/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 3-rule system { 0 1 ⟶ , 0 2 ⟶ 2 2 , 1 2 ⟶ 0 0 1 1 } Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols. After renaming modulo the bijection { (1,↑) ↦ 0, (2,↓) ↦ 1, (0,↑) ↦ 2, (0,↓) ↦ 3, (1,↓) ↦ 4 }, it remains to prove termination of the 7-rule system { 0 1 ⟶ 2 3 4 4 , 0 1 ⟶ 2 4 4 , 0 1 ⟶ 0 4 , 0 1 ⟶ 0 , 3 4 →= , 3 1 →= 1 1 , 4 1 →= 3 3 4 4 } 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 4 ↦ 2, 3 ↦ 3 }, it remains to prove termination of the 5-rule system { 0 1 ⟶ 0 2 , 0 1 ⟶ 0 , 3 2 →= , 3 1 →= 1 1 , 2 1 →= 3 3 2 2 } Applying sparse tiling TROC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (4,0) ↦ 0, (0,1) ↦ 1, (1,1) ↦ 2, (0,2) ↦ 3, (2,1) ↦ 4, (1,5) ↦ 5, (2,5) ↦ 6, (0,5) ↦ 7, (0,3) ↦ 8, (3,2) ↦ 9, (2,2) ↦ 10, (2,3) ↦ 11, (3,3) ↦ 12, (3,1) ↦ 13, (3,5) ↦ 14, (4,3) ↦ 15, (4,1) ↦ 16, (4,2) ↦ 17, (4,5) ↦ 18 }, it remains to prove termination of the 36-rule system { 0 1 2 ⟶ 0 3 4 , 0 1 5 ⟶ 0 3 6 , 0 1 2 ⟶ 0 1 , 0 1 5 ⟶ 0 7 , 8 9 4 →= 1 , 8 9 10 →= 3 , 8 9 11 →= 8 , 8 9 6 →= 7 , 11 9 4 →= 4 , 11 9 10 →= 10 , 11 9 11 →= 11 , 11 9 6 →= 6 , 12 9 4 →= 13 , 12 9 10 →= 9 , 12 9 11 →= 12 , 12 9 6 →= 14 , 15 9 4 →= 16 , 15 9 10 →= 17 , 15 9 11 →= 15 , 15 9 6 →= 18 , 8 13 2 →= 1 2 2 , 8 13 5 →= 1 2 5 , 11 13 2 →= 4 2 2 , 11 13 5 →= 4 2 5 , 12 13 2 →= 13 2 2 , 12 13 5 →= 13 2 5 , 15 13 2 →= 16 2 2 , 15 13 5 →= 16 2 5 , 3 4 2 →= 8 12 9 10 4 , 3 4 5 →= 8 12 9 10 6 , 10 4 2 →= 11 12 9 10 4 , 10 4 5 →= 11 12 9 10 6 , 9 4 2 →= 12 12 9 10 4 , 9 4 5 →= 12 12 9 10 6 , 17 4 2 →= 15 12 9 10 4 , 17 4 5 →= 15 12 9 10 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 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 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 8 ↦ 5, 9 ↦ 6, 10 ↦ 7, 11 ↦ 8, 6 ↦ 9, 7 ↦ 10, 12 ↦ 11, 13 ↦ 12, 14 ↦ 13, 15 ↦ 14, 16 ↦ 15, 17 ↦ 16, 18 ↦ 17, 5 ↦ 18 }, it remains to prove termination of the 30-rule system { 0 1 2 ⟶ 0 3 4 , 0 1 2 ⟶ 0 1 , 5 6 4 →= 1 , 5 6 7 →= 3 , 5 6 8 →= 5 , 5 6 9 →= 10 , 8 6 4 →= 4 , 8 6 7 →= 7 , 8 6 8 →= 8 , 8 6 9 →= 9 , 11 6 4 →= 12 , 11 6 7 →= 6 , 11 6 8 →= 11 , 11 6 9 →= 13 , 14 6 4 →= 15 , 14 6 7 →= 16 , 14 6 8 →= 14 , 14 6 9 →= 17 , 5 12 2 →= 1 2 2 , 5 12 18 →= 1 2 18 , 8 12 2 →= 4 2 2 , 8 12 18 →= 4 2 18 , 