/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 { a ↦ 0, b ↦ 1 }, it remains to prove termination of the 1-rule system { 0 0 1 1 0 0 0 0 0 ⟶ 0 0 0 0 0 0 0 1 1 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, (2,0) ↦ 4 }, it remains to prove termination of the 6-rule system { 0 0 1 2 3 0 0 0 0 0 ⟶ 0 0 0 0 0 0 0 1 2 3 0 1 2 3 , 0 0 1 2 3 0 0 0 0 1 ⟶ 0 0 0 0 0 0 0 1 2 3 0 1 2 2 , 3 0 1 2 3 0 0 0 0 0 ⟶ 3 0 0 0 0 0 0 1 2 3 0 1 2 3 , 3 0 1 2 3 0 0 0 0 1 ⟶ 3 0 0 0 0 0 0 1 2 3 0 1 2 2 , 4 0 1 2 3 0 0 0 0 0 ⟶ 4 0 0 0 0 0 0 1 2 3 0 1 2 3 , 4 0 1 2 3 0 0 0 0 1 ⟶ 4 0 0 0 0 0 0 1 2 3 0 1 2 2 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 11: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 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 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 5-rule system { 0 0 1 2 3 0 0 0 0 0 ⟶ 0 0 0 0 0 0 0 1 2 3 0 1 2 3 , 3 0 1 2 3 0 0 0 0 0 ⟶ 3 0 0 0 0 0 0 1 2 3 0 1 2 3 , 3 0 1 2 3 0 0 0 0 1 ⟶ 3 0 0 0 0 0 0 1 2 3 0 1 2 2 , 4 0 1 2 3 0 0 0 0 0 ⟶ 4 0 0 0 0 0 0 1 2 3 0 1 2 3 , 4 0 1 2 3 0 0 0 0 1 ⟶ 4 0 0 0 0 0 0 1 2 3 0 1 2 2 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 11: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 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 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 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 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 4-rule system { 0 0 1 2 3 0 0 0 0 0 ⟶ 0 0 0 0 0 0 0 1 2 3 0 1 2 3 , 3 0 1 2 3 0 0 0 0 0 ⟶ 3 0 0 0 0 0 0 1 2 3 0 1 2 3 , 3 0 1 2 3 0 0 0 0 1 ⟶ 3 0 0 0 0 0 0 1 2 3 0 1 2 2 , 4 0 1 2 3 0 0 0 0 0 ⟶ 4 0 0 0 0 0 0 1 2 3 0 1 2 3 } 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, (3,↑) ↦ 5, (4,↑) ↦ 6, (4,↓) ↦ 7 }, it remains to prove termination of the 43-rule system { 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 1 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 5 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 5 , 5 1 2 3 4 1 1 1 1 1 ⟶ 5 1 1 1 1 1 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 5 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 5 , 5 1 2 3 4 1 1 1 1 2 ⟶ 5 1 1 1 1 1 1 2 3 4 1 2 3 3 , 5 1 2 3 4 1 1 1 1 2 ⟶ 0 1 1 1 1 1 2 3 4 1 2 3 3 , 5 1 2 3 4 1 1 1 1 2 ⟶ 0 1 1 1 1 2 3 4 1 2 3 3 , 5 1 2 3 4 1 1 1 1 2 ⟶ 0 1 1 1 2 3 4 1 2 3 3 , 5 1 2 3 4 1 1 1 1 2 ⟶ 0 1 1 2 3 4 1 2 3 3 , 5 1 2 3 4 1 1 1 1 2 ⟶ 0 1 2 3 4 1 2 3 3 , 5 1 2 3 4 1 1 1 1 2 ⟶ 0 2 3 4 1 2 3 3 , 5 1 2 3 4 1 1 1 1 2 ⟶ 5 1 2 3 3 , 5 1 2 3 4 1 1 1 1 2 ⟶ 0 2 3 3 , 6 1 2 3 4 1 1 1 1 1 ⟶ 6 1 1 1 1 1 1 2 3 4 1 2 3 4 , 6 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 1 2 3 4 1 2 3 4 , 6 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 2 3 4 1 2 3 4 , 6 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 2 3 4 1 2 3 4 , 6 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 2 3 4 1 2 3 4 , 6 1 2 3 4 1 1 1 1 1 ⟶ 0 1 2 3 4 1 2 3 4 , 6 1 2 3 4 1 1 1 1 1 ⟶ 0 2 3 4 1 2 3 4 , 6 1 2 3 4 1 1 1 1 1 ⟶ 5 1 2 3 4 , 6 1 2 3 4 1 1 1 1 1 ⟶ 0 2 3 4 , 6 1 2 3 4 1 1 1 1 1 ⟶ 5 , 1 1 2 3 4 1 1 1 1 1 →= 1 1 1 1 1 1 1 2 3 4 1 2 3 4 , 4 1 2 3 4 1 1 1 1 1 →= 4 1 1 1 1 1 1 2 3 4 1 2 3 4 , 4 1 2 3 4 1 1 1 1 2 →= 4 1 1 1 1 1 1 2 3 4 1 2 3 3 , 7 1 2 3 4 1 1 1 1 1 →= 7 1 1 1 1 1 1 2 3 4 1 2 3 4 } 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 0 ⎟ ⎜ 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 34-rule system { 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 1 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 