/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 3-rule system { 0 0 0 0 ⟶ 1 1 0 0 , 1 1 1 1 ⟶ 0 1 1 1 , 1 0 1 0 ⟶ 0 0 1 1 } Applying sparse tiling TRFC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (0,0) ↦ 0, (0,1) ↦ 1, (1,1) ↦ 2, (1,0) ↦ 3, (0,3) ↦ 4, (2,0) ↦ 5, (2,1) ↦ 6, (1,3) ↦ 7 }, it remains to prove termination of the 27-rule system { 0 0 0 0 0 ⟶ 1 2 3 0 0 , 0 0 0 0 1 ⟶ 1 2 3 0 1 , 0 0 0 0 4 ⟶ 1 2 3 0 4 , 3 0 0 0 0 ⟶ 2 2 3 0 0 , 3 0 0 0 1 ⟶ 2 2 3 0 1 , 3 0 0 0 4 ⟶ 2 2 3 0 4 , 5 0 0 0 0 ⟶ 6 2 3 0 0 , 5 0 0 0 1 ⟶ 6 2 3 0 1 , 5 0 0 0 4 ⟶ 6 2 3 0 4 , 1 2 2 2 3 ⟶ 0 1 2 2 3 , 1 2 2 2 2 ⟶ 0 1 2 2 2 , 1 2 2 2 7 ⟶ 0 1 2 2 7 , 2 2 2 2 3 ⟶ 3 1 2 2 3 , 2 2 2 2 2 ⟶ 3 1 2 2 2 , 2 2 2 2 7 ⟶ 3 1 2 2 7 , 6 2 2 2 3 ⟶ 5 1 2 2 3 , 6 2 2 2 2 ⟶ 5 1 2 2 2 , 6 2 2 2 7 ⟶ 5 1 2 2 7 , 1 3 1 3 0 ⟶ 0 0 1 2 3 , 1 3 1 3 1 ⟶ 0 0 1 2 2 , 1 3 1 3 4 ⟶ 0 0 1 2 7 , 2 3 1 3 0 ⟶ 3 0 1 2 3 , 2 3 1 3 1 ⟶ 3 0 1 2 2 , 2 3 1 3 4 ⟶ 3 0 1 2 7 , 6 3 1 3 0 ⟶ 5 0 1 2 3 , 6 3 1 3 1 ⟶ 5 0 1 2 2 , 6 3 1 3 4 ⟶ 5 0 1 2 7 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7 }, it remains to prove termination of the 24-rule system { 0 0 0 0 0 ⟶ 1 2 3 0 0 , 0 0 0 0 1 ⟶ 1 2 3 0 1 , 0 0 0 0 4 ⟶ 1 2 3 0 4 , 3 0 0 0 0 ⟶ 2 2 3 0 0 , 3 0 0 0 1 ⟶ 2 2 3 0 1 , 3 0 0 0 4 ⟶ 2 2 3 0 4 , 5 0 0 0 0 ⟶ 6 2 3 0 0 , 5 0 0 0 1 ⟶ 6 2 3 0 1 , 5 0 0 0 4 ⟶ 6 2 3 0 4 , 1 2 2 2 3 ⟶ 0 1 2 2 3 , 1 2 2 2 2 ⟶ 0 1 2 2 2 , 1 2 2 2 7 ⟶ 0 1 2 2 7 , 2 2 2 2 3 ⟶ 3 1 2 2 3 , 2 2 2 2 2 ⟶ 3 1 2 2 2 , 2 2 2 2 7 ⟶ 3 1 2 2 7 , 6 2 2 2 3 ⟶ 5 1 2 2 3 , 6 2 2 2 2 ⟶ 5 1 2 2 2 , 6 2 2 2 7 ⟶ 5 1 2 2 7 , 1 3 1 3 0 ⟶ 0 0 1 2 3 , 1 3 1 3 1 ⟶ 0 0 1 2 2 , 2 3 1 3 0 ⟶ 3 0 1 2 3 , 2 3 1 3 1 ⟶ 3 0 1 2 2 , 6 3 1 3 0 ⟶ 5 0 1 2 3 , 6 3 1 3 1 ⟶ 5 0 1 2 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 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 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 1 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 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 23-rule system { 0 0 0 0 0 ⟶ 1 2 3 0 0 , 0 0 0 0 1 ⟶ 1 2 3 0 1 , 3 0 0 0 0 ⟶ 2 2 3 0 0 , 3 0 0 0 1 ⟶ 2 2 3 0 1 , 3 0 0 0 4 ⟶ 2 2 3 0 4 , 5 0 0 0 0 ⟶ 6 2 3 0 0 , 5 0 0 0 1 ⟶ 6 2 3 0 1 , 5 0 0 0 4 ⟶ 6 2 3 0 4 , 1 2 2 2 3 ⟶ 0 1 2 2 3 , 1 2 2 2 2 ⟶ 0 1 2 2 2 , 1 2 2 2 7 ⟶ 0 1 2 2 7 , 2 2 2 2 3 ⟶ 3 1 2 2 3 , 2 2 2 2 2 ⟶ 3 1 2 2 2 , 2 2 2 2 7 ⟶ 3 1 2 2 7 , 6 2 2 2 3 ⟶ 5 1 2 2 3 , 6 2 2 2 2 ⟶ 5 1 2 2 2 , 6 2 2 2 7 ⟶ 5 1 2 2 7 , 1 3 1 3 0 ⟶ 0 0 1 2 3 , 1 3 1 3 1 ⟶ 0 0 1 2 2 , 2 3 1 3 0 ⟶ 3 0 1 2 3 , 2 3 1 3 1 ⟶ 3 0 1 2 2 , 6 3 1 3 0 ⟶ 5 0 1 2 3 , 6 3 1 3 1 ⟶ 5 0 1 2 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 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 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 1 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 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 22-rule system { 0 0 0 0 0 ⟶ 1 2 3 0 0 , 0 0 0 0 1 ⟶ 1 2 3 0 1 , 3 0 0 0 0 ⟶ 2 2 3 0 0 , 3 0 0 0 1 ⟶ 2 2 3 0 1 , 3 0 0 0 4 ⟶ 2 2 3 0 4 , 5 0 0 0 0 ⟶ 6 2 3 0 0 , 5 0 0 0 1 ⟶ 6 2 3 0 1 , 1 2 2 2 3 ⟶ 0 1 2 2 3 , 1 2 2 2 2 ⟶ 0 1 2 2 2 , 1 2 2 2 7 ⟶ 0 1 2 2 7 , 2 2 2 2 3 ⟶ 3 1 2 2 3 , 2 2 2 2 2 ⟶ 3 1 2 2 2 , 2 2 2 2 7 ⟶ 3 1 2 2 7 , 6 2 2 2 3 ⟶ 5 1 2 2 3 , 6 2 2 2 2 ⟶ 5 1 2 2 2 , 6 2 2 2 7 ⟶ 5 1 2 2 7 , 1 3 1 3 0 ⟶ 0 0 1 2 3 , 1 3 1 3 1 ⟶ 0 0 1 2 2 , 2 3 1 3 0 ⟶ 3 0 1 2 3 , 2 3 1 3 1 ⟶ 3 0 1 2 2 , 6 3 1 3 0 ⟶ 5 0 1 2 3 , 6 3 1 3 1 ⟶ 5 0 1 2 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 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 1 0 0 ⎟ ⎜ 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎜ 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 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎝ ⎠ 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 21-rule system { 0 0 0 0 0 ⟶ 1 2 3 0 0 , 0 0 0 0 1 ⟶ 1 2 3 0 1 , 3 0 0 0 0 ⟶ 2 2 3 0 0 , 3 0 0 0 1 ⟶ 2 2 3 0 1 , 3 0 0 0 4 ⟶ 2 2 3 0 4 , 5 0 0 0 0 ⟶ 6 2 3 0 0 , 5 0 0 0 1 ⟶ 6 2 3 0 1 , 1 2 2 2 3 ⟶ 0 1 2 2 3 , 1 2 2 2 2 ⟶ 0 1 2 2 2 , 1 2 2 2 7 ⟶ 0 1 2 2 7 , 2 2 2 2 3 ⟶ 3 1 2 2 3 , 2 2 2 2 2 ⟶ 3 1 2 2 2 , 2 2 2 2 7 ⟶ 3 1 2 2 7 , 6 2 2 2 3 ⟶ 5 1 2 2 3 , 6 2 2 2 2 ⟶ 5 1 2 2 2 , 1 3 1 3 0 ⟶ 0 0 1 2 3 , 1 3 1 3 1 ⟶ 0 0 1 2 2 , 2 3 1 3 0 ⟶ 3 0 1 2 3 , 2 3 1 3 1 ⟶ 3 0 1 2 2 , 6 3 1 3 0 ⟶ 5 0 1 2 3 , 6 3 1 3 1 ⟶ 5 0 1 2 2 } Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols. After renaming modulo the bijection { (0,↑) ↦ 0, (0,↓) ↦ 1, (1,↑) ↦ 2, (2,↓) ↦ 3, (3,↓) ↦ 4, (2,↑) ↦ 5, (3,↑) ↦ 6, (1,↓) ↦ 7, (4,↓) ↦ 8, (5,↑) ↦ 9, (6,↑) ↦ 10, (7,↓) ↦ 11, (5,↓) ↦ 12, (6,↓) ↦ 13 }, it remains to prove termination of the 88-rule system { 0 1 1 1 1 ⟶ 2 3 4 1 1 , 0 1 1 1 1 ⟶ 5 4 1 1 , 0 1 1 1 1 ⟶ 6 1 1 , 0 1 1 1 7 ⟶ 2 3 4 1 7 , 0 1 1 1 7 ⟶ 5 4 1 7 , 0 1 1 1 7 ⟶ 6 1 7 , 6 1 1 1 1 ⟶ 5 3 4 1 1 , 6 1 1 1 1 ⟶ 5 4 1 1 , 6 1 1 1 1 ⟶ 6 1 1 , 6 1 1 1 7 ⟶ 5 3 4 1 7 , 6 1 1 1 7 ⟶ 5 4 1 7 , 6 1 1 1 7 ⟶ 6 1 7 , 6 1 1 1 8 ⟶ 5 3 4 1 8 , 6 1 1 1 8 ⟶ 5 4 1 8 , 6 1 1 1 8 ⟶ 6 1 8 , 9 1 1 1 1 ⟶ 10 3 4 1 1 , 9 1 1 1 1 ⟶ 5 4 1 1 , 9 1 1 1 1 ⟶ 6 1 1 , 9 1 1 1 7 ⟶ 10 3 4 1 7 , 9 1 1 1 7 ⟶ 5 4 1 7 , 9 1 1 1 7 ⟶ 6 1 7 , 2 3 3 3 4 ⟶ 0 7 3 3 4 , 2 3 3 3 4 ⟶ 2 3 3 4 , 2 3 3 3 3 ⟶ 0 7 3 3 3 , 2 3 3 3 3 ⟶ 2 3 3 3 , 2 3 3 3 11 ⟶ 0 7 3 3 11 , 2 3 3 3 11 ⟶ 2 3 3 11 , 5 3 3 3 4 ⟶ 6 7 3 3 4 , 5 3 3 3 4 ⟶ 2 3 3 4 , 5 3 3 3 3 ⟶ 6 7 3 3 3 , 5 3 3 3 3 ⟶ 2 3 3 3 , 5 3 3 3 11 ⟶ 6 7 3 3 11 , 5 3 3 3 11 ⟶ 2 3 3 11 , 10 3 3 3 4 ⟶ 9 7 3 3 4 , 10 3 3 3 4 ⟶ 2 3 3 4 , 10 3 3 3 3 ⟶ 9 7 3 3 3 , 10 3 3 3 3 ⟶ 2 3 3 3 , 2 4 7 4 1 ⟶ 0 1 7 3 4 , 2 4 7 4 1 ⟶ 0 7 3 4 , 2 4 7 4 1 ⟶ 2 3 4 , 2 4 7 4 1 ⟶ 5 4 , 2 4 7 4 1 ⟶ 6 , 2 4 7 4 7 ⟶ 0 1 7 3 3 , 2 4 7 4 7 ⟶ 0 