11 12 2 →= 12 2 2 , 11 12 18 →= 12 2 18 , 14 12 2 →= 15 2 2 , 14 12 18 →= 15 2 18 , 3 4 2 →= 5 11 6 7 4 , 7 4 2 →= 8 11 6 7 4 , 6 4 2 →= 11 11 6 7 4 , 16 4 2 →= 14 11 6 7 4 } 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 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 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 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, 8 ↦ 8, 9 ↦ 9, 11 ↦ 10, 12 ↦ 11, 14 ↦ 12, 15 ↦ 13, 16 ↦ 14, 18 ↦ 15 }, it remains to prove termination of the 27-rule system { 0 1 2 ⟶ 0 3 4 , 0 1 2 ⟶ 0 1 , 5 6 4 →= 1 , 5 6 7 →= 3 , 5 6 8 →= 5 , 8 6 4 →= 4 , 8 6 7 →= 7 , 8 6 8 →= 8 , 8 6 9 →= 9 , 10 6 4 →= 11 , 10 6 7 →= 6 , 10 6 8 →= 10 , 12 6 4 →= 13 , 12 6 7 →= 14 , 12 6 8 →= 12 , 5 11 2 →= 1 2 2 , 5 11 15 →= 1 2 15 , 8 11 2 →= 4 2 2 , 8 11 15 →= 4 2 15 , 10 11 2 →= 11 2 2 , 10 11 15 →= 11 2 15 , 12 11 2 →= 13 2 2 , 12 11 15 →= 13 2 15 , 3 4 2 →= 5 10 6 7 4 , 7 4 2 →= 8 10 6 7 4 , 6 4 2 →= 10 10 6 7 4 , 14 4 2 →= 12 10 6 7 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 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 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, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 14 ↦ 13, 15 ↦ 14 }, it remains to prove termination of the 24-rule system { 0 1 2 ⟶ 0 3 4 , 0 1 2 ⟶ 0 1 , 5 6 4 →= 1 , 5 6 7 →= 3 , 5 6 8 →= 5 , 8 6 4 →= 4 , 8 6 7 →= 7 , 8 6 8 →= 8 , 8 6 9 →= 9 , 10 6 4 →= 11 , 10 6 7 →= 6 , 10 6 8 →= 10 , 12 6 7 →= 13 , 12 6 8 →= 12 , 5 11 2 →= 1 2 2 , 5 11 14 →= 1 2 14 , 8 11 2 →= 4 2 2 , 8 11 14 →= 4 2 14 , 10 11 2 →= 11 2 2 , 10 11 14 →= 11 2 14 , 3 4 2 →= 5 10 6 7 4 , 7 4 2 →= 8 10 6 7 4 , 6 4 2 →= 10 10 6 7 4 , 13 4 2 →= 12 10 6 7 4 } Applying sparse tiling TROC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (15,0) ↦ 0, (0,1) ↦ 1, (1,2) ↦ 2, (2,16) ↦ 3, (0,3) ↦ 4, (3,4) ↦ 5, (4,16) ↦ 6, (2,2) ↦ 7, (4,2) ↦ 8, (2,14) ↦ 9, (4,14) ↦ 10, (1,16) ↦ 11, (1,14) ↦ 12, (0,5) ↦ 13, (5,6) ↦ 14, (6,4) ↦ 15, (15,5) ↦ 16, (15,1) ↦ 17, (6,7) ↦ 18, (7,16) ↦ 19, (3,16) ↦ 20, (7,4) ↦ 21, (15,3) ↦ 22, (6,8) ↦ 23, (8,16) ↦ 24, (5,16) ↦ 25, (8,6) ↦ 26, (8,10) ↦ 27, (5,10) ↦ 28, (8,11) ↦ 29, (5,11) ↦ 30, (15,8) ↦ 31, (15,4) ↦ 32, (15,7) ↦ 33, (6,9) ↦ 34, (9,16) ↦ 35, (15,9) ↦ 36, (10,6) ↦ 37, (11,16) ↦ 38, (11,2) ↦ 39, (11,14) ↦ 