= 0 3 4 5 1 2 2 8 , 0 1 2 2 2 2 9 →= 0 3 4 5 1 2 2 9 , 5 1 2 2 2 2 2 →= 5 3 4 5 1 2 2 2 , 5 1 2 2 2 2 6 →= 5 3 4 5 1 2 2 6 , 5 1 2 2 2 2 7 →= 5 3 4 5 1 2 2 7 , 5 1 2 2 2 2 8 →= 5 3 4 5 1 2 2 8 , 5 1 2 2 2 2 9 →= 5 3 4 5 1 2 2 9 , 10 1 2 2 2 2 2 →= 10 3 4 5 1 2 2 2 , 10 1 2 2 2 2 6 →= 10 3 4 5 1 2 2 6 , 10 1 2 2 2 2 7 →= 10 3 4 5 1 2 2 7 , 10 1 2 2 2 2 8 →= 10 3 4 5 1 2 2 8 , 10 1 2 2 2 2 9 →= 10 3 4 5 1 2 2 9 , 0 1 2 2 2 6 4 →= 0 3 4 5 1 2 6 4 , 0 1 2 2 2 6 11 →= 0 3 4 5 1 2 6 11 , 0 1 2 2 2 6 12 →= 0 3 4 5 1 2 6 12 , 5 1 2 2 2 6 4 →= 5 3 4 5 1 2 6 4 , 5 1 2 2 2 6 11 →= 5 3 4 5 1 2 6 11 , 5 1 2 2 2 6 12 →= 5 3 4 5 1 2 6 12 , 10 1 2 2 2 6 4 →= 10 3 4 5 1 2 6 4 , 10 1 2 2 2 6 11 →= 10 3 4 5 1 2 6 11 , 10 1 2 2 2 6 12 →= 10 3 4 5 1 2 6 12 , 0 1 2 2 2 8 13 →= 0 3 4 5 1 2 8 13 , 0 1 2 2 2 8 14 →= 0 3 4 5 1 2 8 14 , 5 1 2 2 2 8 13 →= 5 3 4 5 1 2 8 13 , 5 1 2 2 2 8 14 →= 5 3 4 5 1 2 8 14 , 10 1 2 2 2 8 13 →= 10 3 4 5 1 2 8 13 , 10 1 2 2 2 8 14 →= 10 3 4 5 1 2 8 14 , 0 1 2 2 2 9 15 →= 0 3 4 5 1 2 9 15 , 5 1 2 2 2 9 15 →= 5 3 4 5 1 2 9 15 , 10 1 2 2 2 9 15 →= 10 3 4 5 1 2 9 15 , 16 5 1 2 2 2 2 →= 16 17 18 5 1 2 2 2 , 16 5 1 2 2 2 6 →= 16 17 18 5 1 2 2 6 , 16 5 1 2 2 2 7 →= 16 17 18 5 1 2 2 7 , 16 5 1 2 2 2 8 →= 16 17 18 5 1 2 2 8 , 16 5 1 2 2 2 9 →= 16 17 18 5 1 2 2 9 , 19 5 1 2 2 2 2 →= 19 17 18 5 1 2 2 2 , 19 5 1 2 2 2 6 →= 19 17 18 5 1 2 2 6 , 19 5 1 2 2 2 7 →= 19 17 18 5 1 2 2 7 , 19 5 1 2 2 2 8 →= 19 17 18 5 1 2 2 8 , 19 5 1 2 2 2 9 →= 19 17 18 5 1 2 2 9 , 4 5 1 2 2 2 2 →= 4 17 18 5 1 2 2 2 , 4 5 1 2 2 2 6 →= 4 17 18 5 1 2 2 6 , 4 5 1 2 2 2 7 →= 4 17 18 5 1 2 2 7 , 4 5 1 2 2 2 8 →= 4 17 18 5 1 2 2 8 , 4 5 1 2 2 2 9 →= 4 17 18 5 1 2 2 9 , 20 5 1 2 2 2 2 →= 20 17 18 5 1 2 2 2 , 20 5 1 2 2 2 6 →= 20 17 18 5 1 2 2 6 , 20 5 1 2 2 2 7 →= 20 17 18 5 1 2 2 7 , 20 5 1 2 2 2 8 →= 20 17 18 5 1 2 2 8 , 20 5 1 2 2 2 9 →= 20 17 18 5 1 2 2 9 , 21 5 1 2 2 2 2 →= 21 17 18 5 1 2 2 2 , 21 5 1 2 2 2 6 →= 21 17 18 5 1 2 2 6 , 21 5 1 2 2 2 7 →= 21 17 18 5 1 2 2 7 , 21 5 1 2 2 2 8 →= 21 17 18 5 1 2 2 