/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 , 1 0 1 ⟶ 0 0 0 , 2 2 ⟶ 0 } The system was reversed. After renaming modulo the bijection { 0 ↦ 0, 2 ↦ 1, 1 ↦ 2 }, it remains to prove termination of the 3-rule system { 0 ⟶ 1 2 , 2 0 2 ⟶ 0 0 0 , 1 1 ⟶ 0 } Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols. After renaming modulo the bijection { (0,↑) ↦ 0, (1,↑) ↦ 1, (2,↓) ↦ 2, (2,↑) ↦ 3, (0,↓) ↦ 4, (1,↓) ↦ 5 }, it remains to prove termination of the 9-rule system { 0 ⟶ 1 2 , 0 ⟶ 3 , 3 4 2 ⟶ 0 4 4 , 3 4 2 ⟶ 0 4 , 3 4 2 ⟶ 0 , 1 5 ⟶ 0 , 4 →= 5 2 , 2 4 2 →= 4 4 4 , 5 5 →= 4 } Applying sparse tiling TROC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (6,0) ↦ 0, (0,2) ↦ 1, (6,1) ↦ 2, (1,2) ↦ 3, (2,2) ↦ 4, (0,4) ↦ 5, (2,4) ↦ 6, (0,5) ↦ 7, (2,5) ↦ 8, (0,7) ↦ 9, (2,7) ↦ 10, (6,3) ↦ 11, (3,2) ↦ 12, (3,4) ↦ 13, (3,5) ↦ 14, (3,7) ↦ 15, (4,2) ↦ 16, (4,4) ↦ 17, (4,5) ↦ 18, (4,7) ↦ 19, (1,5) ↦ 20, (5,2) ↦ 21, (5,4) ↦ 22, (5,5) ↦ 23, (1,4) ↦ 24, (6,4) ↦ 25, (6,5) ↦ 26 }, it remains to prove termination of the 96-rule system { 0 1 ⟶ 2 3 4 , 0 5 ⟶ 2 3 6 , 0 7 ⟶ 2 3 8 , 0 9 ⟶ 2 3 10 , 0 1 ⟶ 11 12 , 0 5 ⟶ 11 13 , 0 7 ⟶ 11 14 , 0 9 ⟶ 11 15 , 11 13 16 4 ⟶ 0 5 17 16 , 11 13 16 6 ⟶ 0 5 17 17 , 11 13 16 8 ⟶ 0 5 17 18 , 11 13 16 10 ⟶ 0 5 17 19 , 11 13 16 4 ⟶ 0 5 16 , 11 13 16 6 ⟶ 0 5 17 , 11 13 16 8 ⟶ 0 5 18 , 11 13 16 10 ⟶ 0 5 19 , 11 13 16 4 ⟶ 0 1 , 11 13 16 6 ⟶ 0 5 , 11 13 16 8 ⟶ 0 7 , 11 13 16 10 ⟶ 0 9 , 2 20 21 ⟶ 0 1 , 2 20 22 ⟶ 0 5 , 2 20 23 ⟶ 0 7 , 5 16 →= 7 21 4 , 5 17 →= 7 21 6 , 5 18 →= 7 21 8 , 5 19 →= 7 21 10 , 24 16 →= 20 21 4 , 24 17 →= 20 21 6 , 24 18 →= 20 21 8 , 24 19 →= 20 21 10 , 6 16 →= 8 21 4 , 6 17 →= 8 21 6 , 6 18 →= 8 21 8 , 6 19 →= 8 21 10 , 13 16 →= 14 21 4 , 13 17 →= 14 21 6 , 13 18 →= 14 21 8 , 13 19 →= 14 21 10 , 17 16 →= 18 21 4 , 17 17 →= 18 21 6 , 17 18 →= 18 21 8 , 17 19 →= 18 21 10 , 22 16 →= 23 21 4 , 22 17 →= 23 21 6 , 22 18 →= 23 21 8 , 22 19 →= 23 21 10 , 25 16 →= 26 21 4 , 25 17 →= 26 21 6 , 25 18 →= 26 21 8 , 25 19 →= 26 21 10 , 1 6 16 4 →= 5 17 17 16 , 1 6 16 6 →= 5 17 17 17 , 1 6 16 8 →= 5 17 17 18 , 1 6 16 10 →= 5 17 17 19 , 3 6 16 4 →= 24 17 17 16 , 3 6 16 6 →= 24 17 17 17 , 3 6 16 8 →= 24 17 17 18 , 3 6 16 10 →= 24 17 17 19 , 4 6 16 4 →= 6 17 17 16 , 4 6 16 6 →= 6 17 17 17 , 4 6 16 8 →= 6 17 17 18 , 4 6 16 10 →= 6 17 17 19 , 12 6 16 4 →= 13 17 17 16 , 12 6 16 6 →= 13 17 17 17 , 12 6 16 8 →= 13 17 17 18 , 12 6 16 10 →= 13 17 17 19 , 16 6 16 4 →= 17 17 17 16 , 16 6 16 6 →= 17 17 17 17 , 16 6 16 8 →= 17 17 17 18 , 16 6 16 10 →= 17 17 17 19 , 21 6 16 4 →= 22 17 17 16 , 21 6 16 6 →= 22 17 17 17 , 21 6 16 8 →= 22 17 17 18 , 21 6 16 10 →= 22 17 17 19 , 7 23 21 →= 5 16 , 7 23 22 →= 5 17 , 7 23 23 →= 5 18 , 20 23 21 →= 24 16 , 20 23 22 →= 24 17 , 20 23 23 →= 24 18 , 8 23 21 →= 6 16 , 8 23 22 →= 6 17 , 8 23 23 →= 6 18 , 14 23 21 →= 13 16 , 14 23 22 →= 13 17 , 14 23 23 →= 13 18 , 18 23 21 →= 17 16 , 18 23 22 →= 17 17 , 18 23 23 →= 17 18 , 23 23 21 →= 22 16 , 23 23 22 →= 22 17 , 23 23 23 →= 22 18 , 26 23 21 →= 25 16 , 26 23 22 →= 25 17 , 26 23 23 →= 25 18 } Applying sparse untiling TROCU(2) after reversal [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { 1 ↦ 0, 0 ↦ 1, 4 ↦ 2, 3 ↦ 3, 2 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 12 ↦ 11, 11 ↦ 12, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 21 ↦ 20, 20 ↦ 21, 22 ↦ 22, 23 ↦ 23, 24 ↦ 24, 25 ↦ 25, 26 ↦ 26 }, it remains to prove termination of the 93-rule system { 0 1 ⟶ 2 3 4 , 0 5 ⟶ 2 3 6 , 0 7 ⟶ 2 3 8 , 0 9 ⟶ 2 3 10 , 0 1 ⟶ 11 12 , 0 5 ⟶ 11 13 , 0 7 ⟶ 11 14 , 0 9 ⟶ 11 15 , 11 13 16 4 ⟶ 0 5 17 16 , 11 13 16 6 ⟶ 0 5 17 17 , 11 13 16 8 ⟶ 0 5 17 18 , 11 13 16 10 ⟶ 0 5 17 19 , 11 13 16 4 ⟶ 0 5 16 , 11 13 16 6 ⟶ 0 5 17 , 11 13 16 8 ⟶ 0 5 18 , 11 13 16 10 ⟶ 0 5 19 , 11 13 16 4 ⟶ 0 1 , 11 13 16 6 ⟶ 0 5 , 11 13 16 8 ⟶ 0 7 , 11 13 16 10 ⟶ 0 9 , 2 20 21 ⟶ 0 1 , 2 20 22 ⟶ 0 5 , 2 20 23 ⟶ 0 7 , 5 16 →= 7 21 4 , 5 17 →= 7 21 6 , 5 18 →= 7 21 8 , 5 19 →= 7 21 10 , 24 16 →= 20 21 4 , 24 17 →= 20 21 6 , 24 18 →= 20 21 8 , 6 16 →= 8 21 4 , 6 17 →= 8 21 6 , 6 18 →= 8 21 8 , 6 19 →= 8 21 10 , 13 16 →= 14 21 4 , 13 17 →= 14 21 6 , 13 18 →= 14 21 8 , 13 19 →= 14 21 10 , 17 16 →= 18 21 4 , 17 17 →= 18 21 6 , 17 18 →= 18 21 8 , 17 19 →= 18 21 10 , 22 16 →= 23 21 4 , 22 17 →= 23 21 6 , 22 18 →= 23 21 8 , 25 16 →= 26 21 4 , 25 17 →= 26 21 6 , 25 18 →= 26 21 8 , 1 6 16 4 →= 5 17 17 16 , 1 6 16 6 →= 5 17 17 17 , 1 6 16 8 →= 5 17 17 18 , 1 6 16 10 →= 5 17 17 19 , 3 6 16 4 →= 24 17 17 16 , 3 6 16 6 →= 24 17 17 17 , 3 6 16 8 →= 24 17 17 18 , 3 6 16 10 →= 24 17 17 19 , 4 6 16 4 →= 6 17 17 16 , 4 6 16 6 →= 6 17 17 17 , 4 6 16 8 →= 6 17 17 18 , 4 6 16 10 →= 6 17 17 19 , 12 6 16 4 →= 13 17 17 16 , 12 6 16 6 →= 13 17 17 17 , 12 6 16 8 →= 13 17 17 18 , 12 6 16 10 →= 13 17 17 19 , 16 6 16 4 →= 17 17 17 16 , 16 6 16 6 →= 17 17 17 17 , 16 6 16 8 →= 17 17 17 18 , 16 6 16 10 →= 17 17 17 19 , 21 6 16 4 →= 22 17 17 16 , 21 6 16 6 →= 22 17 17 17 , 21 6 16 8 →= 22 17 17 18 , 21 6 16 10 →= 22 17 17 19 , 7 23 