7 3 3 , 2 4 7 4 7 ⟶ 2 3 3 , 2 4 7 4 7 ⟶ 5 3 , 2 4 7 4 7 ⟶ 5 , 5 4 7 4 1 ⟶ 6 1 7 3 4 , 5 4 7 4 1 ⟶ 0 7 3 4 , 5 4 7 4 1 ⟶ 2 3 4 , 5 4 7 4 1 ⟶ 5 4 , 5 4 7 4 1 ⟶ 6 , 5 4 7 4 7 ⟶ 6 1 7 3 3 , 5 4 7 4 7 ⟶ 0 7 3 3 , 5 4 7 4 7 ⟶ 2 3 3 , 5 4 7 4 7 ⟶ 5 3 , 5 4 7 4 7 ⟶ 5 , 10 4 7 4 1 ⟶ 9 1 7 3 4 , 10 4 7 4 1 ⟶ 0 7 3 4 , 10 4 7 4 1 ⟶ 2 3 4 , 10 4 7 4 1 ⟶ 5 4 , 10 4 7 4 1 ⟶ 6 , 10 4 7 4 7 ⟶ 9 1 7 3 3 , 10 4 7 4 7 ⟶ 0 7 3 3 , 10 4 7 4 7 ⟶ 2 3 3 , 10 4 7 4 7 ⟶ 5 3 , 10 4 7 4 7 ⟶ 5 , 1 1 1 1 1 →= 7 3 4 1 1 , 1 1 1 1 7 →= 7 3 4 1 7 , 4 1 1 1 1 →= 3 3 4 1 1 , 4 1 1 1 7 →= 3 3 4 1 7 , 4 1 1 1 8 →= 3 3 4 1 8 , 12 1 1 1 1 →= 13 3 4 1 1 , 12 1 1 1 7 →= 13 3 4 1 7 , 7 3 3 3 4 →= 1 7 3 3 4 , 7 3 3 3 3 →= 1 7 3 3 3 , 7 3 3 3 11 →= 1 7 3 3 11 , 3 3 3 3 4 →= 4 7 3 3 4 , 3 3 3 3 3 →= 4 7 3 3 3 , 3 3 3 3 11 →= 4 7 3 3 11 , 13 3 3 3 4 →= 12 7 3 3 4 , 13 3 3 3 3 →= 12 7 3 3 3 , 7 4 7 4 1 →= 1 1 7 3 4 , 7 4 7 4 7 →= 1 1 7 3 3 , 3 4 7 4 1 →= 4 1 7 3 4 , 3 4 7 4 7 →= 4 1 7 3 3 , 13 4 7 4 1 →= 12 1 7 3 4 , 13 4 7 4 7 →= 12 1 7 3 3 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 7 ↦ 5, 6 ↦ 6, 5 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 42-rule system { 0 1 1 1 1 ⟶ 2 3 4 1 1 , 0 1 1 1 5 ⟶ 2 3 4 1 5 , 6 1 1 1 1 ⟶ 7 3 4 1 1 , 6 1 1 1 5 ⟶ 7 3 4 1 5 , 6 1 1 1 8 ⟶ 7 3 4 1 8 , 9 1 1 1 1 ⟶ 10 3 4 1 1 , 9 1 1 1 5 ⟶ 10 3 4 1 5 , 2 3 3 3 4 ⟶ 0 5 3 3 4 , 2 3 3 3 3 ⟶ 0 5 3 3 3 , 2 3 3 3 11 ⟶ 0 5 3 3 11 , 7 3 3 3 4 ⟶ 6 5 3 3 4 , 7 3 3 3 3 ⟶ 6 5 3 3 3 , 7 3 3 3 11 ⟶ 6 5 3 3 11 , 10 3 3 3 4 ⟶ 9 5 3 3 4 , 10 3 3 3 3 ⟶ 9 5 3 3 3 , 2 4 5 4 1 ⟶ 0 1 5 3 4 , 2 4 5 4 5 ⟶ 0 1 5 3 3 , 7 4 5 4 1 ⟶ 6 1 5 3 4 , 7 4 5 4 5 ⟶ 6 1 5 3 3 , 10 4 5 4 1 ⟶ 9 1 5 3 4 , 10 4 5 4 5 ⟶ 9 1 5 3 3 , 1 1 1 1 1 →= 5 3 4 1 1 , 1 1 1 1 5 →= 5 3 4 1 5 , 4 1 1 1 1 →= 3 3 4 1 1 , 4 1 1 1 5 →= 3 3 4 1 5 , 4 1 1 1 8 →= 3 3 4 1 8 , 12 1 1 1 1 →= 13 3 4 1 1 , 12 1 1 1 5 →= 13 3 4 1 5 , 5 3 3 3 4 →= 1 5 3 3 4 , 5 3 3 3 3 →= 1 5 3 3 3 , 5 3 3 3 11 →= 1 5 3 3 11 , 3 3 3 3 4 →= 4 5 3 3 4 , 3 3 3 3 3 →= 4 5 3 3 3 , 3 3 3 3 11 →= 4 5 3 3 11 , 13 3 3 3 4 →= 12 5 3 3 4 , 13 3 3 3 3 →= 12 5 3 3 3 , 5 4 5 4 1 →= 1 1 5 3 4 , 5 4 5 4 5 →= 1 1 5 3 3 , 3 4 5 4 1 →= 4 1 5 3 4 , 3 4 5 4 5 →= 4 1 5 3 3 , 13 4 5 4 1 →= 12 1 5 3 4 , 13 4 5 4 5 →= 12 1 5 3 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 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 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 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 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, 9 ↦ 8, 10 ↦ 9, 11 ↦ 10, 8 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 41-rule system { 0 1 1 1 1 ⟶ 2 3 4 1 1 , 0 1 1 1 5 ⟶ 2 3 4 1 5 , 6 1 1 1 1 ⟶ 7 3 4 1 1 , 6 1 1 1 5 ⟶ 7 3 4 1 5 , 8 1 1 1 1 ⟶ 9 3 4 1 1 , 8 1 1 1 5 ⟶ 9 3 4 1 5 , 2 3 3 3 4 ⟶ 0 5 3 3 4 , 2 3 3 3 3 ⟶ 0 5 3 3 3 , 2 3 3 3 10 ⟶ 0 5 3 3 10 , 7 3 3 3 4 ⟶ 6 5 3 3 4 , 7 3 3 3 3 ⟶ 6 5 3 3 3 , 7 3 3 3 10 ⟶ 6 5 3 3 10 , 9 3 3 3 4 ⟶ 8 5 3 3 4 , 9 3 3 3 3 ⟶ 8 5 3 3 3 , 2 4 5 4 1 ⟶ 0 1 5 3 4 , 2 4 5 4 5 ⟶ 0 1 5 3 3 , 7 4 5 4 1 ⟶ 6 1 5 3 4 , 7 4 5 4 5 ⟶ 6 1 5 3 3 , 9 4 5 4 1 ⟶ 8 1 5 3 4 , 9 4 5 4 5 ⟶ 8 1 5 3 3 , 1 1 1 1 1 →= 5 3 4 1 1 , 1 1 1 1 5 →= 5 3 4 1 5 , 4 1 1 1 1 →= 3 3 4 1 1 , 4 1 1 1 5 →= 3 3 4 1 5 , 4 1 1 1 11 →= 3 3 4 1 11 , 12 1 1 1 1 →= 13 3 4 1 1 , 12 1 1 1 5 →= 13 3 4 1 5 , 5 3 3 3 4 →= 1 5 3 3 4 , 5 3 3 3 3 →= 1 5 3 3 3 , 5 3 3 3 10 →= 1 5 3 3 10 , 3 3 3 3 4 →= 4 5 3 3 4 , 3 3 3 3 3 →= 4 5 3 3 3 , 3 3 3 3 10 →= 4 5 3 3 10 , 13 3 3 3 4 →= 12 5 3 3 4 , 13 3 3 3 3 →= 12 5 3 3 3 , 5 4 5 4 1 →= 1 1 5 3 4 , 5 4 5 4 5 →= 1 1 5 3 3 , 3 4 5 4 1 →= 4 1 5 3 4 , 3 4 5 4 5 →= 4 1 5 3 3 , 13 4 5 4 1 →= 12 1 5 3 4 , 13 4 5 4 5 →= 12 1 5 3 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 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 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 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 40-rule system { 0 1 1 1 1 ⟶ 2 3 4 1 1 , 0 1 1 1 5 ⟶ 2 3 4 1 5 , 6 1 1 1 1 ⟶ 7 3 4 1 1 , 6 1 1 1 5 ⟶ 7 3 4 1 5 , 8 1 1 1 1 ⟶ 9 3 4 1 1 , 8 1 1 1 5 ⟶ 9 3 4 1 5 , 2 3 3 3 4 ⟶ 0 5 3 3 4 , 2 3 3 3 3 ⟶ 0 5 3 3 3 , 7 3 3 3 4 ⟶ 6 5 3 3 4 , 7 3 3 3 3 ⟶ 6 5 3 3 3 , 7 3 3 3 10 ⟶ 6 5 3 3 10 , 9 3 3 3 4 ⟶ 8 5 3 3 4 , 9 3 3 3 3 ⟶ 8 5 3 3 3 , 2 4 5 4 1 ⟶ 0 1 5 3 4 , 2 4 5 4 5 ⟶ 0 1 5 3 3 , 7 4 5 4 1 ⟶ 6 1 5 3 4 , 7 4 5 4 5 ⟶ 6 1 5 3 3 , 9 4 5 4 1 ⟶ 8 1 5 3 4 , 9 4 5 4 5 ⟶ 8 1 5 3 3 , 1 1 1 1 1 →= 5 3 4 1 1 , 1 1 1 1 5 →= 5 3 4 1 5 , 4 1 1 1 1 →= 3 3 4 1 1 , 4 1 1 1 5 →= 3 3 4 1 5 , 4 1 1 1 11 →= 3 3 4 1 11 , 12 1 1 1 1 →= 13 3 4 1 1 , 12 1 1 1 5 →= 13 3 4 1 5 , 5 3 3 3 4 →= 1 5 3 3 4 , 5 3 3 3 3 →= 1 5 3 3 3 , 5 3 3 3 10 →= 1 5 3 3 10 , 3 3 3 3 4 →= 4 5 3 3 4 , 3 3 3 3 3 →= 4 5 3 3 3 , 3 3 3 3 10 →= 4 5 3 3 10 , 13 3 3 3 4 →= 12 5 3 3 4 , 13 3 3 3 3 →= 12 5 3 3 3 , 5 4 5 4 1 →= 1 1 5 3 4 , 5 4 5 4 5 →= 1 1 5 3 3 , 3 4 5 4 1 →= 4 1 5 3 4 , 3 4 5 4 5 →= 4 1 5 3 3 , 13 4 5 4 1 →= 12 1 5 3 4 , 13 4 5 4 5 →= 12 1 5 3 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 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 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 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 11 ↦ 10, 12 ↦ 11, 13 ↦ 12, 10 ↦ 13 }, it remains to prove termination of the 39-rule system { 0 1 1 1 1 ⟶ 2 3 4 1 1 , 0 1 1 1 5 ⟶ 2 3 4 1 5 , 6 1 1 1 1 ⟶ 7 3 4 1 1 , 6 1 1 1 5 ⟶ 7 3 4 1 5 , 8 1 1 1 1 ⟶ 9 3 4 1 1 , 8 1 1 1 5 ⟶ 9 3 4 1 5 , 2 3 3 3 4 ⟶ 0 5 3 3 4 , 2 3 3 3 3 ⟶ 0 5 3 3 3 , 7 3 3 3 4 ⟶ 6 5 3 3 4 , 7 3 3 3 3 ⟶ 6 5 3 3 3 , 9 3 3 3 4 ⟶ 8 5 3 3 4 , 9 3 3 3 3 ⟶ 8 5 3 3 3 , 2 4 5 4 1 ⟶ 0 1 5 3 4 , 2 4 5 4 5 ⟶ 0 1 5 3 3 , 7 4 5 4 1 ⟶ 6 1 5 3 4 , 7 4 5 4 5 ⟶ 6 1 5 3 3 , 9 4 5 4 1 ⟶ 8 1 5 3 4 , 9 4 5 4 5 ⟶ 8 1 5 3 3 , 1 1 1 1 1 →= 5 3 4 1 1 , 1 1 1 1 5 →= 5 3 4 1 5 , 4 1 1 1 1 →= 3 3 4 1 1 , 4 1 1 1 5 →= 3 3 4 1 5 , 4 1 1 1 10 →= 3 3 4 1 10 , 11 1 1 1 1 →= 12 3 4 1 1 , 11 1 1 1 5 →= 12 3 4 1 5 , 5 3 3 3 4 →= 1 5 3 3 4 , 5 3 3 3 3 →= 1 5 3 3 3 , 5 3 3 3 13 →= 1 5 3 3 13 , 3 3 3 3 4 →= 4 5 3 3 4 , 3 3 3 3 3 →= 4 5 3 3 3 , 3 3 3 3 13 →= 4 5 3 3 13 , 12 3 3 3 4 →= 11 5 3 3 4 , 12 3 3 3 3 →= 11 5 3 3 3 , 5 4 5 4 1 →= 1 1 5 3 4 , 5 4 5 4 5 →= 1 1 5 3 3 , 3 4 5 4 1 →= 4 1 5 3 4 , 3 4 5 4 5 →= 4 1 5 3 3 , 12 4 5 4 1 →= 11 1 5 3 4 , 12 4 5 4 5 →= 11 1 5 3 3 } Applying sparse tiling TROC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (14,0) ↦ 0, (0,1) ↦ 1, (1,1) ↦ 2, (14,2) ↦ 3, (2,3) ↦ 4, (3,4) ↦ 5, (4,1) ↦ 6, (1,5) ↦ 7, (1,10) ↦ 8, (1,15) ↦ 9, (5,3) ↦ 10, (5,4) ↦ 11, (5,15) ↦ 12, (14,6) ↦ 13, (6,1) ↦ 14, (14,7) ↦ 15, (7,3) ↦ 16, (14,8) ↦ 17, (8,1) ↦ 18, (14,9) ↦ 19, (9,3) ↦ 20, (3,3) ↦ 21, (0,5) ↦ 22, (4,5) ↦ 23, (4,10) ↦ 24, (4,15) ↦ 25, (3,13) ↦ 26, (3,15) ↦ 27, (6,5) ↦ 28, (8,5) ↦ 29, (2,4) ↦ 30, (7,4) ↦ 31, (9,4) ↦ 32, (11,1) ↦ 33, (11,5) ↦ 34, (14,1) ↦ 35, (14,5) ↦ 36, (12,4) ↦ 37, (12,3) ↦ 38, (14,4) ↦ 39, (14,3) ↦ 40, (10,15) ↦ 41, (14,11) ↦ 42, (14,12) ↦ 43, (13,15) ↦ 44 }, it remains to prove termination of the 417-rule system { 0 1 2 2 2 2 ⟶ 3 4 5 6 2 2 , 0 1 2 2 2 7 ⟶ 3 4 5 6 2 7 , 0 1 2 2 2 8 ⟶ 3 4 5 6 2 8 , 0 1 2 2 2 9 ⟶ 3 4 5 6 2 9 , 0 1 2 2 7 10 ⟶ 3 4 5 6 7 10 , 0 1 2 2 7 11 ⟶ 3 4 5 6 7 11 , 0 1 2 2 7 12 ⟶ 3 4 5 6 7 12 , 13 14 2 2 2 2 ⟶ 15 16 5 6 2 2 , 13 14 2 2 2 7 ⟶ 15 16 5 6 2 7 , 13 14 2 2 2 8 ⟶ 15 16 5 6 2 8 , 13 14 2 2 2 9 ⟶ 15 16 5 6 2 9 , 13 14 2 2 7 10 ⟶ 15 16 5 6 7 10 , 13 14 2 2 7 11 ⟶ 15 16 5 6 7 11 , 13 14 2 2 7 12 ⟶ 15 16 5 6 7 12 , 17 18 2 2 2 2 ⟶ 19 20 5 6 2 2 , 17 18 2 2 2 7 ⟶ 19 20 5 6 2 7 , 17 18 2 2 2 8 ⟶ 19 20 5 6 2 8 , 17 18 2 2 2 9 ⟶ 19 20 5 6 2 9 , 17 18 2 2 7 10 ⟶ 19 20 5 6 7 10 , 17 18 2 2 7 11 ⟶ 19 20 5 6 7 11 , 17 18 2 2 7 12 ⟶ 19 20 5 6 7 12 , 3 4 21 21 5 6 ⟶ 0 22 10 21 5 6 , 3 4 21 21 5 23 ⟶ 0 22 10 21 5 23 , 3 4 21 21 5 24 ⟶ 0 22 10 21 5 24 , 3 4 21 21 5 25 ⟶ 0 22 10 21 5 25 , 3 4 21 21 21 21 ⟶ 0 22 10 21 21 21 , 3 4 21 21 21 5 ⟶ 0 22 10 21 21 5 , 3 4 21 21 21 26 ⟶ 0 22 10 21 21 26 , 3 4 21 21 21 27 ⟶ 0 22 10 21 21 27 , 15 16 21 21 5 6 ⟶ 13 28 10 21 5 6 , 15 16 21 21 5 23 ⟶ 13 28 10 21 5 23 , 15 16 21 21 5 24 ⟶ 13 28 10 21 5 24 , 15 16 21 21 5 25 ⟶ 13 28 10 21 5 25 , 15 16 21 21 21 21 ⟶ 13 28 10 21 21 21 , 15 16 21 21 21 5 ⟶ 13 28 10 21 21 5 , 15 16 21 21 21 26 ⟶ 13 28 10 21 21 26 , 15 16 21 21 21 27 ⟶ 13 28 10 21 21 27 , 19 20 21 21 5 6 ⟶ 17 29 10 21 5 6 , 19 20 21 21 5 23 ⟶ 17 29 10 21 5 23 , 19 20 21 21 5 24 ⟶ 17 29 10 21 5 24 , 19 20 21 21 5 25 ⟶ 17 29 10 21 5 25 , 19 20 21 21 21 21 ⟶ 17 29 10 21 21 21 , 19 20 21 21 21 5 ⟶ 17 29 10 21 21 5 , 19 20 21 21 21 26 ⟶ 17 29 10 21 21 26 , 19 20 21 21 21 27 ⟶ 17 29 10 21 21 27 , 3 30 23 11 6 2 ⟶ 0 1 7 10 5 6 , 3 30 23 11 6 7 ⟶ 0 1 7 10 5 23 , 3 30 23 11 6 8 ⟶ 0 1 7 10 5 24 , 3 30 23 11 6 9 ⟶ 0 1 7 10 5 25 , 3 30 23 11 23 10 ⟶ 0 1 7 10 21 21 , 3 30 23 11 23 11 ⟶ 0 1 7 10 21 5 , 3 30 23 11 23 12 ⟶ 0 1 7 10 21 27 , 15 31 23 11 6 2 ⟶ 13 14 7 10 5 6 , 15 31 23 11 6 7 ⟶ 13 14 7 10 5 23 , 15 31 23 11 6 8 ⟶ 13 14 7 10 5 24 , 15 31 23 11 6 9 ⟶ 13 14 7 10 5 25 , 15 31 23 11 23 10 ⟶ 13 14 7 10 21 21 , 15 31 23 11 23 11 ⟶ 13 14 7 10 21 5 , 15 31 23 11 23 12 ⟶ 13 14 7 10 21 27 , 19 32 23 11 6 2 ⟶ 17 18 7 10 5 6 , 19 32 23 11 6 7 ⟶ 17 18 7 10 5 23 , 19 32 23 11 6 8 ⟶ 17 18 7 10 5 24 , 19 32 23 11 6 9 ⟶ 17 18 7 10 5 25 , 19 32 23 11 23 10 ⟶ 17 18 7 10 21 21 , 19 32 23 11 23 11 ⟶ 17 18 7 10 21 5 , 19 32 23 11 23 12 ⟶ 17 18 7 10 21 27 , 1 2 2 2 2 2 →= 22 10 5 6 2 2 , 1 2 2 2 2 7 →= 22 10 5 6 2 7 , 1 2 2 2 2 8 →= 22 10 5 6 2 8 , 1 2 2 2 2 9 →= 22 10 5 6 2 9 , 2 2 2 2 2 2 →= 7 10 5 6 2 2 , 2 2 2 2 2 7 →= 7 10 5 6 2 7 , 2 2 2 2 2 8 →= 7 10 5 6 2 8 , 2 2 2 2 2 9 →= 7 10 5 6 2 9 , 6 2 2 2 2 2 →= 23 10 5 6 2 2 , 6 2 2 2 2 7 →= 23 10 5 6 2 7 , 6 2 2 2 2 8 →= 23 10 5 6 2 8 , 6 2 2 2 2 9 →= 23 10 5 6 2 9 , 14 2 2 2 2 2 →= 28 10 5 6 2 2 , 14 2 2 2 2 7 →= 28 10 5 6 2 7 , 14 2 2 2 2 8 →= 28 10 5 6 2 8 , 14 2 2 2 2 9 →= 28 10 5 6 2 9 , 18 2 2 2 2 2 →= 29 10 5 6 2 2 , 18 2 2 2 2 7 →= 29 10 5 6 2 7 , 18 2 2 2 2 8 →= 29 10 5 6 2 8 , 18 2 2 2 2 9 →= 29 10 5 6 2 9 , 33 2 2 2 2 2 →= 34 10 5 6 2 2 , 33 2 2 2 2 7 →= 34 10 5 6 2 7 , 33 2 2 2 2 8 →= 34 10 5 6 2 8 , 33 2 2 2 2 9 →= 34 10 5 6 2 9 , 35 2 2 2 2 2 →= 36 10 5 6 2 2 , 35 2 2 2 2 7 →= 36 10 5 6 2 7 , 35 2 2 2 2 8 →= 36 10 5 6 2 8 , 35 2 2 2 2 9 →= 36 10 5 6 2 9 , 1 2 2 2 7 10 →= 22 10 5 6 7 10 , 1 2 2 2 7 11 →= 22 10 5 6 7 11 , 1 2 2 2 7 12 →= 22 10 5 6 7 12 , 2 2 2 2 7 10 →= 7 10 5 6 7 10 , 2 2 2 2 7 11 →= 7 10 5 6 7 11 , 2 2 2 2 7 12 →= 7 10 5 6 7 12 , 6 2 2 2 7 10 →= 23 10 5 6 7 10 , 6 2 2 2 7 11 →= 23 10 5 6 7 11 , 6 2 2 2 7 12 →= 23 10 5 6 7 12 , 14 2 2 2 7 10 →= 28 10 5 6 7 10 , 14 2 2 2 7 11 →= 28 10 5 6 7 11 , 14 2 2 2 7 12 →= 28 10 5 6 7 12 , 18 2 2 2 7 10 →= 29 10 5 6 7 10 , 18 2 2 2 7 11 →= 29 10 5 6 7 11 , 18 2 2 2 7 12 →= 29 10 5 6 7 12 , 33 2 2 2 7 10 →= 34 10 5 6 7 10 , 33 2 2 2 7 11 →= 34 10 5 6 7 11 , 33 2 2 2 7 12 →= 34 10 5 6 7 12 , 35 2 2 2 7 10 →= 36 10 5 6 7 10 , 35 2 2 2 7 11 →= 36 10 5 6 7 11 , 35 2 2 2 7 12 →= 36 10 5 6 7 12 , 30 6 2 2 2 2 →= 4 21 5 6 2 2 , 30 6 2 2 2 7 →= 4 21 5 6 2 7 , 30 6 2 2 2 8 →= 4 21 5 6 2 8 , 30 6 2 2 2 9 →= 4 21 5 6 2 9 , 5 6 2 2 2 2 →= 21 21 5 6 2 2 , 5 6 2 2 2 7 →= 21 21 5 6 2 7 , 5 6 2 2 2 8 →= 21 21 5 6 2 8 , 5 6 2 2 2 9 →= 21 21 5 6 2 9 , 11 6 2 2 2 2 →= 10 21 5 6 2 2 , 11 6 2 2 2 7 →= 10 21 5 6 2 7 , 11 6 2 2 2 8 →= 10 21 5 6 2 8 , 11 6 2 2 2 9 →= 10 21 5 6 2 9 , 31 6 2 2 2 2 →= 16 21 5 6 2 2 , 31 6 2 2 2 7 →= 16 21 5 6 2 7 , 31 6 2 2 2 8 →= 16 21 5 6 2 8 , 31 6 2 2 2 9 →= 16 21 5 6 2 9 , 32 6 2 2 2 2 →= 20 21 5 6 2 2 , 32 6 2 2 2 7 →= 20 21 5 6 2 7 , 32 6 2 2 2 8 →= 20 21 5 6 2 8 , 32 6 2 2 2 9 →= 20 21 5 6 2 9 , 37 6 2 2 2 2 →= 38 21 5 6 2 2 , 37 6 2 2 2 7 →= 38 21 5 6 2 7 , 37 6 2 2 2 8 →= 38 21 5 6 2 8 , 37 6 2 2 2 9 →= 38 21 5 6 2 9 , 39 6 2 2 2 2 →= 40 21 5 6 2 2 , 39 6 2 2 2 7 →= 40 21 5 6 2 7 , 39 6 2 2 2 8 →= 40 21 5 6 2 8 , 39 6 2 2 2 9 →= 40 21 5 6 2 9 , 30 6 2 2 7 10 →= 4 21 5 6 7 10 , 30 6 2 2 7 11 →= 4 21 5 6 7 11 , 30 6 2 2 7 12 →= 4 21 5 6 7 12 , 5 6 2 2 7 10 →= 21 21 5 6 7 10 , 5 6 2 2 7 11 →= 21 21 5 6 7 11 , 5 6 2 2 7 12 →= 21 21 5 6 7 12 , 11 6 2 2 7 10 →= 10 21 5 6 7 10 , 11 6 2 2 7 11 →= 10 21 5 6 7 11 , 11 6 2 2 7 12 →= 10 21 5 6 7 12 , 31 6 2 2 7 10 →= 16 21 5 6 7 10 , 31 6 2 2 7 11 →= 16 21 5 6 7 11 , 31 6 2 2 7 12 →= 16 21 5 6 7 12 , 32 6 2 2 7 10 →= 20 21 5 6 7 10 , 32 6 2 2 7 11 →= 20 21 5 6 7 11 , 32 6 2 2 7 12 →= 20 21 5 6 7 12 , 37 6 2 2 7 10 →= 38 21 5 6 7 10 , 37 6 2 2 7 11 →= 38 21 5 6 7 11 , 37 6 2 2 7 12 →= 38 21 5 6 7 12 , 39 6 2 2 7 10 →= 40 21 5 6 7 10 , 39 6 2 2 7 11 →= 40 21 5 6 7 11 , 39 6 2 2 7 12 →= 40 21 5 6 7 12 , 30 6 2 2 8 41 →= 4 21 5 6 8 41 , 5 6 2 2 8 41 →= 21 21 5 6 8 41 , 11 6 2 2 8 41 →= 10 21 5 6 8 41 , 31 6 2 2 8 41 →= 16 21 5 6 8 41 , 32 6 2 2 8 41 →= 20 21 5 6 8 41 , 37 6 2 2 8 41 →= 38 21 5 6 8 41 , 39 6 2 2 8 41 →= 40 21 5 6 8 41 , 42 33 2 2 2 2 →= 43 38 5 6 2 2 , 42 33 2 2 2 7 →= 43 38 5 6 2 7 , 42 33 2 2 2 8 →= 43 38 5 6 2 8 , 42 33 2 2 2 9 →= 43 38 5 6 2 9 , 42 33 2 2 7 10 →= 43 38 5 6 7 10 , 42 33 2 2 7 11 →= 43 38 5 6 7 11 , 42 33 2 2 7 12 →= 43 38 5 6 7 12 , 22 10 21 21 5 6 →= 1 7 10 21 5 6 , 22 10 21 21 5 23 →= 1 7 10 21 5 23 , 22 10 21 21 5 24 →= 1 7 10 21 5 24 , 22 10 21 21 5 25 →= 1 7 10 21 5 25 , 7 10 21 21 5 6 →= 2 7 10 21 5 6 , 7 10 21 21 5 23 →= 2 7 10 21 5 23 , 7 10 21 21 5 24 →= 2 7 10 21 5 24 , 7 10 21 21 5 25 →= 2 7 10 21 5 25 , 23 10 21 21 5 6 →= 6 7 10 21 5 6 , 23 10 21 21 5 23 →= 6 7 10 21 5 23 , 23 10 21 21 5 24 →= 6 7 10 21 5 24 , 23 10 21 21 5 25 →= 6 7 10 21 5 25 , 28 10 21 21 5 6 →= 14 7 10 21 5 6 , 28 10 21 21 5 23 →= 14 7 10 21 5 23 , 28 10 21 21 5 24 →= 14 7 10 21 5 24 , 28 10 21 21 5 25 →= 14 7 10 21 5 25 , 29 10 21 21 5 6 →= 18 7 10 21 5 6 , 29 10 21 21 5 23 →= 18 7 10 21 5 23 , 29 10 21 21 5 24 →= 18 7 10 21 5 24 , 29 10 21 21 5 25 →= 18 7 10 21 5 25 , 34 10 21 21 5 6 →= 33 7 10 21 5 6 , 34 10 21 21 5 23 →= 33 7 10 21 5 23 , 34 10 21 21 5 24 →= 33 7 10 21 5 24 , 34 10 21 21 5 25 →= 33 7 10 21 5 25 , 36 10 21 21 5 6 →= 35 7 10 21 5 6 , 36 10 21 21 5 23 →= 35 7 10 21 5 23 , 36 10 21 21 5 24 →= 35 7 10 21 5 24 , 36 10 21 21 5 25 →= 35 7 10 21 5 25 , 22 10 21 21 21 21 →= 1 7 10 21 21 21 , 22 10 21 21 21 5 →= 1 7 10 21 21 5 , 22 10 21 21 21 26 →= 1 7 10 21 21 26 , 22 10 21 21 21 27 →= 1 7 10 21 21 27 , 7 10 21 21 21 21 →= 2 7 10 21 21 21 , 7 10 21 21 21 5 →= 2 7 10 21 21 5 , 7 10 21 21 21 26 →= 2 7 10 21 21 26 , 7 10 21 21 21 27 →= 2 7 10 21 21 27 , 23 10 21 21 21 21 →= 6 7 10 21 21 21 , 23 10 21 21 21 5 →= 6 7 10 21 21 5 , 23 10 21 21 21 26 →= 6 7 10 21 21 26 , 23 10 21 21 21 27 →= 6 7 10 21 21 27 , 28 10 21 21 21 21 →= 14 7 10 21 21 21 , 28 10 21 21 21 5 →= 14 7 10 21 21 5 , 28 10 21 21 21 26 →= 14 7 10 21 21 26 , 28 10 21 21 21 27 →= 14 7 10 21 21 27 , 29 10 21 21 21 21 →= 18 7 10 21 21 21 , 29 10 21 21 21 5 →= 18 7 10 21 21 5 , 29 10 21 21 21 26 →= 18 7 10 21 21 26 , 29 10 21 21 21 27 →= 18 7 10 21 21 27 , 34 10 21 21 21 21 →= 33 7 10 21 21 21 , 34 10 21 21 21 5 →= 33 7 10 21 21 5 , 34 10 21 21 21 26 →= 33 7 10 21 21 26 , 34 10 21 21 21 27 →= 33 7 10 21 21 27 , 36 10 21 21 21 21 →= 35 7 10 21 21 21 , 36 10 21 21 21 5 →= 35 7 10 21 21 5 , 36 10 21 21 21 26 →= 35 7 10 21 21 26 , 36 10 21 21 21 27 →= 35 7 10 21 21 27 , 22 10 21 21 26 44 →= 1 7 10 21 26 44 , 7 10 21 21 26 44 →= 2 7 10 21 26 44 , 23 10 21 21 26 44 →= 6 7 10 21 26 44 , 28 10 21 21 26 44 →= 14 7 10 21 26 44 , 29 10 21 21 26 44 →= 18 7 10 21 26 44 , 34 10 21 21 26 44 →= 33 7 10 21 26 44 , 36 10 21 21 26 44 →= 35 7 10 21 26 44 , 4 21 21 21 5 6 →= 30 23 10 21 5 6 , 4 21 21 21 5 23 →= 30 23 10 21 5 23 , 4 21 21 21 5 24 →= 30 23 10 21 5 24 , 4 21 21 21 5 25 →= 30 23 10 21 5 25 , 21 21 21 21 5 6 →= 5 23 10 21 5 6 , 21 21 21 21 5 23 →= 5 23 10 21 5 23 , 21 21 21 21 5 24 →= 5 23 10 21 5 24 , 21 21 21 21 5 25 →= 5 23 10 21 5 25 , 10 21 21 21 5 6 →= 11 23 10 21 5 6 , 10 21 21 21 5 23 →= 11 23 10 21 5 23 , 10 21 21 21 5 24 →= 11 23 10 21 5 24 , 10 21 21 21 5 25 →= 11 23 10 21 5 25 , 16 21 21 21 5 6 →= 31 23 10 21 5 6 , 16 21 21 21 5 23 →= 31 23 10 21 5 23 , 16 21 21 21 5 24 →= 31 23 10 21 5 24 , 16 21 21 21 5 25 →= 31 23 10 21 5 25 , 20 21 21 21 5 6 →= 32 23 10 21 5 6 , 20 21 21 21 5 23 →= 32 23 10 21 5 23 , 20 21 21 21 5 24 →= 32 23 10 21 5 24 , 20 21 21 21 5 25 →= 32 23 10 21 5 25 , 38 21 21 21 5 6 →= 37 23 10 21 5 6 , 38 21 21 21 5 23 →= 37 23 10 21 5 23 , 38 21 21 21 5 24 →= 37 23 10 21 5 24 , 38 21 21 21 5 25 →= 37 23 10 21 5 25 , 40 21 21 21 5 6 →= 39 23 10 21 5 6 , 40 21 21 21 5 23 →= 39 23 10 21 5 23 , 40 21 21 21 5 24 →= 39 23 10 21 5 24 , 40 21 21 21 5 25 →= 39 23 10 21 5 25 , 4 21 21 21 21 21 →= 30 23 10 21 21 21 , 4 21 21 21 21 5 →= 30 23 10 21 21 5 , 4 21 21 21 21 26 →= 30 23 10 21 21 26 , 4 21 21 21 21 27 →= 30 23 10 21 21 27 , 21 21 21 21 21 21 →= 5 23 10 21 21 21 , 21 21 21 21 21 5 →= 5 23 10 21 21 