/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 ⟶ , 0 0 ⟶ 0 1 0 2 0 , 2 1 ⟶ 0 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, (2,↑) ↦ 4 }, it remains to prove termination of the 8-rule system { 0 1 ⟶ 0 2 1 3 1 , 0 1 ⟶ 0 3 1 , 0 1 ⟶ 4 1 , 4 2 ⟶ 0 3 , 4 2 ⟶ 4 , 1 →= , 1 1 →= 1 2 1 3 1 , 3 2 →= 1 3 } Applying sparse tiling TROC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (5,0) ↦ 0, (0,1) ↦ 1, (1,1) ↦ 2, (0,2) ↦ 3, (2,1) ↦ 4, (1,3) ↦ 5, (3,1) ↦ 6, (1,2) ↦ 7, (1,6) ↦ 8, (0,3) ↦ 9, (5,4) ↦ 10, (4,1) ↦ 11, (4,2) ↦ 12, (2,2) ↦ 13, (3,2) ↦ 14, (2,3) ↦ 15, (3,3) ↦ 16, (2,6) ↦ 17, (3,6) ↦ 18, (4,3) ↦ 19, (4,6) ↦ 20, (0,6) ↦ 21, (5,1) ↦ 22, (5,2) ↦ 23, (5,3) ↦ 24, (5,6) ↦ 25 }, it remains to prove termination of the 92-rule system { 0 1 2 ⟶ 0 3 4 5 6 2 , 0 1 7 ⟶ 0 3 4 5 6 7 , 0 1 5 ⟶ 0 3 4 5 6 5 , 0 1 8 ⟶ 0 3 4 5 6 8 , 0 1 2 ⟶ 0 9 6 2 , 0 1 7 ⟶ 0 9 6 7 , 0 1 5 ⟶ 0 9 6 5 , 0 1 8 ⟶ 0 9 6 8 , 0 1 2 ⟶ 10 11 2 , 0 1 7 ⟶ 10 11 7 , 0 1 5 ⟶ 10 11 5 , 0 1 8 ⟶ 10 11 8 , 10 12 4 ⟶ 0 9 6 , 10 12 13 ⟶ 0 9 14 , 10 12 15 ⟶ 0 9 16 , 10 12 17 ⟶ 0 9 18 , 10 12 4 ⟶ 10 11 , 10 12 13 ⟶ 10 12 , 10 12 15 ⟶ 10 19 , 10 12 17 ⟶ 10 20 , 1 2 →= 1 , 1 7 →= 3 , 1 5 →= 9 , 1 8 →= 21 , 2 2 →= 2 , 2 7 →= 7 , 2 5 →= 5 , 2 8 →= 8 , 4 2 →= 4 , 4 7 →= 13 , 4 5 →= 15 , 4 8 →= 17 , 6 2 →= 6 , 6 7 →= 14 , 6 5 →= 16 , 6 8 →= 18 , 11 2 →= 11 , 11 7 →= 12 , 11 5 →= 19 , 11 8 →= 20 , 22 2 →= 22 , 22 7 →= 23 , 22 5 →= 24 , 22 8 →= 25 , 1 2 2 →= 1 7 4 5 6 2 , 1 2 7 →= 1 7 4 5 6 7 , 1 2 5 →= 1 7 4 5 6 5 , 1 2 8 →= 1 7 4 5 6 8 , 2 2 2 →= 2 7 4 5 6 2 , 2 2 7 →= 2 7 4 5 6 7 , 2 2 5 →= 2 7 4 5 6 5 , 2 2 8 →= 2 7 4 5 6 8 , 4 2 2 →= 4 7 4 5 6 2 , 4 2 7 →= 4 7 4 5 6 7 , 4 2 5 →= 4 7 4 5 6 5 , 4 2 8 →= 4 7 4 5 6 8 , 6 2 2 →= 6 7 4 5 6 2 , 6 2 7 →= 6 7 4 5 6 7 , 6 2 5 →= 6 7 4 5 6 5 , 6 2 8 →= 6 7 4 5 6 8 , 11 2 2 →= 11 7 4 5 6 2 , 11 2 7 →= 11 7 4 5 6 7 , 11 2 5 →= 11 7 4 5 6 5 , 11 2 8 →= 11 7 4 5 6 8 , 22 2 2 →= 22 7 4 5 6 2 , 22 2 7 →= 22 7 4 5 6 7 , 22 2 5 →= 22 7 4 5 6 5 , 22 2 8 →= 22 7 4 5 6 8 , 9 14 4 →= 1 5 6 , 9 14 13 →= 1 5 14 , 9 14 15 →= 1 5 16 , 9 14 17 →= 1 5 18 , 5 14 4 →= 2 5 6 , 5 14 13 →= 2 5 14 , 5 14 15 →= 2 5 16 , 5 14 17 →= 2 5 18 , 15 14 4 →= 4 5 6 , 15 14 13 →= 4 5 14 , 15 14 15 →= 4 5 16 , 15 14 17 →= 4 5 18 , 16 14 4 →= 6 5 6 , 16 14 13 →= 6 5 14 , 16 14 15 →= 6 5 16 , 16 14 17 →= 6 5 18 , 19 14 4 →= 11 5 6 , 19 14 13 →= 11 5 14 , 19 14 15 →= 11 5 16 , 19 14 17 →= 11 5 18 , 24 14 4 →= 22 5 6 , 24 14 13 →= 22 5 14 , 24 14 15 →= 22 5 16 , 24 14 17 →= 22 5 18 } 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 23 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 24 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 25 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 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, 22 ↦ 21, 23 ↦ 22, 24 ↦ 23 }, it remains to prove termination of the 87-rule system { 0 1 2 ⟶ 0 3 4 5 6 2 , 0 1 7 ⟶ 0 3 4 5 6 7 , 0 1 5 ⟶ 0 3 4 5 6 5 , 0 1 8 ⟶ 0 3 4 5 6 8 , 0 1 2 ⟶ 0 9 6 2 , 0 1 7 ⟶ 0 9 6 7 , 0 1 5 ⟶ 0 9 6 5 , 0 1 8 ⟶ 0 9 6 8 , 0 1 2 ⟶ 10 11 2 , 0 1 7 ⟶ 10 11 7 , 0 1 5 ⟶ 10 11 5 , 0 1 8 ⟶ 10 11 8 , 10 12 4 ⟶ 0 9 6 , 10 12 13 ⟶ 0 9 14 , 10 12 15 ⟶ 0 9 16 , 10 12 17 ⟶ 0 9 18 , 10 12 4 ⟶ 10 11 , 10 12 13 ⟶ 10 12 , 10 12 15 ⟶ 10 19 , 10 12 17 ⟶ 10 20 , 1 2 →= 1 , 1 7 →= 3 , 1 5 →= 9 , 2 2 →= 2 , 2 7 →= 7 , 2 5 →= 5 , 2 8 →= 8 , 4 2 →= 4 , 4 7 →= 13 , 4 5 →= 15 , 6 2 →= 6 , 6 7 →= 14 , 6 5 →= 16 , 11 2 →= 11 , 11 7 →= 12 , 11 5 →= 19 , 21 2 →= 21 , 21 7 →= 22 , 21 5 →= 23 , 1 2 2 →= 1 7 4 5 6 2 , 1 2 7 →= 1 7 4 5 6 7 , 1 2 5 →= 1 7 4 5 6 5 , 1 2 8 →= 1 7 4 5 6 8 , 2 2 2 →= 2 7 4 5 6 2 , 2 2 7 →= 2 7 4 5 6 7 , 2 2 5 →= 2 7 4 5 6 5 , 2 2 8 →= 