21 →= 5 16 , 7 23 22 →= 5 17 , 7 23 23 →= 5 18 , 20 23 21 →= 24 16 , 20 23 22 →= 24 17 , 20 23 23 →= 24 18 , 8 23 21 →= 6 16 , 8 23 22 →= 6 17 , 8 23 23 →= 6 18 , 14 23 21 →= 13 16 , 14 23 22 →= 13 17 , 14 23 23 →= 13 18 , 18 23 21 →= 17 16 , 18 23 22 →= 17 17 , 18 23 23 →= 17 18 , 23 23 21 →= 22 16 , 23 23 22 →= 22 17 , 23 23 23 →= 22 18 , 26 23 21 →= 25 16 , 26 23 22 →= 25 17 , 26 23 23 →= 25 18 } 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 0 ⎟ ⎜ 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 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 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 23 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 24 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 25 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 26 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 20 ↦ 5, 21 ↦ 6, 5 ↦ 7, 16 ↦ 8, 7 ↦ 9, 17 ↦ 10, 6 ↦ 11, 18 ↦ 12, 8 ↦ 13, 19 ↦ 14, 10 ↦ 15, 24 ↦ 16, 13 ↦ 17, 14 ↦ 18, 22 ↦ 19, 23 ↦ 20, 25 ↦ 21, 26 ↦ 22 }, it remains to prove termination of the 42-rule system { 0 1 ⟶ 2 3 4 , 2 5 6 ⟶ 0 1 , 7 8 →= 9 6 4 , 7 10 →= 9 6 11 , 7 12 →= 9 6 13 , 7 14 →= 9 6 15 , 16 8 →= 5 6 4 , 16 10 →= 5 6 11 , 16 12 →= 5 6 13 , 11 8 →= 13 6 4 , 11 10 →= 13 6 11 , 11 12 →= 13 6 13 , 11 14 →= 13 6 15 , 17 8 →= 18 6 4 , 17 10 →= 18 6 11 , 17 12 →= 18 6 13 , 17 14 →= 18 6 15 , 10 8 →= 12 6 4 , 10 10 →= 12 6 11 , 10 12 →= 12 6 13 , 10 14 →= 12 6 15 , 19 8 →= 20 6 4 , 19 10 →= 20 6 11 , 19 12 →= 20 6 13 , 21 8 →= 22 6 4 , 21 10 →= 22 6 11 , 21 12 →= 22 6 13 , 3 11 8 4 →= 16 10 10 8 , 3 11 8 11 →= 16 10 10 10 , 3 11 8 13 →= 16 10 10 12 , 3 11 8 15 →= 16 10 10 14 , 6 11 8 4 →= 19 10 10 8 , 6 11 8 11 →= 19 10 10 10 , 6 11 8 13 →= 19 10 10 12 , 6 11 8 15 →= 19 10 10 14 , 9 20 6 →= 7 8 , 5 20 6 →= 16 8 , 13 20 6 →= 11 8 , 18 20 6 →= 17 8 , 12 20 6 →= 10 8 , 20 20 6 →= 19 8 , 22 20 6 →= 21 8 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 3: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 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 1 ⎟ ⎜ 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 1 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 1 0 0 ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 1 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, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13, 16 ↦ 14, 14 ↦ 15, 15 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 41-rule system { 0 1 ⟶ 2 3 4 , 2 5 6 ⟶ 0 1 , 7 8 →= 9 6 4 , 7 10 →= 