/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES After renaming modulo the bijection { a ↦ 0, b ↦ 1, c ↦ 2 }, it remains to prove termination of the 3-rule system { 0 1 2 ⟶ 0 0 1 , 0 1 2 ⟶ 1 2 1 2 , 0 1 2 ⟶ 2 1 2 0 } The system was reversed. After renaming modulo the bijection { 2 ↦ 0, 1 ↦ 1, 0 ↦ 2 }, it remains to prove termination of the 3-rule system { 0 1 2 ⟶ 1 2 2 , 0 1 2 ⟶ 0 1 0 1 , 0 1 2 ⟶ 2 0 1 0 } Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols. After renaming modulo the bijection { (0,↑) ↦ 0, (1,↓) ↦ 1, (2,↓) ↦ 2, (0,↓) ↦ 3 }, it remains to prove termination of the 7-rule system { 0 1 2 ⟶ 0 1 3 1 , 0 1 2 ⟶ 0 1 , 0 1 2 ⟶ 0 1 3 , 0 1 2 ⟶ 0 , 3 1 2 →= 1 2 2 , 3 1 2 →= 3 1 3 1 , 3 1 2 →= 2 3 1 3 } Applying sparse tiling TROC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (4,0) ↦ 0, (0,1) ↦ 1, (1,2) ↦ 2, (2,1) ↦ 3, (1,3) ↦ 4, (3,1) ↦ 5, (1,1) ↦ 6, (2,2) ↦ 7, (2,3) ↦ 8, (2,5) ↦ 9, (1,5) ↦ 10, (3,2) ↦ 11, (3,3) ↦ 12, (3,5) ↦ 13, (0,2) ↦ 14, (0,3) ↦ 15, (0,5) ↦ 16, (4,3) ↦ 17, (4,1) ↦ 18, (4,2) ↦ 19 }, it remains to prove termination of the 76-rule system { 0 1 2 3 ⟶ 0 1 4 5 6 , 0 1 2 7 ⟶ 0 1 4 5 2 , 0 1 2 8 ⟶ 0 1 4 5 4 , 0 1 2 9 ⟶ 0 1 4 5 10 , 0 1 2 3 ⟶ 0 1 6 , 0 1 2 7 ⟶ 0 1 2 , 0 1 2 8 ⟶ 0 1 4 , 0 1 2 9 ⟶ 0 1 10 , 0 1 2 3 ⟶ 0 1 4 5 , 0 1 2 7 ⟶ 0 1 4 11 , 0 1 2 8 ⟶ 0 1 4 12 , 0 1 2 9 ⟶ 0 1 4 13 , 0 1 2 3 ⟶ 0 1 , 0 1 2 7 ⟶ 0 14 , 0 1 2 8 ⟶ 0 15 , 0 1 2 9 ⟶ 0 16 , 15 5 2 3 →= 1 2 7 3 , 15 5 2 7 →= 1 2 7 7 , 15 5 2 8 →= 1 2 7 8 , 15 5 2 9 →= 1 2 7 9 , 4 5 2 3 →= 6 2 7 3 , 4 5 2 7 →= 6 2 7 7 , 4 5 2 8 →= 6 2 7 8 , 4 5 2 9 →= 6 2 7 9 , 8 5 2 3 →= 3 2 7 3 , 8 5 2 7 →= 3 2 7 7 , 8 5 2 8 →= 3 2 7 8 , 8 5 2 9 →= 3 2 7 9 , 12 5 2 3 →= 5 2 7 3 , 12 5 2 7 →= 5 2 7 7 , 12 5 2 8 →= 5 2 7 8 , 12 5 2 9 →= 5 2 7 9 , 17 5 2 3 →= 18 2 7 3 , 17 5 2 7 →= 18 2 7 7 , 17 5 2 8 →= 18 2 7 8 , 17 5 2 9 →= 18 2 7 9 , 15 5 2 3 →= 15 5 4 5 6 , 15 5 2 7 →= 15 5 4 5 