5 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 5 , 5 1 2 3 4 1 1 1 1 1 ⟶ 5 1 1 1 1 1 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 5 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 5 , 5 1 2 3 4 1 1 1 1 2 ⟶ 5 1 1 1 1 1 1 2 3 4 1 2 3 3 , 5 1 2 3 4 1 1 1 1 2 ⟶ 0 1 1 1 1 1 2 3 4 1 2 3 3 , 5 1 2 3 4 1 1 1 1 2 ⟶ 0 1 1 1 1 2 3 4 1 2 3 3 , 5 1 2 3 4 1 1 1 1 2 ⟶ 0 1 1 1 2 3 4 1 2 3 3 , 5 1 2 3 4 1 1 1 1 2 ⟶ 0 1 1 2 3 4 1 2 3 3 , 5 1 2 3 4 1 1 1 1 2 ⟶ 0 1 2 3 4 1 2 3 3 , 5 1 2 3 4 1 1 1 1 2 ⟶ 0 2 3 4 1 2 3 3 , 5 1 2 3 4 1 1 1 1 2 ⟶ 5 1 2 3 3 , 5 1 2 3 4 1 1 1 1 2 ⟶ 0 2 3 3 , 6 1 2 3 4 1 1 1 1 1 ⟶ 6 1 1 1 1 1 1 2 3 4 1 2 3 4 , 1 1 2 3 4 1 1 1 1 1 →= 1 1 1 1 1 1 1 2 3 4 1 2 3 4 , 4 1 2 3 4 1 1 1 1 1 →= 4 1 1 1 1 1 1 2 3 4 1 2 3 4 , 4 1 2 3 4 1 1 1 1 2 →= 4 1 1 1 1 1 1 2 3 4 1 2 3 3 , 7 1 2 3 4 1 1 1 1 1 →= 7 1 1 1 1 1 1 2 3 4 1 2 3 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 11: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 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 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 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 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 25-rule system { 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 1 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 5 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 5 , 5 1 2 3 4 1 1 1 1 1 ⟶ 5 1 1 1 1 1 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 5 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 5 , 6 1 2 3 4 1 1 1 1 1 ⟶ 6 1 1 1 1 1 1 2 3 4 1 2 3 4 , 1 1 2 3 4 1 1 1 1 1 →= 1 1 1 1 1 1 1 2 3 4 1 2 3 4 , 4 1 2 3 4 1 1 1 1 1 →= 4 1 1 1 1 1 1 2 3 4 1 2 3 4 , 4 1 2 3 4 1 1 1 1 2 →= 4 1 1 1 1 1 1 2 3 4 1 2 3 3 , 7 1 2 3 4 1 1 1 1 1 →= 7 1 1 1 1 1 1 2 3 4 1 2 3 4 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 11: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 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 1 0 0 0 ⎟ ⎜ 0 0 1 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 1 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 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 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 }, it remains to prove termination of the 24-rule system { 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 1 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 5 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 5 , 5 1 2 3 4 1 1 1 1 1 ⟶ 5 1 1 1 1 1 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 5 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 5 , 6 1 2 3 4 1 1 1 1 1 ⟶ 6 1 1 1 1 1 1 2 3 4 1 2 3 4 , 1 1 2 3 4 1 1 1 1 1 →= 1 1 1 1 1 1 1 2 3 4 1 2 3 4 , 4 1 2 3 4 1 1 1 1 1 →= 4 1 1 1 1 1 1 2 3 4 1 2 3 4 , 4 1 2 3 4 1 1 1 1 2 →= 4 1 1 1 1 1 1 2 3 4 1 2 3 3 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 11: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 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 1 0 0 0 ⎟ ⎜ 0 0 1 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 1 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 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 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 }, it remains to prove termination of the 23-rule system { 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 1 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 5 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 0 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 