2 7 4 5 6 8 , 4 2 2 →= 4 7 4 5 6 2 , 4 2 7 →= 4 7 4 5 6 7 , 4 2 5 →= 4 7 4 5 6 5 , 4 2 8 →= 4 7 4 5 6 8 , 6 2 2 →= 6 7 4 5 6 2 , 6 2 7 →= 6 7 4 5 6 7 , 6 2 5 →= 6 7 4 5 6 5 , 6 2 8 →= 6 7 4 5 6 8 , 11 2 2 →= 11 7 4 5 6 2 , 11 2 7 →= 11 7 4 5 6 7 , 11 2 5 →= 11 7 4 5 6 5 , 11 2 8 →= 11 7 4 5 6 8 , 21 2 2 →= 21 7 4 5 6 2 , 21 2 7 →= 21 7 4 5 6 7 , 21 2 5 →= 21 7 4 5 6 5 , 21 2 8 →= 21 7 4 5 6 8 , 9 14 4 →= 1 5 6 , 9 14 13 →= 1 5 14 , 9 14 15 →= 1 5 16 , 9 14 17 →= 1 5 18 , 5 14 4 →= 2 5 6 , 5 14 13 →= 2 5 14 , 5 14 15 →= 2 5 16 , 5 14 17 →= 2 5 18 , 15 14 4 →= 4 5 6 , 15 14 13 →= 4 5 14 , 15 14 15 →= 4 5 16 , 15 14 17 →= 4 5 18 , 16 14 4 →= 6 5 6 , 16 14 13 →= 6 5 14 , 16 14 15 →= 6 5 16 , 16 14 17 →= 6 5 18 , 19 14 4 →= 11 5 6 , 19 14 13 →= 11 5 14 , 19 14 15 →= 11 5 16 , 19 14 17 →= 11 5 18 , 23 14 4 →= 21 5 6 , 23 14 13 →= 21 5 14 , 23 14 15 →= 21 5 16 , 23 14 17 →= 21 5 18 } 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 23 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 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, 19 ↦ 17, 21 ↦ 18, 22 ↦ 19, 23 ↦ 20 }, it remains to prove termination of the 79-rule system { 0 1 2 ⟶ 0 3 4 5 6 2 , 0 1 7 ⟶ 0 3 4 5 6 7 , 0 1 5 ⟶ 0 3 4 5 6 5 , 0 1 8 ⟶ 0 3 4 5 6 8 , 0 1 2 ⟶ 0 9 6 2 , 0 1 7 ⟶ 0 9 6 7 , 0 1 5 ⟶ 0 9 6 5 , 0 1 8 ⟶ 0 9 6 8 , 0 1 2 ⟶ 10 11 2 , 0 1 7 ⟶ 10 11 7 , 0 1 5 ⟶ 10 11 5 , 0 1 8 ⟶ 10 11 8 , 10 12 4 ⟶ 0 9 6 , 10 12 13 ⟶ 0 9 14 , 10 12 15 ⟶ 0 9 16 , 10 12 4 ⟶ 10 11 , 10 12 13 ⟶ 10 12 , 10 12 15 ⟶ 10 17 , 1 2 →= 1 , 1 7 →= 3 , 1 5 →= 9 , 2 2 →= 2 , 2 7 →= 7 , 2 5 →= 5 , 2 8 →= 8 , 4 2 →= 4 , 4 7 →= 13 , 4 5 →= 15 , 6 2 →= 6 , 6 7 →= 14 , 6 5 →= 16 , 11 2 →= 11 , 11 7 →= 12 , 11 5 →= 17 , 18 2 →= 18 , 18 7 →= 19 , 18 5 →= 20 , 1 2 2 →= 1 7 4 5 6 2 , 1 2 7 →= 1 7 4 5 6 7 , 1 2 5 →= 1 7 4 5 6 5 , 1 2 8 →= 1 7 4 5 6 8 , 2 2 