/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, c ↦ 2 }, it remains to prove termination of the 4-rule system { 0 ⟶ , 0 ⟶ 1 2 , 1 1 ⟶ 0 0 , 2 2 2 ⟶ 1 } The system was reversed. After renaming modulo the bijection { 0 ↦ 0, 2 ↦ 1, 1 ↦ 2 }, it remains to prove termination of the 4-rule system { 0 ⟶ , 0 ⟶ 1 2 , 2 2 ⟶ 0 0 , 1 1 1 ⟶ 2 } Applying sparse tiling TRFC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (0,0) ↦ 0, (0,1) ↦ 1, (0,2) ↦ 2, (0,4) ↦ 3, (1,0) ↦ 4, (1,1) ↦ 5, (1,2) ↦ 6, (1,4) ↦ 7, (2,0) ↦ 8, (2,1) ↦ 9, (2,2) ↦ 10, (2,4) ↦ 11, (3,0) ↦ 12, (3,1) ↦ 13, (3,2) ↦ 14, (3,4) ↦ 15 }, it remains to prove termination of the 64-rule system { 0 0 ⟶ 0 , 0 1 ⟶ 1 , 0 2 ⟶ 2 , 0 3 ⟶ 3 , 4 0 ⟶ 4 , 4 1 ⟶ 5 , 4 2 ⟶ 6 , 4 3 ⟶ 7 , 8 0 ⟶ 8 , 8 1 ⟶ 9 , 8 2 ⟶ 10 , 8 3 ⟶ 11 , 12 0 ⟶ 12 , 12 1 ⟶ 13 , 12 2 ⟶ 14 , 12 3 ⟶ 15 , 0 0 ⟶ 1 6 8 , 0 1 ⟶ 1 6 9 , 0 2 ⟶ 1 6 10 , 0 3 ⟶ 1 6 11 , 4 0 ⟶ 5 6 8 , 4 1 ⟶ 5 6 9 , 4 2 ⟶ 5 6 10 , 4 3 ⟶ 5 6 11 , 8 0 ⟶ 9 6 8 , 8 1 ⟶ 9 6 9 , 8 2 ⟶ 9 6 10 , 8 3 ⟶ 9 6 11 , 12 0 ⟶ 13 6 8 , 12 1 ⟶ 13 6 9 , 12 2 ⟶ 13 6 10 , 12 3 ⟶ 13 6 11 , 2 10 8 ⟶ 0 0 0 , 2 10 9 ⟶ 0 0 1 , 2 10 10 ⟶ 0 0 2 , 2 10 11 ⟶ 0 0 3 , 6 10 8 ⟶ 4 0 0 , 6 10 9 ⟶ 4 0 1 , 6 10 10 ⟶ 4 0 2 , 6 10 11 ⟶ 4 0 3 , 10 10 8 ⟶ 8 0 0 , 10 10 9 ⟶ 8 0 1 , 10 10 10 ⟶ 8 0 2 , 10 10 11 ⟶ 8 0 3 , 14 10 8 ⟶ 12 0 0 , 14 10 9 ⟶ 12 0 1 , 14 10 10 ⟶ 12 0 2 , 14 10 11 ⟶ 12 0 3 , 1 5 5 4 ⟶ 2 8 , 1 5 5 5 ⟶ 2 9 , 1 5 5 6 ⟶ 2 10 , 1 5 5 7 ⟶ 2 11 , 5 5 5 4 ⟶ 6 8 , 5 5 5 5 ⟶ 6 9 , 5 5 5 6 ⟶ 6 10 , 5 5 5 7 ⟶ 6 11 , 9 5 5 4 ⟶ 10 8 , 9 5 5 5 ⟶ 10 9 , 9 5 5 6 ⟶ 10 10 , 9 5 5 7 ⟶ 10 11 , 13 5 5 4 ⟶ 14 8 , 13 5 5 5 ⟶ 14 9 , 13 5 5 6 ⟶ 14 10 , 13 5 5 7 ⟶ 14 11 } 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 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 8 ↦ 6, 9 ↦ 7, 10 ↦ 8, 11 ↦ 9, 12 ↦ 10, 6 ↦ 11 }, it remains to prove termination of the 29-rule system { 0 0 ⟶ 0 , 0 1 ⟶ 1 , 0 2 ⟶ 2 , 0 3 ⟶ 3 , 4 0 ⟶ 4 , 4 1 ⟶ 5 , 6 0 ⟶ 6 , 6 1 ⟶ 7 , 6 2 ⟶ 8 , 6 3 ⟶ 9 , 10 0 ⟶ 10 , 0 0 ⟶ 1 11 6 , 0 1 ⟶ 1 11 7 , 0 2 ⟶ 1 11 8 , 0 3 ⟶ 1 11 9 , 4 0 ⟶ 5 11 6 , 4 1 ⟶ 5 11 7 , 4 2 ⟶ 5 11 8 , 4 3 ⟶ 5 11 9 , 6 0 ⟶ 7 11 6 , 6 1 ⟶ 7 11 7 , 6 2 ⟶ 7 11 8 , 6 3 ⟶ 7 11 9 , 11 8 6 ⟶ 4 0 0 , 11 8 7 ⟶ 4 0 1 , 11 8 8 ⟶ 4 0 2 , 11 8 9 ⟶ 4 0 3 , 1 5 5 11 ⟶ 2 8 , 7 5 5 11 ⟶ 8 8 } Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols. After renaming modulo the bijection { (4,↑) ↦ 0, (0,↓) ↦ 1, (6,↑) ↦ 2, (1,↓) ↦ 3, (7,↑) ↦ 4, (10,↑) ↦ 5, (0,↑) ↦ 6, (1,↑) ↦ 7, (11,↓) ↦ 8, (6,↓) ↦ 9, (11,↑) ↦ 10, (7,↓) ↦ 11, (2,↓) ↦ 12, (8,↓) ↦ 13, (3,↓) ↦ 14, (9,↓) ↦ 15, (4,↓) ↦ 16, (5,↓) ↦ 17, (10,↓) ↦ 18 }, it remains to prove termination of the 67-rule system { 0 1 ⟶ 0 , 2 1 ⟶ 2 , 2 3 ⟶ 4 , 5 1 ⟶ 5 , 6 1 ⟶ 7 8 9 , 6 1 ⟶ 10 9 , 6 1 ⟶ 2 , 6 3 ⟶ 7 8 11 , 6 3 ⟶ 10 11 , 6 3 ⟶ 4 , 6 12 ⟶ 7 8 13 , 6 12 ⟶ 10 13 , 6 14 ⟶ 7 8 15 , 6 14 ⟶ 10 15 , 0 1 ⟶ 10 9 , 0 1 ⟶ 2 , 0 3 ⟶ 10 11 , 0 3 ⟶ 4 , 0 12 ⟶ 10 13 , 0 14 ⟶ 10 15 , 2 1 ⟶ 4 8 9 , 2 1 ⟶ 10 9 , 2 3 ⟶ 4 8 11 , 2 3 ⟶ 10 11 , 2 12 ⟶ 4 8 13 , 2 12 ⟶ 10 13 , 2 14 ⟶ 4 8 15 , 2 14 ⟶ 10 15 , 10 13 9 ⟶ 0 1 1 , 10 13 9 ⟶ 6 1 , 10 13 9 ⟶ 6 , 10 13 11 ⟶ 0 1 3 , 10 13 11 ⟶ 6 3 , 10 13 11 ⟶ 7 , 10 13 13 ⟶ 0 1 12 , 10 13 13 ⟶ 6 12 , 10 13 15 ⟶ 0 1 14 , 10 13 15 ⟶ 6 14 , 1 1 →= 1 , 1 3 →= 3 , 1 12 →= 12 , 1 14 →= 14 , 16 1 →= 16 , 16 3 →= 17 , 9 1 →= 9 , 9 3 →= 11 , 9 12 →= 13 , 9 14 →= 15 , 18 1 →= 18 , 1 1 →= 3 8 9 , 1 3 →= 3 8 11 , 1 12 →= 3 8 13 , 1 14 →= 3 8 15 , 16 1 →= 17 8 9 , 16 3 →= 17 8 11 , 16 12 →= 17 8 13 , 16 14 →= 17 8 15 , 9 1 →= 11 8 9 , 9 3 →= 11 8 11 , 9 12 →= 11 8 13 , 9 14 →= 11 8 15 , 8 13 9 →= 16 1 1 , 8 13 11 →= 16 1 3 , 8 13 13 →= 16 1 12 , 8 13 15 →= 16 1 14 , 3 17 17 8 →= 12 13 , 11 17 17 8 →= 13 13 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 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 2 ⎟ ⎜ 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 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 5 ↦ 3, 3 ↦ 4, 12 ↦ 5, 14 ↦ 6, 16 ↦ 7, 17 ↦ 8, 9 ↦ 9, 11 ↦ 10, 13 ↦ 11, 15 ↦ 12, 18 ↦ 13, 8 ↦ 14 }, it remains to prove termination of the 32-rule system { 0 1 ⟶ 0 , 2 1 ⟶ 2 , 3 1 ⟶ 3 , 1 1 →= 1 , 1 4 →= 4 , 1 5 →= 5 , 1 6 →= 6 , 7 1 →= 7 , 7 4 →= 8 , 9 1 →= 9 , 9 4 →= 10 , 9 5 →= 11 , 9 6 →= 12 , 13 1 →= 13 , 1 1 →= 4 14 9 , 1 4 →= 4 14 10 , 1 5 →= 4 14 11 , 1 6 →= 4 14 12 , 7 1 →= 8 14 9 , 7 4 →= 8 14 10 , 7 5 →= 8 14 11 , 7 6 →= 8 14 12 , 9 1 →= 10 14 9 , 9 4 →= 10 14 10 , 9 5 →= 10 14 11 , 9 6 →= 10 14 12 , 14 11 9 →= 7 1 1 , 14 11 10 →= 7 1 4 , 14 11 11 →= 7 1 5 , 14 11 12 →= 7 1 6 , 4 8 8 14 →= 5 11 , 10 8 8 14 →= 11 11 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 