8 , 21 5 1 2 2 2 9 →= 21 17 18 5 1 2 2 9 , 22 5 1 2 2 2 2 →= 22 17 18 5 1 2 2 2 , 22 5 1 2 2 2 6 →= 22 17 18 5 1 2 2 6 , 22 5 1 2 2 2 7 →= 22 17 18 5 1 2 2 7 , 22 5 1 2 2 2 8 →= 22 17 18 5 1 2 2 8 , 22 5 1 2 2 2 9 →= 22 17 18 5 1 2 2 9 , 18 5 1 2 2 2 2 →= 18 17 18 5 1 2 2 2 , 18 5 1 2 2 2 6 →= 18 17 18 5 1 2 2 6 , 18 5 1 2 2 2 7 →= 18 17 18 5 1 2 2 7 , 18 5 1 2 2 2 8 →= 18 17 18 5 1 2 2 8 , 18 5 1 2 2 2 9 →= 18 17 18 5 1 2 2 9 , 16 5 1 2 2 6 4 →= 16 17 18 5 1 2 6 4 , 16 5 1 2 2 6 11 →= 16 17 18 5 1 2 6 11 , 16 5 1 2 2 6 12 →= 16 17 18 5 1 2 6 12 , 19 5 1 2 2 6 4 →= 19 17 18 5 1 2 6 4 , 19 5 1 2 2 6 11 →= 19 17 18 5 1 2 6 11 , 19 5 1 2 2 6 12 →= 19 17 18 5 1 2 6 12 , 4 5 1 2 2 6 4 →= 4 17 18 5 1 2 6 4 , 4 5 1 2 2 6 11 →= 4 17 18 5 1 2 6 11 , 4 5 1 2 2 6 12 →= 4 17 18 5 1 2 6 12 , 20 5 1 2 2 6 4 →= 20 17 18 5 1 2 6 4 , 20 5 1 2 2 6 11 →= 20 17 18 5 1 2 6 11 , 20 5 1 2 2 6 12 →= 20 17 18 5 1 2 6 12 , 21 5 1 2 2 6 4 →= 21 17 18 5 1 2 6 4 , 21 5 1 2 2 6 11 →= 21 17 18 5 1 2 6 11 , 21 5 1 2 2 6 12 →= 21 17 18 5 1 2 6 12 , 22 5 1 2 2 6 4 →= 22 17 18 5 1 2 6 4 , 22 5 1 2 2 6 11 →= 22 17 18 5 1 2 6 11 , 22 5 1 2 2 6 12 →= 22 17 18 5 1 2 6 12 , 18 5 1 2 2 6 4 →= 18 17 18 5 1 2 6 4 , 18 5 1 2 2 6 11 →= 18 17 18 5 1 2 6 11 , 18 5 1 2 2 6 12 →= 18 17 18 5 1 2 6 12 , 16 5 1 2 2 8 13 →= 16 17 18 5 1 2 8 13 , 16 5 1 2 2 8 14 →= 16 17 18 5 1 2 8 14 , 19 5 1 2 2 8 13 →= 19 17 18 5 1 2 8 13 , 19 5 1 2 2 8 14 →= 19 17 18 5 1 2 8 14 , 4 5 1 2 2 8 13 →= 4 17 18 5 1 2 8 13 , 4 5 1 2 2 8 14 →= 4 17 18 5 1 2 8 14 , 20 5 1 2 2 8 13 →= 20 17 18 5 1 2 8 13 , 20 5 1 2 2 8 14 →= 20 17 18 5 1 2 8 14 , 21 5 1 2 2 8 13 →= 21 17 18 5 1 2 8 13 , 21 5 1 2 2 8 14 →= 21 17 18 5 1 2 8 14 , 22 5 1 2 2 8 13 →= 22 17 18 5 1 2 8 13 , 22 5 1 2 2 8 14 →= 22 17 18 5 1 2 8 14 , 18 5 1 2 2 8 13 →= 18 17 18 5 1 2 8 13 , 18 5 1 2 2 8 14 →= 18 17 18 5 1 2 8 14 , 16 5 1 2 2 9 15 →= 16 17 18 5 1 2 9 15 , 19 5 1 2 2 9 15 →= 19 17 18 5 1 2 9 15 , 4 5 1 2 2 9 15 →= 4 17 18 5 1 2 9 15 , 20 5 1 2 2 9 15 →= 20 17 18 5 1 2 9 15 , 21 