5 , 21 21 21 21 21 26 →= 5 23 10 21 21 26 , 21 21 21 21 21 27 →= 5 23 10 21 21 27 , 10 21 21 21 21 21 →= 11 23 10 21 21 21 , 10 21 21 21 21 5 →= 11 23 10 21 21 5 , 10 21 21 21 21 26 →= 11 23 10 21 21 26 , 10 21 21 21 21 27 →= 11 23 10 21 21 27 , 16 21 21 21 21 21 →= 31 23 10 21 21 21 , 16 21 21 21 21 5 →= 31 23 10 21 21 5 , 16 21 21 21 21 26 →= 31 23 10 21 21 26 , 16 21 21 21 21 27 →= 31 23 10 21 21 27 , 20 21 21 21 21 21 →= 32 23 10 21 21 21 , 20 21 21 21 21 5 →= 32 23 10 21 21 5 , 20 21 21 21 21 26 →= 32 23 10 21 21 26 , 20 21 21 21 21 27 →= 32 23 10 21 21 27 , 38 21 21 21 21 21 →= 37 23 10 21 21 21 , 38 21 21 21 21 5 →= 37 23 10 21 21 5 , 38 21 21 21 21 26 →= 37 23 10 21 21 26 , 38 21 21 21 21 27 →= 37 23 10 21 21 27 , 40 21 21 21 21 21 →= 39 23 10 21 21 21 , 40 21 21 21 21 5 →= 39 23 10 21 21 5 , 40 21 21 21 21 26 →= 39 23 10 21 21 26 , 40 21 21 21 21 27 →= 39 23 10 21 21 27 , 4 21 21 21 26 44 →= 30 23 10 21 26 44 , 21 21 21 21 26 44 →= 5 23 10 21 26 44 , 10 21 21 21 26 44 →= 11 23 10 21 26 44 , 16 21 21 21 26 44 →= 31 23 10 21 26 44 , 20 21 21 21 26 44 →= 32 23 10 21 26 44 , 38 21 21 21 26 44 →= 37 23 10 21 26 44 , 40 21 21 21 26 44 →= 39 23 10 21 26 44 , 43 38 21 21 5 6 →= 42 34 10 21 5 6 , 43 38 21 21 5 23 →= 42 34 10 21 5 23 , 43 38 21 21 5 24 →= 42 34 10 21 5 24 , 43 38 21 21 5 25 →= 42 34 10 21 5 25 , 43 38 21 21 21 21 →= 42 34 10 21 21 21 , 43 38 21 21 21 5 →= 42 34 10 21 21 5 , 43 38 21 21 21 26 →= 42 34 10 21 21 26 , 43 38 21 21 21 27 →= 42 34 10 21 21 27 , 22 11 23 11 6 2 →= 1 2 7 10 5 6 , 22 11 23 11 6 7 →= 1 2 7 10 5 23 , 22 11 23 11 6 8 →= 1 2 7 10 5 24 , 22 11 23 11 6 9 →= 1 2 7 10 5 25 , 7 11 23 11 6 2 →= 2 2 7 10 5 6 , 7 11 23 11 6 7 →= 2 2 7 10 5 23 , 7 11 23 11 6 8 →= 2 2 7 10 5 24 , 7 11 23 11 6 9 →= 2 2 7 10 5 25 , 23 11 23 11 6 2 →= 6 2 7 10 5 6 , 23 11 23 11 6 7 →= 6 2 7 10 5 23 , 23 11 23 11 6 8 →= 6 2 7 10 5 24 , 23 11 23 11 6 9 →= 6 2 7 10 5 25 , 28 11 23 11 6 2 →= 14 2 7 10 5 6 , 28 11 23 11 6 7 →= 14 2 7 10 5 23 , 28 11 23 11 6 8 →= 14 2 7 10 5 24 , 28 11 23 11 6 9 →= 14 2 7 10 5 25 , 29 11 23 11 6 2 →= 18 2 7 10 5 6 , 29 11 23 11 6 7 →= 18 2 7 10 5 23 , 29 11 23 11 6 8 →= 18 2 7 10 5 24 , 29 11 23 11 6 9 →= 18 2 7 10 5 25 , 34 11 23 11 6 2 →= 33 2 7 10 5 6 , 34 11 23 11 6 7 →= 33 2 7 10 5 23 , 34 11 23 11 6 8 →= 33 2 7 10 5 24 , 34 11 23 11 6 9 →= 33 2 7 10 5 25 , 36 11 23 11 6 2 →= 35 2 7 10 5 6 , 36 11 23 11 6 7 →= 35 2 7 10 5 23 , 36 11 23 11 6 8 →= 35 2 7 10 5 24 , 36 11 23 11 6 9 →= 35 2 7 10 5 25 , 22 11 23 11 23 10 →= 1 2 7 10 21 21 , 22 11 23 11 23 11 →= 1 2 7 10 21 5 , 22 11 23 11 23 12 →= 1 2 7 10 21 27 , 7 11 23 11 23 10 →= 2 2 7 10 21 21 , 7 11 23 11 23 11 →= 2 2 7 10 21 5 , 7 11 23 11 23 12 →= 2 2 7 10 21 27 , 23 11 23 11 23 10 →= 6 2 7 10 21 21 , 23 11 23 11 23 11 →= 6 2 7 10 21 5 , 23 11 23 11 23 12 →= 6 2 7 10 21 27 , 28 11 23 11 23 10 →= 14 2 7 10 21 21 , 28 11 23 11 23 11 →= 14 2 7 10 21 5 , 28 11 23 11 23 12 →= 14 2 7 10 21 27 , 29 11 23 11 23 10 →= 18 2 7 10 21 21 , 29 11 23 11 23 11 →= 18 2 7 10 21 5 , 29 11 23 11 23 12 →= 18 2 7 10 21 27 , 34 11 23 11 23 10 →= 33 2 7 10 21 21 , 34 11 23 11 23 11 →= 33 2 7 10 21 5 , 34 11 23 11 23 12 →= 33 2 7 10 21 27 , 36 11 23 11 23 10 →= 35 2 7 10 21 21 , 36 11 23 11 23 11 →= 35 2 7 10 21 5 , 36 11 23 11 23 12 →= 35 2 7 10 21 27 , 4 5 23 11 6 2 →= 30 6 7 10 5 6 , 4 5 23 11 6 7 →= 30 6 7 10 5 23 , 4 5 23 11 6 8 →= 30 6 7 10 5 24 , 4 5 23 11 6 9 →= 30 6 7 10 5 25 , 21 5 23 11 6 2 →= 5 6 7 10 5 6 , 21 5 23 11 6 7 →= 5 6 7 10 5 23 , 21 5 23 11 6 8 →= 5 6 7 10 5 24 , 21 5 23 11 6 9 →= 5 6 7 10 5 25 , 10 5 23 11 6 2 →= 11 6 7 10 5 6 , 10 5 23 11 6 7 →= 11 6 7 10 5 23 , 10 5 23 11 6 8 →= 11 6 7 10 5 24 , 10 5 23 11 6 9 →= 11 6 7 10 5 25 , 16 5 23 11 6 2 →= 31 6 7 10 5 6 , 16 5 23 11 6 7 →= 31 6 7 10 5 23 , 16 5 23 11 6 8 →= 31 6 7 10 5 24 , 16 5 23 11 6 9 →= 31 6 7 10 5 25 , 20 5 23 11 6 2 →= 32 6 7 10 5 6 , 20 5 23 11 6 7 →= 32 6 7 10 5 23 , 20 5 23 11 6 8 →= 32 6 7 10 5 24 , 20 5 23 11 6 9 →= 32 6 7 10 5 25 , 38 5 23 11 6 2 →= 37 6 7 10 5 6 , 38 5 23 11 6 7 →= 37 6 7 10 5 23 , 38 5 23 11 6 8 →= 37 6 7 10 5 24 , 38 5 23 11 6 9 →= 37 6 7 10 5 25 , 40 5 23 11 6 2 →= 39 6 7 10 5 6 , 40 5 23 11 6 7 →= 39 6 7 10 5 23 , 40 5 23 11 6 8 →= 39 6 7 10 5 24 , 40 5 23 11 6 9 →= 39 6 7 10 5 25 , 4 5 23 11 23 10 →= 30 6 7 10 21 21 , 4 5 23 11 23 11 →= 30 6 7 10 21 5 , 4 5 23 11 23 12 →= 30 6 7 10 21 27 , 21 5 23 11 23 10 →= 5 6 7 10 21 21 , 21 5 23 11 23 11 →= 5 6 7 10 21 5 , 21 5 23 11 23 12 →= 5 6 7 10 21 27 , 10 5 23 11 23 10 →= 11 6 7 10 21 21 , 10 5 23 11 23 11 →= 11 6 7 10 21 5 , 10 5 23 11 23 12 →= 11 6 7 10 21 27 , 16 5 23 11 23 10 →= 31 6 7 10 21 21 , 16 5 23 11 23 11 →= 31 6 7 10 21 5 , 16 5 23 11 23 12 →= 31 6 7 10 21 27 , 20 5 23 11 23 10 →= 32 6 7 10 21 21 , 20 5 23 11 23 11 →= 32 6 7 10 21 5 , 20 5 23 11 23 12 →= 32 6 7 10 21 27 , 38 5 23 11 23 10 →= 37 6 7 10 21 21 , 38 5 23 11 23 11 →= 37 6 7 10 21 5 , 38 5 23 11 23 12 →= 37 6 7 10 21 27 , 40 5 23 11 23 10 →= 39 6 7 10 21 21 , 40 5 23 11 23 11 →= 39 6 7 10 21 5 , 40 5 23 11 23 12 →= 39 6 7 10 21 27 , 43 37 23 11 6 2 →= 42 33 7 10 5 6 , 43 37 23 11 6 7 →= 42 33 7 10 5 23 , 43 37 23 11 6 8 →= 42 33 7 10 5 24 , 43 37 23 11 6 9 →= 42 33 7 10 5 25 , 43 37 23 11 23 10 →= 42 33 7 10 21 21 , 43 37 23 11 23 11 →= 42 33 7 10 21 5 , 43 37 23 11 23 12 →= 42 33 7 10 21 27 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 23 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 24 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 25 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 