9 6 11 , 7 12 →= 9 6 13 , 14 8 →= 5 6 4 , 14 10 →= 5 6 11 , 14 12 →= 5 6 13 , 11 8 →= 13 6 4 , 11 10 →= 13 6 11 , 11 12 →= 13 6 13 , 11 15 →= 13 6 16 , 17 8 →= 18 6 4 , 17 10 →= 18 6 11 , 17 12 →= 18 6 13 , 17 15 →= 18 6 16 , 10 8 →= 12 6 4 , 10 10 →= 12 6 11 , 10 12 →= 12 6 13 , 10 15 →= 12 6 16 , 19 8 →= 20 6 4 , 19 10 →= 20 6 11 , 19 12 →= 20 6 13 , 21 8 →= 22 6 4 , 21 10 →= 22 6 11 , 21 12 →= 22 6 13 , 3 11 8 4 →= 14 10 10 8 , 3 11 8 11 →= 14 10 10 10 , 3 11 8 13 →= 14 10 10 12 , 3 11 8 16 →= 14 10 10 15 , 6 11 8 4 →= 19 10 10 8 , 6 11 8 11 →= 19 10 10 10 , 6 11 8 13 →= 19 10 10 12 , 6 11 8 16 →= 19 10 10 15 , 9 20 6 →= 7 8 , 5 20 6 →= 14 8 , 13 20 6 →= 11 8 , 18 20 6 →= 17 8 , 12 20 6 →= 10 8 , 20 20 6 →= 19 8 , 22 20 6 →= 21 8 } Applying sparse untiling TROCU(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 7 ↦ 5, 8 ↦ 6, 9 ↦ 7, 6 ↦ 8, 14 ↦ 9, 5 ↦ 10, 10 ↦ 11, 11 ↦ 12, 12 ↦ 13, 13 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 29-rule system { 0 1 ⟶ 2 3 4 , 5 6 →= 7 8 4 , 9 6 →= 10 8 4 , 9 11 →= 10 8 12 , 9 13 →= 10 8 14 , 12 6 →= 14 8 4 , 12 11 →= 14 8 12 , 12 13 →= 14 8 14 , 12 15 →= 14 8 16 , 17 6 →= 18 8 4 , 11 6 →= 13 8 4 , 11 11 →= 13 8 12 , 11 13 →= 13 8 14 , 11 15 →= 13 8 16 , 19 6 →= 20 8 4 , 19 11 →= 20 8 12 , 19 13 →= 20 8 14 , 21 6 →= 22 8 4 , 8 12 6 4 →= 19 11 11 6 , 8 12 6 12 →= 19 11 11 11 , 8 12 6 14 →= 19 11 11 13 , 8 12 6 16 →= 19 11 11 15 , 7 20 8 →= 5 6 , 10 20 8 →= 9 6 , 14 20 8 →= 12 6 , 18 20 8 →= 17 6 , 13 20 8 →= 11 6 , 20 20 8 →= 19 6 , 22 20 8 →= 21 6 } 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 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 0 ⎟ ⎜ 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 ⎟ ⎝ ⎠ After renaming modulo the bijection { 5 ↦ 0, 6 ↦ 1, 7 ↦ 2, 8 ↦ 3, 4 ↦ 4, 9 ↦ 5, 10 ↦ 6, 11 ↦ 7, 12 ↦ 8, 13 ↦ 9, 14 ↦ 10, 15 ↦ 11, 16 ↦ 12, 17 ↦ 13, 18 ↦ 14, 19 ↦ 15, 20 ↦ 16, 21 ↦ 17, 22 ↦ 18 }, it remains to prove termination of the 28-rule system { 0 1 →= 2 3 4 , 5 1 →= 6 3 4 , 5 7 →= 6 3 8 , 5 9 →= 6 3 10 , 8 1 →= 10 3 4 , 8 7 →= 10 3 8 , 8 9 →= 10 3 10 , 8 11 →= 10 3 12 , 13 1 →= 14 3 4 , 7 1 →= 9 3 4 , 7 7 →= 9 3 8 , 7 9 →= 9 3 10 , 7 11 →= 9 3 12 , 15 1 →= 16 3 4 , 15 7 →= 16 3 8 , 15 9 →= 16 3 10 , 17 1 →= 18 3 4 , 3 8 1 4 →= 15 7 7 1 , 3 8 1 8 →= 15 7 7 7 , 3 8 1 10 →= 15 7 7 9 , 3 8 1 12 →= 15 7 7 11 , 2 16 3 →= 0 1 , 6 16 3 →= 5 1 , 10 16 3 →= 8 1 , 14 16 3 →= 13 1 , 9 16 3 →= 7 1 , 16 16 3 →= 15 1 , 18 16 3 →= 17 1 } The system is trivially terminating.