2 →= 2 7 4 5 6 2 , 2 2 7 →= 2 7 4 5 6 7 , 2 2 5 →= 2 7 4 5 6 5 , 2 2 8 →= 2 7 4 5 6 8 , 4 2 2 →= 4 7 4 5 6 2 , 4 2 7 →= 4 7 4 5 6 7 , 4 2 5 →= 4 7 4 5 6 5 , 4 2 8 →= 4 7 4 5 6 8 , 6 2 2 →= 6 7 4 5 6 2 , 6 2 7 →= 6 7 4 5 6 7 , 6 2 5 →= 6 7 4 5 6 5 , 6 2 8 →= 6 7 4 5 6 8 , 11 2 2 →= 11 7 4 5 6 2 , 11 2 7 →= 11 7 4 5 6 7 , 11 2 5 →= 11 7 4 5 6 5 , 11 2 8 →= 11 7 4 5 6 8 , 18 2 2 →= 18 7 4 5 6 2 , 18 2 7 →= 18 7 4 5 6 7 , 18 2 5 →= 18 7 4 5 6 5 , 18 2 8 →= 18 7 4 5 6 8 , 9 14 4 →= 1 5 6 , 9 14 13 →= 1 5 14 , 9 14 15 →= 1 5 16 , 5 14 4 →= 2 5 6 , 5 14 13 →= 2 5 14 , 5 14 15 →= 2 5 16 , 15 14 4 →= 4 5 6 , 15 14 13 →= 4 5 14 , 15 14 15 →= 4 5 16 , 16 14 4 →= 6 5 6 , 16 14 13 →= 6 5 14 , 16 14 15 →= 6 5 16 , 17 14 4 →= 11 5 6 , 17 14 13 →= 11 5 14 , 17 14 15 →= 11 5 16 , 20 14 4 →= 18 5 6 , 20 14 13 →= 18 5 14 , 20 14 15 →= 18 5 16 } 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 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 4 ↦ 0, 7 ↦ 1, 13 ↦ 2, 5 ↦ 3, 15 ↦ 4, 6 ↦ 5, 14 ↦ 6, 16 ↦ 7, 1 ↦ 8, 2 ↦ 9, 8 ↦ 10, 11 ↦ 11, 18 ↦ 12 }, it remains to prove termination of the 31-rule system { 0 1 →= 2 , 0 3 →= 4 , 5 1 →= 6 , 5 3 →= 7 , 8 9 9 →= 8 1 0 3 5 9 , 8 9 1 →= 8 1 0 3 5 1 , 8 9 3 →= 8 1 0 3 5 3 , 8 9 10 →= 8 1 0 3 5 10 , 9 9 9 →= 9 1 0 3 5 9 , 9 9 1 →= 9 1 0 3 5 1 , 9 9 3 →= 9 1 0 3 5 3 , 9 9 10 →= 9 1 0 3 5 10 , 0 9 9 →= 0 1 0 3 5 9 , 0 9 1 →= 0 1 0 3 5 1 , 0 9 3 →= 0 1 0 3 5 3 , 0 9 10 →= 0 1 0 3 5 10 , 5 9 9 →= 5 1 0 3 5 9 , 5 9 1 →= 5 1 0 3 5 1 , 5 9 3 →= 5 1 0 3 5 3 , 5 9 10 →= 5 1 0 3 5 10 , 11 9 9 →= 11 1 0 3 5 9 , 11 9 1 →= 11 1 0 3 5 1 , 11 9 3 →= 11 1 0 3 5 3 , 11 9 10 →= 11 1 0 3 5 10 , 12 9 9 →= 12 1 0 3 5 9 , 12 9 1 →= 12 1 0 3 5 1 , 12 9 3 →= 12 1 0 3 5 3 , 12 9 10 →= 12 1 0 3 5 10 , 3 6 0 →= 9 3 5 , 3 6 2 →= 9 3 6 , 3 6 4 →= 9 3 7 } The system is trivially terminating.