26 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 27 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 28 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 29 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 30 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 31 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 32 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 33 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 34 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 35 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 36 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 37 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 38 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 39 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 40 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 41 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 42 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 43 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 44 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22, 23 ↦ 23, 24 ↦ 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 }, it remains to prove termination of the 363-rule system { 0 1 2 2 2 2 ⟶ 3 4 5 6 2 2 , 0 1 2 2 2 7 ⟶ 3 4 5 6 2 7 , 0 1 2 2 2 8 ⟶ 3 4 5 6 2 8 , 0 1 2 2 2 9 ⟶ 3 4 5 6 2 9 , 0 1 2 2 7 10 ⟶ 3 4 5 6 7 10 , 0 1 2 2 7 11 ⟶ 3 4 5 6 7 11 , 0 1 2 2 7 12 ⟶ 3 4 5 6 7 12 , 13 14 2 2 2 2 ⟶ 15 16 5 6 2 2 , 13 14 2 2 2 7 ⟶ 15 16 5 6 2 7 , 13 14 2 2 2 8 ⟶ 15 16 5 6 2 8 , 13 14 2 2 2 9 ⟶ 15 16 5 6 2 9 , 13 14 2 2 7 10 ⟶ 15 16 5 6 7 10 , 13 14 2 2 7 11 ⟶ 15 16 5 6 7 11 , 13 14 2 2 7 12 ⟶ 15 16 5 6 7 12 , 17 18 2 2 2 2 ⟶ 19 20 5 6 2 2 , 17 18 2 2 2 7 ⟶ 19 20 5 6 2 7 , 17 18 2 2 2 8 ⟶ 19 20 5 6 2 8 , 17 18 2 2 2 9 ⟶ 19 20 5 6 2 9 , 17 18 2 2 7 10 ⟶ 19 20 5 6 7 10 , 17 18 2 2 7 11 ⟶ 19 20 5 6 7 11 , 17 18 2 2 7 12 ⟶ 19 20 5 6 7 12 , 3 4 21 21 5 6 ⟶ 0 22 10 21 5 6 , 3 4 21 21 5 23 ⟶ 0 22 10 21 5 23 , 3 4 21 21 5 24 ⟶ 0 22 10 21 5 24 , 3 4 21 21 5 25 ⟶ 0 22 10 21 5 25 , 3 4 21 21 21 21 ⟶ 0 22 10 21 21 21 , 3 4 21 21 21 5 ⟶ 0 22 10 21 21 5 , 3 4 21 21 21 26 ⟶ 0 22 10 21 21 26 , 3 4 21 21 21 27 ⟶ 0 22 10 21 21 27 , 15 16 21 21 5 6 ⟶ 13 28 10 21 5 6 , 15 16 21 21 5 23 ⟶ 13 28 10 21 5 23 , 15 16 21 21 5 24 ⟶ 13 28 10 21 5 24 , 15 16 21 21 5 25 ⟶ 13 28 10 21 5 25 , 15 16 21 21 21 21 ⟶ 13 28 10 21 21 21 , 15 16 21 21 21 5 ⟶ 13 28 10 21 21 5 , 15 16 21 21 21 26 ⟶ 13 28 10 21 21 26 , 15 16 21 21 21 27 ⟶ 13 28 10 21 21 27 , 19 20 21 21 5 6 ⟶ 17 29 10 21 5 6 , 19 20 21 21 5 23 ⟶ 17 29 10 21 5 23 , 19 20 21 21 5 24 ⟶ 17 29 10 21 5 24 , 19 20 21 21 5 25 ⟶ 17 29 10 21 5 25 , 19 20 21 21 21 21 ⟶ 17 29 10 21 21 21 , 19 20 21 21 21 5 ⟶ 17 29 10 21 21 5 , 19 20 21 21 21 26 ⟶ 17 29 10 21 21 26 , 19 20 21 21 21 27 ⟶ 17 29 10 21 21 27 , 3 30 23 11 6 2 ⟶ 0 1 7 10 5 6 , 3 30 23 11 6 7 ⟶ 0 1 7 10 5 23 , 3 30 23 11 23 10 ⟶ 0 1 7 10 21 21 , 3 30 23 11 23 11 ⟶ 0 1 7 10 21 5 , 15 31 23 11 6 2 ⟶ 13 14 7 10 5 6 , 15 31 23 11 6 7 ⟶ 13 14 7 10 5 23 , 15 31 23 11 23 10 ⟶ 13 14 7 10 21 21 , 15 31 23 11 23 11 ⟶ 13 14 7 10 21 5 , 19 32 23 11 6 2 ⟶ 17 18 7 10 5 6 , 19 32 23 11 6 7 ⟶ 17 18 7 10 5 23 , 19 32 23 11 23 10 ⟶ 17 18 7 10 21 21 , 19 32 23 11 23 11 ⟶ 17 18 7 10 21 5 , 1 2 2 2 2 2 →= 22 10 5 6 2 2 , 1 2 2 2 2 7 →= 22 10 5 6 2 7 , 1 2 2 2 2 8 →= 22 10 5 6 2 8 , 1 2 2 2 2 9 →= 22 10 5 6 2 9 , 2 2 2 2 2 2 →= 7 10 5 6 2 2 , 2 2 2 2 2 7 →= 7 10 5 6 2 7 , 2 2 2 2 2 8 →= 7 10 5 6 2 8 , 2 2 2 2 2 9 →= 7 10 5 6 2 9 , 6 2 2 2 2 2 →= 23 10 5 6 2 2 , 6 2 2 2 2 7 →= 23 10 5 6 2 7 , 6 2 2 2 2 8 →= 23 10 5 6 2 8 , 6 2 2 2 2 9 →= 23 10 5 6 2 9 , 14 2 2 2 2 2 →= 28 10 5 6 2 2 , 14 2 2 2 2 7 →= 28 10 5 6 2 7 , 14 2 2 2 2 8 →= 28 10 5 6 2 8 , 14 2 2 2 2 9 →= 28 10 5 6 2 9 , 18 2 2 2 2 2 →= 29 10 5 6 2 2 , 18 2 2 2 2 7 →= 29 10 5 6 2 7 , 18 2 2 2 2 8 →= 29 10 5 6 2 8 , 18 2 2 2 2 9 →= 29 10 5 6 2 9 , 33 2 2 2 2 2 →= 34 10 5 6 2 2 , 33 2 2 2 2 7 →= 34 10 5 6 2 7 , 33 2 2 2 2 8 →= 34 10 5 6 2 8 , 33 2 2 2 2 9 →= 34 10 5 6 2 9 , 35 2 2 2 2 2 →= 36 10 5 6 2 2 , 35 2 2 2 2 7 →= 36 10 5 6 2 7 , 35 2 2 2 2 8 →= 36 10 5 6 2 8 , 35 2 2 2 2 9 →= 36 10 5 6 2 9 , 1 2 2 2 7 10 →= 22 10 5 6 7 10 , 1 2 2 2 7 11 →= 22 10 5 6 7 11 , 1 2 2 2 7 12 →= 22 10 5 6 7 12 , 2 2 2 2 7 10 →= 7 10 5 6 7 10 , 2 2 2 2 7 11 →= 7 10 5 6 7 11 , 2 2 2 2 7 12 →= 7 10 5 6 7 12 , 6 2 2 2 7 10 →= 23 10 5 6 7 10 , 6 2 2 2 7 11 →= 23 10 5 6 7 11 , 6 2 2 2 7 12 →= 23 10 5 6 7 12 , 14 2 2 2 7 10 →= 28 10 5 6 7 10 , 14 2 2 2 7 11 →= 28 10 5 6 7 11 , 14 2 2 2 7 12 →= 28 10 5 6 7 12 , 18 2 2 2 7 10 →= 29 10 5 6 7 10 , 18 2 2 2 7 11 →= 29 10 5 6 7 11 , 18 2 2 2 7 12 →= 29 10 5 6 7 12 , 33 2 2 2 7 10 →= 34 10 5 6 7 10 , 33 2 2 2 7 11 →= 34 10 5 6 7 11 , 33 2 2 2 7 12 →= 34 10 5 6 7 12 , 35 2 2 2 7 10 →= 36 10 5 6 7 10 , 35 2 2 2 7 11 →= 36 10 5 6 7 11 , 35 2 2 2 7 12 →= 36 10 5 6 7 12 , 30 6 2 2 2 2 →= 4 21 5 6 2 2 , 30 6 2 2 2 7 →= 4 21 5 6 2 7 , 30 6 2 2 2 8 →= 4 21 5 6 2 8 , 30 6 2 2 2 9 →= 4 21 5 6 2 9 , 5 6 2 2 2 2 →= 21 21 5 6 2 2 , 5 6 2 2 2 7 →= 21 21 5 6 2 7 , 5 6 2 2 2 8 →= 21 21 5 6 2 8 , 5 6 2 2 2 9 →= 21 21 5 6 2 9 , 11 6 2 2 2 2 →= 10 21 5 6 2 2 , 11 6 2 2 2 7 →= 10 21 5 6 2 7 , 11 6 2 2 2 8 →= 10 21 5 6 2 8 , 11 6 2 2 2 9 →= 10 21 5 6 2 9 , 31 6 2 2 2 2 →= 16 21 5 6 2 2 , 31 6 2 2 2 7 →= 16 21 5 6 2 7 , 31 6 2 2 2 8 →= 16 21 5 6 2 8 , 31 6 2 2 2 9 →= 16 21 5 6 2 9 , 32 6 2 2 2 2 →= 20 21 5 6 2 2 , 32 6 2 2 2 7 →= 20 21 5 6 2 7 , 32 6 2 2 2 8 →= 20 21 5 6 2 8 , 32 6 2 2 2 9 →= 20 21 5 6 2 9 , 37 6 2 2 2 2 →= 38 21 5 6 2 2 , 37 6 2 2 2 7 →= 38 21 5 6 2 7 , 37 6 2 2 2 8 →= 38 21 5 6 2 8 , 37 6 2 2 2 9 →= 38 21 5 6 2 9 , 39 6 2 2 2 2 →= 40 21 5 6 2 2 , 39 6 2 2 2 7 →= 40 21 5 6 2 7 , 39 6 2 2 2 8 →= 40 21 5 6 2 8 , 39 6 2 2 2 9 →= 40 21 5 6 2 9 , 30 6 2 2 7 10 →= 4 21 5 6 7 10 , 30 6 2 2 7 11 →= 4 21 5 6 7 11 , 30 6 2 2 7 12 →= 4 21 5 6 7 12 , 5 6 2 2 7 10 →= 21 21 5 6 7 10 , 5 6 2 2 7 11 →= 21 21 5 6 7 11 , 5 6 2 2 7 12 →= 21 21 5 6 7 12 , 11 6 2 2 7 10 →= 10 21 5 6 7 10 , 11 6 2 2 7 11 →= 10 21 5 6 7 11 , 11 6 2 2 7 12 →= 10 21 5 6 7 12 , 31 6 2 2 7 10 →= 16 21 5 6 7 10 , 31 6 2 2 7 11 →= 16 21 5 6 7 11 , 31 6 2 2 7 12 →= 16 21 5 6 7 12 , 32 6 2 2 7 10 →= 20 21 5 6 7 10 , 32 6 2 2 7 11 →= 20 21 5 6 7 11 , 32 6 2 2 7 12 →= 20 21 5 6 7 12 , 37 6 2 2 7 10 →= 38 21 5 6 7 10 , 37 6 2 2 7 11 →= 38 21 5 6 7 11 , 37 6 2 2 7 12 →= 38 21 5 6 7 12 , 39 6 2 2 7 10 →= 40 21 5 6 7 10 , 39 6 2 2 7 11 →= 40 21 5 6 7 11 , 39 6 2 2 7 12 →= 40 21 5 6 7 12 , 30 6 2 2 8 41 →= 4 21 5 6 8 41 , 5 6 2 2 8 41 →= 21 21 5 6 8 41 , 11 6 2 2 8 41 →= 10 21 5 6 8 41 , 31 6 2 2 8 41 →= 16 21 5 6 8 41 , 32 6 2 2 8 41 →= 20 21 5 6 8 41 , 37 6 2 2 8 41 →= 38 21 5 6 8 41 , 39 6 2 2 8 41 →= 40 21 5 6 8 41 , 42 33 2 2 2 2 →= 43 38 5 6 2 2 , 42 33 2 2 2 7 →= 43 38 5 6 2 7 , 42 33 2 2 2 8 →= 43 38 5 6 2 8 , 42 33 2 2 2 9 →= 43 38 5 6 2 9 , 42 33 2 2 7 10 →= 43 38 5 6 7 10 , 42 33 2 2 7 11 →= 43 38 5 6 7 11 , 42 33 2 2 7 12 →= 43 38 5 6 7 12 , 22 10 21 21 5 6 →= 1 7 10 21 5 6 , 22 10 21 21 5 23 →= 1 7 10 21 5 23 , 22 10 21 21 5 24 →= 1 7 10 21 5 24 , 22 10 21 21 5 25 →= 1 7 10 21 5 25 , 7 10 21 21 5 6 →= 2 7 10 21 5 6 , 7 10 21 21 5 23 →= 2 7 10 21 5 23 , 7 10 21 21 5 24 →= 2 7 10 21 5 24 , 7 10 21 21 5 25 →= 2 7 10 21 5 25 , 23 10 21 21 5 6 →= 6 7 10 21 5 6 , 23 10 21 21 5 23 →= 6 7 10 21 5 23 , 23 10 21 21 5 24 →= 6 7 10 21 5 24 , 23 10 21 21 5 25 →= 6 7 10 21 5 25 , 28 10 21 21 5 6 →= 14 7 10 21 5 6 , 28 10 21 21 5 23 →= 14 7 10 21 5 23 , 28 10 21 21 5 24 →= 14 7 10 21 5 24 , 28 10 21 21 5 25 →= 14 7 10 21 5 25 , 29 10 21 21 5 6 →= 18 7 10 21 5 6 , 29 10 21 21 5 23 →= 18 7 10 21 5 23 , 29 10 21 21 5 24 →= 18 7 10 21 5 24 , 29 10 21 21 5 25 →= 18 7 10 21 5 25 , 34 10 21 21 5 6 →= 33 7 10 21 5 6 , 34 10 21 21 5 23 →= 33 7 10 21 5 23 , 34 10 21 21 5 24 →= 33 7 10 21 5 24 , 34 10 21 21 5 25 →= 33 7 10 21 5 25 , 36 10 21 21 5 6 →= 35 7 10 21 5 6 , 36 10 21 21 5 23 →= 35 7 10 21 5 23 , 36 10 21 21 5 24 →= 35 7 10 21 5 24 , 36 10 21 21 5 25 →= 35 7 10 21 5 25 , 22 10 21 21 21 21 →= 1 7 10 21 21 21 , 22 10 21 21 21 5 →= 1 7 10 21 21 5 , 22 10 21 21 21 26 →= 1 7 10 21 21 26 , 22 10 21 21 21 27 →= 1 7 10 21 21 27 , 7 10 21 21 21 21 →= 2 7 10 21 21 21 , 7 10 21 21 21 5 →= 2 7 10 21 21 5 , 7 10 21 21 21 26 →= 2 7 10 21 21 26 , 7 10 21 21 21 27 →= 2 7 10 21 21 27 , 23 10 21 21 21 21 →= 6 7 10 21 21 21 , 23 10 21 21 21 5 →= 6 7 10 21 21 5 , 23 10 21 21 21 26 →= 6 7 10 21 21 26 , 23 10 21 21 21 27 →= 6 7 10 21 21 27 , 28 10 21 21 21 21 →= 14 7 10 21 21 21 , 28 10 21 21 21 5 →= 14 7 10 21 21 5 , 28 10 21 21 21 26 →= 14 7 10 21 21 26 , 28 10 21 21 21 27 →= 14 7 10 21 21 27 , 29 10 21 21 21 21 →= 18 7 10 21 21 21 , 29 10 21 21 21 5 →= 18 7 10 21 21 5 , 29 10 21 21 21 26 →= 18 7 10 21 21 26 , 29 10 21 21 21 27 →= 18 7 10 21 21 27 , 34 10 21 21 21 21 →= 33 7 10 21 21 21 , 34 10 21 21 21 5 →= 33 7 10 21 21 5 , 34 10 21 21 21 26 →= 33 7 10 21 21 26 , 34 10 21 21 21 27 →= 33 7 10 21 21 27 , 36 10 21 21 21 21 →= 35 7 10 21 21 21 , 36 10 21 21 21 5 →= 35 7 10 21 21 5 , 36 10 21 21 21 26 →= 35 7 10 21 21 26 , 36 10 21 21 21 27 →= 35 7 10 21 21 27 , 22 10 21 21 26 44 →= 1 7 10 21 26 44 , 7 10 21 21 26 44 →= 2 7 10 21 26 44 , 23 10 21 21 26 44 →= 6 7 10 21 26 44 , 28 10 21 21 26 44 →= 14 7 10 21 26 44 , 29 10 21 21 26 44 →= 18 7 10 21 26 44 , 34 10 21 21 26 44 →= 33 7 10 21 26 44 , 36 10 21 21 26 44 →= 35 7 10 21 26 44 , 4 21 21 21 5 6 →= 30 23 10 21 5 6 , 4 21 21 21 5 23 →= 30 23 10 21 5 23 , 4 21 21 21 5 24 →= 30 23 10 21 5 24 , 4 21 21 21 5 25 →= 30 23 10 21 5 25 , 21 21 21 21 5 6 →= 5 23 10 21 5 6 , 21 21 21 21 5 23 →= 5 23 10 21 5 23 , 21 21 21 21 5 24 →= 5 23 10 21 5 24 , 21 21 21 21 5 25 →= 5 23 10 21 5 25 , 10 21 21 21 5 6 →= 11 23 10 21 5 6 , 10 21 21 21 5 23 →= 11 23 10 21 5 23 , 10 21 21 21 5 24 →= 11 23 10 21 5 24 , 10 21 21 21 5 25 →= 11 23 10 21 5 25 , 16 21 21 21 5 6 →= 31 23 10 21 5 6 , 16 21 21 21 5 23 →= 31 23 10 21 5 23 , 16 21 21 21 5 24 →= 31 23 10 21 5 24 , 16 21 21 21 5 25 →= 31 23 10 21 5 25 , 20 21 21 21 5 6 →= 32 23 10 21 5 6 , 20 21 21 21 5 23 →= 32 23 10 21 5 23 , 20 21 21 21 5 24 →= 32 23 10 21 5 24 , 20 21 21 21 5 25 →= 32 23 10 21 5 25 , 38 21 21 21 5 6 →= 37 23 10 21 5 6 , 38 21 21 21 5 23 →= 37 23 10 21 5 23 , 38 21 21 21 5 24 →= 37 23 10 21 5 24 , 38 21 21 21 5 25 →= 37 23 10 21 5 25 , 40 21 21 21 5 6 →= 39 23 10 21 5 6 , 40 21 21 21 5 23 →= 39 23 10 21 5 23 , 40 21 21 21 5 24 →= 39 23 10 21 5 24 , 40 21 21 21 5 25 →= 39 23 10 21 5 25 , 4 21 21 21 21 21 →= 30 23 10 21 21 21 , 4 21 21 21 21 5 →= 30 23 10 21 21 5 , 4 21 21 21 21 26 →= 30 23 10 21 21 26 , 4 21 21 21 21 27 →= 30 23 10 21 21 27 , 21 21 21 21 21 21 →= 5 23 10 21 21 21 , 21 21 21 21 21 5 →= 5 23 10 21 21 5 , 21 21 21 21 21 26 →= 5 23 10 21 21 26 , 21 21 21 21 21 27 →= 5 23 10 21 21 27 , 10 21 21 21 21 21 →= 11 23 10 21 21 21 , 10 21 21 21 21 5 →= 11 23 10 21 21 5 , 10 21 21 21 21 26 →= 11 23 10 21 21 26 , 10 21 21 21 21 27 →= 11 23 10 21 21 27 , 16 21 21 21 21 21 →= 31 23 10 21 21 21 , 16 21 21 21 21 5 →= 31 23 10 21 21 5 , 16 21 21 21 21 26 →= 31 23 10 21 21 26 , 16 21 21 21 21 27 →= 31 23 10 21 