5 1 2 2 9 15 →= 21 17 18 5 1 2 9 15 , 22 5 1 2 2 9 15 →= 22 17 18 5 1 2 9 15 , 18 5 1 2 2 9 15 →= 18 17 18 5 1 2 9 15 , 23 0 1 2 2 2 2 →= 23 24 18 5 1 2 2 2 , 23 0 1 2 2 2 6 →= 23 24 18 5 1 2 2 6 , 23 0 1 2 2 2 7 →= 23 24 18 5 1 2 2 7 , 23 0 1 2 2 2 8 →= 23 24 18 5 1 2 2 8 , 23 0 1 2 2 2 9 →= 23 24 18 5 1 2 2 9 , 23 0 1 2 2 6 4 →= 23 24 18 5 1 2 6 4 , 23 0 1 2 2 6 11 →= 23 24 18 5 1 2 6 11 , 23 0 1 2 2 6 12 →= 23 24 18 5 1 2 6 12 , 0 1 2 2 6 4 5 →= 0 3 4 5 1 6 4 5 , 0 1 2 2 6 4 25 →= 0 3 4 5 1 6 4 25 , 0 1 2 2 6 4 17 →= 0 3 4 5 1 6 4 17 , 5 1 2 2 6 4 5 →= 5 3 4 5 1 6 4 5 , 5 1 2 2 6 4 25 →= 5 3 4 5 1 6 4 25 , 5 1 2 2 6 4 17 →= 5 3 4 5 1 6 4 17 , 10 1 2 2 6 4 5 →= 10 3 4 5 1 6 4 5 , 10 1 2 2 6 4 25 →= 10 3 4 5 1 6 4 25 , 10 1 2 2 6 4 17 →= 10 3 4 5 1 6 4 17 , 0 1 2 2 6 11 26 →= 0 3 4 5 1 6 11 26 , 5 1 2 2 6 11 26 →= 5 3 4 5 1 6 11 26 , 10 1 2 2 6 11 26 →= 10 3 4 5 1 6 11 26 , 16 5 1 2 6 4 5 →= 16 17 18 5 1 6 4 5 , 16 5 1 2 6 4 25 →= 16 17 18 5 1 6 4 25 , 16 5 1 2 6 4 17 →= 16 17 18 5 1 6 4 17 , 19 5 1 2 6 4 5 →= 19 17 18 5 1 6 4 5 , 19 5 1 2 6 4 25 →= 19 17 18 5 1 6 4 25 , 19 5 1 2 6 4 17 →= 19 17 18 5 1 6 4 17 , 4 5 1 2 6 4 5 →= 4 17 18 5 1 6 4 5 , 4 5 1 2 6 4 25 →= 4 17 18 5 1 6 4 25 , 4 5 1 2 6 4 17 →= 4 17 18 5 1 6 4 17 , 20 5 1 2 6 4 5 →= 20 17 18 5 1 6 4 5 , 20 5 1 2 6 4 25 →= 20 17 18 5 1 6 4 25 , 20 5 1 2 6 4 17 →= 20 17 18 5 1 6 4 17 , 21 5 1 2 6 4 5 →= 21 17 18 5 1 6 4 5 , 21 5 1 2 6 4 25 →= 21 17 18 5 1 6 4 25 , 21 5 1 2 6 4 17 →= 21 17 18 5 1 6 4 17 , 22 5 1 2 6 4 5 →= 22 17 18 5 1 6 4 5 , 22 5 1 2 6 4 25 →= 22 17 18 5 1 6 4 25 , 22 5 1 2 6 4 17 →= 22 17 18 5 1 6 4 17 , 18 5 1 2 6 4 5 →= 18 17 18 5 1 6 4 5 , 18 5 1 2 6 4 25 →= 18 17 18 5 1 6 4 25 , 18 5 1 2 6 4 17 →= 18 17 18 5 1 6 4 17 , 16 5 1 2 6 11 26 →= 16 17 18 5 1 6 11 26 , 19 5 1 2 6 11 26 →= 19 17 18 5 1 6 11 26 , 4 5 1 2 6 11 26 →= 4 17 18 5 1 6 11 26 , 20 5 1 2 6 11 26 →= 20 17 18 5 1 6 11 26 , 21 5 1 2 6 11 26 →= 21 17 18 5 1 6 11 26 , 22 5 1 2 6 11 26 →= 22 17 18 5 1 6 11 26 , 18 5 1 2 6 11 26 →= 18 17 