3: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 1 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 2 ↦ 0, 1 ↦ 1, 3 ↦ 2, 4 ↦ 3, 5 ↦ 4, 6 ↦ 5, 7 ↦ 6, 8 ↦ 7, 9 ↦ 8, 10 ↦ 9, 11 ↦ 10, 12 ↦ 11, 13 ↦ 12, 14 ↦ 13 }, it remains to prove termination of the 31-rule system { 0 1 ⟶ 0 , 2 1 ⟶ 2 , 1 1 →= 1 , 1 3 →= 3 , 1 4 →= 4 , 1 5 →= 5 , 6 1 →= 6 , 6 3 →= 7 , 8 1 →= 8 , 8 3 →= 9 , 8 4 →= 10 , 8 5 →= 11 , 12 1 →= 12 , 1 1 →= 3 13 8 , 1 3 →= 3 13 9 , 1 4 →= 3 13 10 , 1 5 →= 3 13 11 , 6 1 →= 7 13 8 , 6 3 →= 7 13 9 , 6 4 →= 7 13 10 , 6 5 →= 7 13 11 , 8 1 →= 9 13 8 , 8 3 →= 9 13 9 , 8 4 →= 9 13 10 , 8 5 →= 9 13 11 , 13 10 8 →= 6 1 1 , 13 10 9 →= 6 1 3 , 13 10 10 →= 6 1 4 , 13 10 11 →= 6 1 5 , 3 7 7 13 →= 4 10 , 9 7 7 13 →= 10 10 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 3: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 1 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 2 ↦ 0, 1 ↦ 1, 3 ↦ 2, 4 ↦ 3, 5 ↦ 4, 6 ↦ 5, 7 ↦ 6, 8 ↦ 7, 9 ↦ 8, 10 ↦ 9, 11 ↦ 10, 12 ↦ 11, 13 ↦ 12 }, it remains to prove termination of the 30-rule system { 0 1 ⟶ 0 , 1 1 →= 1 , 1 2 →= 2 , 1 3 →= 3 , 1 4 →= 4 , 5 1 →= 5 , 5 2 →= 6 , 7 1 →= 7 , 7 2 →= 8 , 7 3 →= 9 , 7 4 →= 10 , 11 1 →= 11 , 1 1 →= 2 12 7 , 1 2 →= 2 12 8 , 1 3 →= 2 12 9 , 1 4 →= 2 12 10 , 5 1 →= 6 12 7 , 5 2 →= 6 12 8 , 5 3 →= 6 12 9 , 5 4 →= 6 12 10 , 7 1 →= 8 12 7 , 7 2 →= 8 12 8 , 7 3 →= 8 12 9 , 7 4 →= 8 12 10 , 12 9 7 →= 5 1 1 , 12 9 8 →= 5 1 2 , 12 9 9 →= 5 1 3 , 12 9 10 →= 5 1 4 , 2 6 6 12 →= 3 9 , 8 6 6 12 →= 9 9 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 3: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 1 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 1 ↦ 0, 2 ↦ 1, 3 ↦ 2, 4 ↦ 3, 5 ↦ 4, 6 ↦ 5, 7 ↦ 6, 8 ↦ 7, 9 ↦ 8, 10 ↦ 9, 11 ↦ 10, 12 ↦ 11 }, it remains to prove termination of the 29-rule system { 0 0 →= 0 , 0 1 →= 1 , 0 2 →= 2 , 0 3 →= 3 , 4 0 →= 4 , 4 1 →= 5 , 6 0 →= 6 , 6 1 →= 7 , 6 2 →= 8 , 6 3 →= 9 , 10 0 →= 10 , 0 0 →= 1 11 6 , 0 1 →= 1 11 7 , 0 2 →= 1 11 8 , 0 3 →= 1 11 9 , 4 0 →= 5 11 6 , 4 1 →= 5 11 7 , 4 2 →= 5 11 8 , 4 3 →= 5 11 9 , 6 0 →= 7 11 6 , 6 1 →= 7 11 7 , 6 2 →= 7 11 8 , 6 3 →= 7 11 9 , 11 8 6 →= 4 0 0 , 11 8 7 →= 4 0 1 , 11 8 8 →= 4 0 2 , 11 8 9 →= 4 0 3 , 1 5 5 11 →= 2 8 , 7 5 5 11 →= 8 8 } The system is trivially terminating.