21 27 , 20 21 21 21 21 21 →= 32 23 10 21 21 21 , 20 21 21 21 21 5 →= 32 23 10 21 21 5 , 20 21 21 21 21 26 →= 32 23 10 21 21 26 , 20 21 21 21 21 27 →= 32 23 10 21 21 27 , 38 21 21 21 21 21 →= 37 23 10 21 21 21 , 38 21 21 21 21 5 →= 37 23 10 21 21 5 , 38 21 21 21 21 26 →= 37 23 10 21 21 26 , 38 21 21 21 21 27 →= 37 23 10 21 21 27 , 40 21 21 21 21 21 →= 39 23 10 21 21 21 , 40 21 21 21 21 5 →= 39 23 10 21 21 5 , 40 21 21 21 21 26 →= 39 23 10 21 21 26 , 40 21 21 21 21 27 →= 39 23 10 21 21 27 , 4 21 21 21 26 44 →= 30 23 10 21 26 44 , 21 21 21 21 26 44 →= 5 23 10 21 26 44 , 10 21 21 21 26 44 →= 11 23 10 21 26 44 , 16 21 21 21 26 44 →= 31 23 10 21 26 44 , 20 21 21 21 26 44 →= 32 23 10 21 26 44 , 38 21 21 21 26 44 →= 37 23 10 21 26 44 , 40 21 21 21 26 44 →= 39 23 10 21 26 44 , 43 38 21 21 5 6 →= 42 34 10 21 5 6 , 43 38 21 21 5 23 →= 42 34 10 21 5 23 , 43 38 21 21 5 24 →= 42 34 10 21 5 24 , 43 38 21 21 5 25 →= 42 34 10 21 5 25 , 43 38 21 21 21 21 →= 42 34 10 21 21 21 , 43 38 21 21 21 5 →= 42 34 10 21 21 5 , 43 38 21 21 21 26 →= 42 34 10 21 21 26 , 43 38 21 21 21 27 →= 42 34 10 21 21 27 , 22 11 23 11 6 2 →= 1 2 7 10 5 6 , 22 11 23 11 6 7 →= 1 2 7 10 5 23 , 7 11 23 11 6 2 →= 2 2 7 10 5 6 , 7 11 23 11 6 7 →= 2 2 7 10 5 23 , 23 11 23 11 6 2 →= 6 2 7 10 5 6 , 23 11 23 11 6 7 →= 6 2 7 10 5 23 , 28 11 23 11 6 2 →= 14 2 7 10 5 6 , 28 11 23 11 6 7 →= 14 2 7 10 5 23 , 29 11 23 11 6 2 →= 18 2 7 10 5 6 , 29 11 23 11 6 7 →= 18 2 7 10 5 23 , 34 11 23 11 6 2 →= 33 2 7 10 5 6 , 34 11 23 11 6 7 →= 33 2 7 10 5 23 , 36 11 23 11 6 2 →= 35 2 7 10 5 6 , 36 11 23 11 6 7 →= 35 2 7 10 5 23 , 22 11 23 11 23 10 →= 1 2 7 10 21 21 , 22 11 23 11 23 11 →= 1 2 7 10 21 5 , 7 11 23 11 23 10 →= 2 2 7 10 21 21 , 7 11 23 11 23 11 →= 2 2 7 10 21 5 , 23 11 23 11 23 10 →= 6 2 7 10 21 21 , 23 11 23 11 23 11 →= 6 2 7 10 21 5 , 28 11 23 11 23 10 →= 14 2 7 10 21 21 , 28 11 23 11 23 11 →= 14 2 7 10 21 5 , 29 11 23 11 23 10 →= 18 2 7 10 21 21 , 29 11 23 11 23 11 →= 18 2 7 10 21 5 , 34 11 23 11 23 10 →= 33 2 7 10 21 21 , 34 11 23 11 23 11 →= 33 2 7 10 21 5 , 36 11 23 11 23 10 →= 35 2 7 10 21 21 , 36 11 23 11 23 11 →= 35 2 7 10 21 5 , 4 5 23 11 6 2 →= 30 6 7 10 5 6 , 4 5 23 11 6 7 →= 30 6 7 10 5 23 , 21 5 23 11 6 2 →= 5 6 7 10 5 6 , 21 5 23 11 6 7 →= 5 6 7 10 5 23 , 10 5 23 11 6 2 →= 11 6 7 10 5 6 , 10 5 23 11 6 7 →= 11 6 7 10 5 23 , 16 5 23 11 6 2 →= 31 6 7 10 5 6 , 16 5 23 11 6 7 →= 31 6 7 10 5 23 , 20 5 23 11 6 2 →= 32 6 7 10 5 6 , 20 5 23 11 6 7 →= 32 6 7 10 5 23 , 38 5 23 11 6 2 →= 37 6 7 10 5 6 , 38 5 23 11 6 7 →= 37 6 7 10 5 23 , 40 5 23 11 6 2 →= 39 6 7 10 5 6 , 40 5 23 11 6 7 →= 39 6 7 10 5 23 , 4 5 23 11 23 10 →= 30 6 7 10 21 21 , 4 5 23 11 23 11 →= 30 6 7 10 21 5 , 21 5 23 11 23 10 →= 5 6 7 10 21 21 , 21 5 23 11 23 11 →= 5 6 7 10 21 5 , 10 5 23 11 23 10 →= 11 6 7 10 21 21 , 10 5 23 11 23 11 →= 11 6 7 10 21 5 , 16 5 23 11 23 10 →= 31 6 7 10 21 21 , 16 5 23 11 23 11 →= 31 6 7 10 21 5 , 20 5 23 11 23 10 →= 32 6 7 10 21 21 , 20 5 23 11 23 11 →= 32 6 7 10 21 5 , 38 5 23 11 23 10 →= 37 6 7 10 21 21 , 38 5 23 11 23 11 →= 37 6 7 10 21 5 , 40 5 23 11 23 10 →= 39 6 7 10 21 21 , 40 5 23 11 23 11 →= 39 6 7 10 21 5 , 43 37 23 11 6 2 →= 42 33 7 10 5 6 , 43 37 23 11 6 7 →= 42 33 7 10 5 23 , 43 37 23 11 23 10 →= 42 33 7 10 21 21 , 43 37 23 11 23 11 →= 42 33 7 10 21 5 } 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 23 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 24 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 25 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 26 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 27 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 28 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 29 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 30 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 31 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 32 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 33 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 34 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 35 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 36 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 37 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 38 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 39 ↦ ⎛ ⎞ ⎜ 1 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 ⎟ ⎝ ⎠ After renaming modulo the bijection { 5 ↦ 0, 6 ↦ 1, 2 ↦ 2, 21 ↦ 3, 7 ↦ 4, 8 ↦ 5, 9 ↦ 6, 10 ↦ 7, 11 ↦ 8, 12 ↦ 9, 41 ↦ 10, 23 ↦ 11, 24 ↦ 12, 25 ↦ 13, 26 ↦ 14, 27 ↦ 15, 44 ↦ 16 }, it remains to prove termination of the 30-rule system { 0 1 2 2 2 2 →= 3 3 0 1 2 2 , 0 1 2 2 2 4 →= 3 3 0 1 2 4 , 0 1 2 2 2 5 →= 3 3 0 1 2 5 , 0 1 2 2 2 6 →= 3 3 0 1 2 6 , 0 1 2 2 4 7 →= 3 3 0 1 4 7 , 0 1 2 2 4 8 →= 3 3 0 1 4 8 , 0 1 2 2 4 9 →= 3 3 0 1 4 9 , 0 1 2 2 5 10 →= 3 3 0 1 5 10 , 4 7 3 3 0 1 →= 2 4 7 3 0 1 , 4 7 3 3 0 11 →= 2 4 7 3 0 11 , 4 7 3 3 0 12 →= 2 4 7 3 0 12 , 4 7 3 3 0 13 →= 2 4 7 3 0 13 , 4 7 3 3 3 3 →= 2 4 7 3 3 3 , 4 7 3 3 3 0 →= 2 4 7 3 3 0 , 4 7 3 3 3 14 →= 2 4 7 3 3 14 , 4 7 3 3 3 15 →= 2 4 7 3 3 15 , 4 7 3 3 14 16 →= 2 4 7 3 14 16 , 7 3 3 3 0 1 →= 8 11 7 3 0 1 , 7 3 3 3 0 11 →= 8 11 7 3 0 11 , 7 3 3 3 0 12 →= 8 11 7 3 0 12 , 7 3 3 3 0 13 →= 8 11 7 3 0 13 , 7 3 3 3 3 3 →= 8 11 7 3 3 3 , 7 3 3 3 3 0 →= 8 11 7 3 3 0 , 7 3 3 3 3 14 →= 8 11 7 3 3 14 , 7 3 3 3 3 15 →= 8 11 7 3 3 15 , 7 3 3 3 14 16 →= 8 11 7 3 14 16 , 4 8 11 8 1 4 →= 2 2 4 7 0 11 , 4 8 11 8 11 7 →= 2 2 4 7 3 3 , 7 0 11 8 1 4 →= 8 1 4 7 0 11 , 7 0 11 8 11 7 →= 8 1 4 7 3 3 } The system is trivially terminating.