5 , 5 1 2 3 4 1 1 1 1 1 ⟶ 5 1 1 1 1 1 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 5 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 5 , 1 1 2 3 4 1 1 1 1 1 →= 1 1 1 1 1 1 1 2 3 4 1 2 3 4 , 4 1 2 3 4 1 1 1 1 1 →= 4 1 1 1 1 1 1 2 3 4 1 2 3 4 , 4 1 2 3 4 1 1 1 1 2 →= 4 1 1 1 1 1 1 2 3 4 1 2 3 3 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 11: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 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 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 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 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 1 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 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 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 1 0 0 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 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 }, it remains to prove termination of the 10-rule system { 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 2 3 4 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 5 1 2 3 4 , 0 1 2 3 4 1 1 1 1 1 ⟶ 5 , 5 1 2 3 4 1 1 1 1 1 ⟶ 5 1 1 1 1 1 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 0 1 2 3 4 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 5 1 2 3 4 , 5 1 2 3 4 1 1 1 1 1 ⟶ 5 , 1 1 2 3 4 1 1 1 1 1 →= 1 1 1 1 1 1 1 2 3 4 1 2 3 4 , 4 1 2 3 4 1 1 1 1 1 →= 4 1 1 1 1 1 1 2 3 4 1 2 3 4 , 4 1 2 3 4 1 1 1 1 2 →= 4 1 1 1 1 1 1 2 3 4 1 2 3 3 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 11: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 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 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 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 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 1 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 1 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 1 0 0 0 0 1 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 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 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 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 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 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 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 { 5 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4 }, it remains to prove termination of the 4-rule system { 0 1 2 3 4 1 1 1 1 1 ⟶ 0 1 1 1 1 1 1 2 3 4 1 2 3 4 , 1 1 2 3 4 1 1 1 1 1 →= 1 1 1 1 1 1 1 2 3 4 1 2 3 4 , 4 1 2 3 4 1 1 1 1 1 →= 4 1 1 1 1 1 1 2 3 4 1 2 3 4 , 4 1 2 3 4 1 1 1 1 2 →= 4 1 1 1 1 1 1 2 3 4 1 2 3 3 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 11: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 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 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 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 1 0 0 0 ⎟ ⎜ 0 0 1 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 1 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 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 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 { 1 ↦ 0, 2 ↦ 1, 3 ↦ 2, 4 ↦ 3 }, it remains to prove termination of the 3-rule system { 0 0 1 2 3 0 0 0 0 0 →= 0 0 0 0 0 0 0 1 2 3 0 1 2 3 , 3 0 1 2 3 0 0 0 0 0 →= 3 0 0 0 0 0 0 1 2 3 0 1 2 3 , 3 0 1 2 3 0 0 0 0 1 →= 3 0 0 0 0 0 0 1 2 3 0 1 2 2 } The system is trivially terminating.