18 5 1 6 11 26 , 23 0 1 2 6 4 5 →= 23 24 18 5 1 6 4 5 , 23 0 1 2 6 4 25 →= 23 24 18 5 1 6 4 25 , 23 0 1 2 6 4 17 →= 23 24 18 5 1 6 4 17 , 0 1 2 2 8 13 27 →= 0 3 4 5 1 8 13 27 , 5 1 2 2 8 13 27 →= 5 3 4 5 1 8 13 27 , 10 1 2 2 8 13 27 →= 10 3 4 5 1 8 13 27 , 16 5 1 2 8 13 27 →= 16 17 18 5 1 8 13 27 , 19 5 1 2 8 13 27 →= 19 17 18 5 1 8 13 27 , 4 5 1 2 8 13 27 →= 4 17 18 5 1 8 13 27 , 20 5 1 2 8 13 27 →= 20 17 18 5 1 8 13 27 , 21 5 1 2 8 13 27 →= 21 17 18 5 1 8 13 27 , 22 5 1 2 8 13 27 →= 22 17 18 5 1 8 13 27 , 18 5 1 2 8 13 27 →= 18 17 18 5 1 8 13 27 , 28 6 4 5 3 4 5 1 2 →= 28 2 6 4 5 1 2 2 , 28 6 4 5 3 4 5 1 6 →= 28 2 6 4 5 1 2 6 , 28 6 4 5 3 4 5 1 8 →= 28 2 6 4 5 1 2 8 , 29 6 4 5 3 4 5 1 2 →= 29 2 6 4 5 1 2 2 , 29 6 4 5 3 4 5 1 6 →= 29 2 6 4 5 1 2 6 , 29 6 4 5 3 4 5 1 8 →= 29 2 6 4 5 1 2 8 , 30 6 4 5 3 4 5 1 2 →= 30 2 6 4 5 1 2 2 , 30 6 4 5 3 4 5 1 6 →= 30 2 6 4 5 1 2 6 , 30 6 4 5 3 4 5 1 8 →= 30 2 6 4 5 1 2 8 , 2 6 4 5 3 4 5 1 2 →= 2 2 6 4 5 1 2 2 , 2 6 4 5 3 4 5 1 6 →= 2 2 6 4 5 1 2 6 , 2 6 4 5 3 4 5 1 8 →= 2 2 6 4 5 1 2 8 , 1 6 4 5 3 4 5 1 2 →= 1 2 6 4 5 1 2 2 , 1 6 4 5 3 4 5 1 6 →= 1 2 6 4 5 1 2 6 , 1 6 4 5 3 4 5 1 8 →= 1 2 6 4 5 1 2 8 , 31 6 4 5 3 4 5 1 2 →= 31 2 6 4 5 1 2 2 , 31 6 4 5 3 4 5 1 6 →= 31 2 6 4 5 1 2 6 , 31 6 4 5 3 4 5 1 8 →= 31 2 6 4 5 1 2 8 , 32 6 4 5 3 4 5 1 2 →= 32 2 6 4 5 1 2 2 , 32 6 4 5 3 4 5 1 6 →= 32 2 6 4 5 1 2 6 , 32 6 4 5 3 4 5 1 8 →= 32 2 6 4 5 1 2 8 , 33 6 4 5 3 4 5 1 2 →= 33 2 6 4 5 1 2 2 , 33 6 4 5 3 4 5 1 6 →= 33 2 6 4 5 1 2 6 , 33 6 4 5 3 4 5 1 8 →= 33 2 6 4 5 1 2 8 , 16 17 18 5 3 4 5 1 2 →= 16 5 3 4 5 1 2 2 , 16 17 18 5 3 4 5 1 6 →= 16 5 3 4 5 1 2 6 , 16 17 18 5 3 4 5 1 8 →= 16 5 3 4 5 1 2 8 , 19 17 18 5 3 4 5 1 2 →= 19 5 3 4 5 1 2 2 , 19 17 18 5 3 4 5 1 6 →= 19 5 3 4 5 1 2 6 , 19 17 18 5 3 4 5 1 8 →= 19 5 3 4 5 1 2 8 , 4 17 18 5 3 4 5 1 2 →= 4 5 3 4 5 1 2 2 , 4 17 18 5 3 4 5 1 6 →= 4 5 3 4 5 1 2 6 , 4 17 18 5 3 4 5 1 8 →= 4 5 3 4 5 1 2 8 , 20 17 18 5 3 4 5 1 2 →= 20 5 3 4 5 1 2 2 , 20 17 18 5 3 4 5 1 6 →= 20 5 3 4 5 1 2 6 , 20 17 18 5 3 