2 , 15 5 2 8 →= 15 5 4 5 4 , 15 5 2 9 →= 15 5 4 5 10 , 4 5 2 3 →= 4 5 4 5 6 , 4 5 2 7 →= 4 5 4 5 2 , 4 5 2 8 →= 4 5 4 5 4 , 4 5 2 9 →= 4 5 4 5 10 , 8 5 2 3 →= 8 5 4 5 6 , 8 5 2 7 →= 8 5 4 5 2 , 8 5 2 8 →= 8 5 4 5 4 , 8 5 2 9 →= 8 5 4 5 10 , 12 5 2 3 →= 12 5 4 5 6 , 12 5 2 7 →= 12 5 4 5 2 , 12 5 2 8 →= 12 5 4 5 4 , 12 5 2 9 →= 12 5 4 5 10 , 17 5 2 3 →= 17 5 4 5 6 , 17 5 2 7 →= 17 5 4 5 2 , 17 5 2 8 →= 17 5 4 5 4 , 17 5 2 9 →= 17 5 4 5 10 , 15 5 2 3 →= 14 8 5 4 5 , 15 5 2 7 →= 14 8 5 4 11 , 15 5 2 8 →= 14 8 5 4 12 , 15 5 2 9 →= 14 8 5 4 13 , 4 5 2 3 →= 2 8 5 4 5 , 4 5 2 7 →= 2 8 5 4 11 , 4 5 2 8 →= 2 8 5 4 12 , 4 5 2 9 →= 2 8 5 4 13 , 8 5 2 3 →= 7 8 5 4 5 , 8 5 2 7 →= 7 8 5 4 11 , 8 5 2 8 →= 7 8 5 4 12 , 8 5 2 9 →= 7 8 5 4 13 , 12 5 2 3 →= 11 8 5 4 5 , 12 5 2 7 →= 11 8 5 4 11 , 12 5 2 8 →= 11 8 5 4 12 , 12 5 2 9 →= 11 8 5 4 13 , 17 5 2 3 →= 19 8 5 4 5 , 17 5 2 7 →= 19 8 5 4 11 , 17 5 2 8 →= 19 8 5 4 12 , 17 5 2 9 →= 19 8 5 4 13 } 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 0 ⎟ ⎜ 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 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 7 ↦ 3, 4 ↦ 4, 5 ↦ 5, 15 ↦ 6, 3 ↦ 7, 8 ↦ 8, 9 ↦ 9, 6 ↦ 10, 12 ↦ 11, 17 ↦ 12, 11 ↦ 13 }, it remains to prove termination of the 26-rule system { 0 1 2 3 ⟶ 0 1 4 5 2 , 0 1 2 3 ⟶ 0 1 2 , 6 5 2 7 →= 1 2 3 7 , 6 5 2 3 →= 1 2 3 3 , 6 5 2 8 →= 1 2 3 8 , 6 5 2 9 →= 1 2 3 9 , 4 5 2 7 →= 10 2 3 7 , 4 5 2 3 →= 10 2 3 3 , 4 5 2 8 →= 10 2 3 8 , 4 5 2 9 →= 10 2 3 9 , 8 5 2 7 →= 7 2 3 7 , 8 5 2 3 →= 7 2 3 3 , 8 5 2 8 →= 7 2 3 8 , 8 5 2 9 →= 7 2 3 9 , 11 5 2 7 →= 5 2 3 7 , 11 5 2 3 →= 5 2 3 3 , 11 5 2 8 →= 5 2 3 8 , 11 5 2 9 →= 5 2 3 9 , 6 5 2 3 →= 6 5 4 5 2 , 4 5 2 3 →= 4 5 4 5 2 , 8 5 2 3 →= 8 5 4 5 2 , 11 5 2 3 →= 11 5 4 5 2 , 12 5 2 3 →= 12 5 4 5 2 , 4 5 2 7 →= 2 8 5 4 5 , 4 5 2 3 →= 2 8 5 4 13 , 4 5 2 8 →= 2 8 5 4 11 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 7 ↦ 6, 10 ↦ 7, 8 ↦ 8, 9 ↦ 9, 11 ↦ 10, 6 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 22-rule system { 0 1 2 3 ⟶ 0 1 4 5 2 , 0 1 2 3 ⟶ 0 1 2 , 4 5 2 6 →= 7 2 3 6 , 4 5 2 3 →= 7 2 3 3 , 4 5 2 8 →= 7 2 3 8 , 4 5 2 9 →= 7 2 3 9 , 8 5 2 6 →= 6 2 3 6 , 8 5 2 3 →= 6 2 3 3 , 8 5 2 8 →= 6 2 3 8 , 8 5 2 9 →= 6 2 3 9 , 10 5 2 6 →= 5 2 3 6 , 10 5 2 3 →= 5 2 3 3 , 10 5 2 8 →= 5 2 3 8 , 10 5 2 9 →= 5 2 3 9 , 11 5 2 3 →= 11 5 4 5 2 , 4 5 2 3 →= 4 5 4 5 2 , 8 5 2 3 →= 8 5 4 5 2 , 10 5 2 3 →= 10 5 4 5 2 , 12 5 2 3 →= 12 5 4 5 2 , 4 5 2 6 →= 2 8 5 4 5 , 4 5 2 3 →= 2 8 5 4 13 , 4 5 2 8 →= 2 8 5 4 10 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 5: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 21-rule system { 0 1 2 3 ⟶ 0 1 4 5 2 , 0 1 2 3 ⟶ 0 1 2 , 4 5 2 6 →= 7 2 3 6 , 4 5 2 3 →= 7 2 3 3 , 4 5 2 8 →= 7 2 3 8 , 4 5 2 9 →= 7 2 3 9 , 8 5 2 6 →= 6 2 3 6 , 8 5 2 3 →= 6 2 3 3 , 8 5 2 8 →= 6 2 3 8 , 10 5 2 6 →= 5 2 3 6 , 10 5 2 3 →= 5 2 3 3 , 10 5 2 8 →= 5 2 3 8 , 10 5 2 9 →= 5 2 3 9 , 11 5 2 3 →= 11 5 4 5 2 , 4 5 2 3 →= 4 5 4 5 2 , 8 5 2 3 →= 8 5 4 5 2 , 10 5 2 3 →= 10 5 4 5 2 , 12 5 2 3 →= 12 5 4 5 2 , 4 5 2 6 →= 2 8 5 4 5 , 4 5 2 3 →= 2 8 5 4 13 , 4 5 2 8 →= 2 8 5 4 10 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 5: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 4 ↦ 0, 5 ↦ 1, 2 ↦ 2, 6 ↦ 3, 7 ↦ 4, 3 ↦ 5, 8 ↦ 6, 9 ↦ 7, 10 ↦ 8, 11 ↦ 9, 12 ↦ 10, 13 ↦ 11 }, it remains to prove termination of the 19-rule system { 0 1 2 3 →= 4 2 5 3 , 0 1 2 5 →= 4 2 5 5 , 0 1 2 6 →= 4 2 5 6 , 0 1 2 7 →= 4 2 5 7 , 6 1 2 3 →= 3 2 5 3 , 6 1 2 5 →= 3 2 5 5 , 6 1 2 6 →= 3 2 5 6 , 8 1 2 3 →= 1 2 5 3 , 8 1 2 5 →= 1 2 5 5 , 8 1 2 6 →= 1 2 5 6 , 8 1 2 7 →= 1 2 5 7 , 9 1 2 5 →= 9 1 0 1 2 , 0 1 2 5 →= 0 1 0 1 2 , 6 1 2 5 →= 6 1 0 1 2 , 8 1 2 5 →= 8 1 0 1 2 , 10 1 2 5 →= 10 1 0 1 2 , 0 1 2 3 →= 2 6 1 0 1 , 0 1 2 5 →= 2 6 1 0 11 , 0 1 2 6 →= 2 6 1 0 8 } The system is trivially terminating.