4 5 1 8 →= 20 5 3 4 5 1 2 8 , 21 17 18 5 3 4 5 1 2 →= 21 5 3 4 5 1 2 2 , 21 17 18 5 3 4 5 1 6 →= 21 5 3 4 5 1 2 6 , 21 17 18 5 3 4 5 1 8 →= 21 5 3 4 5 1 2 8 , 22 17 18 5 3 4 5 1 2 →= 22 5 3 4 5 1 2 2 , 22 17 18 5 3 4 5 1 6 →= 22 5 3 4 5 1 2 6 , 22 17 18 5 3 4 5 1 8 →= 22 5 3 4 5 1 2 8 , 18 17 18 5 3 4 5 1 2 →= 18 5 3 4 5 1 2 2 , 18 17 18 5 3 4 5 1 6 →= 18 5 3 4 5 1 2 6 , 18 17 18 5 3 4 5 1 8 →= 18 5 3 4 5 1 2 8 , 23 24 18 5 3 4 5 1 2 →= 23 0 3 4 5 1 2 2 , 23 24 18 5 3 4 5 1 6 →= 23 0 3 4 5 1 2 6 , 23 24 18 5 3 4 5 1 8 →= 23 0 3 4 5 1 2 8 , 28 6 4 5 3 4 17 18 5 →= 28 2 6 4 5 3 4 5 , 28 6 4 5 3 4 17 18 17 →= 28 2 6 4 5 3 4 17 , 29 6 4 5 3 4 17 18 5 →= 29 2 6 4 5 3 4 5 , 29 6 4 5 3 4 17 18 17 →= 29 2 6 4 5 3 4 17 , 30 6 4 5 3 4 17 18 5 →= 30 2 6 4 5 3 4 5 , 30 6 4 5 3 4 17 18 17 →= 30 2 6 4 5 3 4 17 , 2 6 4 5 3 4 17 18 5 →= 2 2 6 4 5 3 4 5 , 2 6 4 5 3 4 17 18 17 →= 2 2 6 4 5 3 4 17 , 1 6 4 5 3 4 17 18 5 →= 1 2 6 4 5 3 4 5 , 1 6 4 5 3 4 17 18 17 →= 1 2 6 4 5 3 4 17 , 31 6 4 5 3 4 17 18 5 →= 31 2 6 4 5 3 4 5 , 31 6 4 5 3 4 17 18 17 →= 31 2 6 4 5 3 4 17 , 32 6 4 5 3 4 17 18 5 →= 32 2 6 4 5 3 4 5 , 32 6 4 5 3 4 17 18 17 →= 32 2 6 4 5 3 4 17 , 33 6 4 5 3 4 17 18 5 →= 33 2 6 4 5 3 4 5 , 33 6 4 5 3 4 17 18 17 →= 33 2 6 4 5 3 4 17 , 16 17 18 5 3 4 17 18 5 →= 16 5 3 4 5 3 4 5 , 16 17 18 5 3 4 17 18 17 →= 16 5 3 4 5 3 4 17 , 19 17 18 5 3 4 17 18 5 →= 19 5 3 4 5 3 4 5 , 19 17 18 5 3 4 17 18 17 →= 19 5 3 4 5 3 4 17 , 4 17 18 5 3 4 17 18 5 →= 4 5 3 4 5 3 4 5 , 4 17 18 5 3 4 17 18 17 →= 4 5 3 4 5 3 4 17 , 20 17 18 5 3 4 17 18 5 →= 20 5 3 4 5 3 4 5 , 20 17 18 5 3 4 17 18 17 →= 20 5 3 4 5 3 4 17 , 21 17 18 5 3 4 17 18 5 →= 21 5 3 4 5 3 4 5 , 21 17 18 5 3 4 17 18 17 →= 21 5 3 4 5 3 4 17 , 22 17 18 5 3 4 17 18 5 →= 22 5 3 4 5 3 4 5 , 22 17 18 5 3 4 17 18 17 →= 22 5 3 4 5 3 4 17 , 18 17 18 5 3 4 17 18 5 →= 18 5 3 4 5 3 4 5 , 18 17 18 5 3 4 17 18 17 →= 18 5 3 4 5 3 4 17 , 23 24 18 5 3 4 17 18 5 →= 23 0 3 4 5 3 4 5 , 23 24 18 5 3 4 17 18 17 →= 23 0 3 4 5 3 4 17 } The system is trivially terminating.