40, (10,10) ↦ 41, (10,11) ↦ 42, (12,10) ↦ 43, (12,11) ↦ 44, (15,10) ↦ 45, (15,11) ↦ 46, (6,16) ↦ 47, (12,6) ↦ 48, (15,6) ↦ 49, (10,16) ↦ 50, (15,12) ↦ 51, (15,13) ↦ 52, (13,16) ↦ 53, (13,4) ↦ 54, (12,16) ↦ 55, (14,16) ↦ 56 }, it remains to prove termination of the 161-rule system { 0 1 2 3 ⟶ 0 4 5 6 , 0 1 2 7 ⟶ 0 4 5 8 , 0 1 2 9 ⟶ 0 4 5 10 , 0 1 2 3 ⟶ 0 1 11 , 0 1 2 7 ⟶ 0 1 2 , 0 1 2 9 ⟶ 0 1 12 , 13 14 15 6 →= 1 11 , 13 14 15 8 →= 1 2 , 13 14 15 10 →= 1 12 , 16 14 15 6 →= 17 11 , 16 14 15 8 →= 17 2 , 16 14 15 10 →= 17 12 , 13 14 18 19 →= 4 20 , 13 14 18 21 →= 4 5 , 16 14 18 19 →= 22 20 , 16 14 18 21 →= 22 5 , 13 14 23 24 →= 13 25 , 13 14 23 26 →= 13 14 , 13 14 23 27 →= 13 28 , 13 14 23 29 →= 13 30 , 16 14 23 24 →= 16 25 , 16 14 23 26 →= 16 14 , 16 14 23 27 →= 16 28 , 16 14 23 29 →= 16 30 , 23 26 15 6 →= 15 6 , 23 26 15 8 →= 15 8 , 23 26 15 10 →= 15 10 , 31 26 15 6 →= 32 6 , 31 26 15 8 →= 32 8 , 31 26 15 10 →= 32 10 , 23 26 18 19 →= 18 19 , 23 26 18 21 →= 18 21 , 31 26 18 19 →= 33 19 , 31 26 18 21 →= 33 21 , 23 26 23 24 →= 23 24 , 23 26 23 26 →= 23 26 , 23 26 23 27 →= 23 27 , 23 26 23 29 →= 23 29 , 31 26 23 24 →= 31 24 , 31 26 23 26 →= 31 26 , 31 26 23 27 →= 31 27 , 31 26 23 29 →= 31 29 , 23 26 34 35 →= 34 35 , 31 26 34 35 →= 36 35 , 28 37 15 6 →= 30 38 , 28 37 15 8 →= 30 39 , 28 37 15 10 →= 30 40 , 27 37 15 6 →= 29 38 , 27 37 15 8 →= 29 39 , 27 37 15 10 →= 29 40 , 41 37 15 6 →= 42 38 , 41 37 15 8 →= 42 39 , 41 37 15 10 →= 42 40 , 43 37 15 6 →= 44 38 , 43 37 15 8 →= 44 39 , 43 37 15 10 →= 44 40 , 45 37 15 6 →= 46 38 , 45 37 15 8 →= 46 39 , 45 37 15 10 →= 46 40 , 28 37 18 19 →= 14 47 , 28 37 18 21 →= 14 15 , 27 37 18 19 →= 26 47 , 27 37 18 21 →= 26 15 , 41 37 18 19 →= 37 47 , 41 37 18 21 →= 37 15 , 43 37 18 19 →= 48 47 , 43 37 18 21 →= 48 15 , 45 37 18 19 →= 49 47 , 45 37 18 21 →= 49 15 , 28 37 23 24 →= 28 50 , 28 37 23 26 →= 28 37 , 28 37 23 27 →= 28 41 , 28 37 23 29 →= 28 42 , 27 37 23 24 →= 27 50 , 27 37 23 26 →= 27 37 , 27 37 23 27 →= 27 41 , 27 37 23 29 →= 27 42 , 41 37 23 24 →= 41 50 , 41 37 23 26 →= 41 37 , 41 37 23 27 →= 41 41 , 41 37 23 29 →= 41 42 , 43 37 23 24 →= 43 50 , 43 37 23 26 →= 43 37 , 43 37 23 27 →= 43 41 , 43 37 23 29 →= 43 42 , 45 37 23 24 →= 45 50 , 45 37 23 26 →= 45 37 , 45 37 23 27 →= 45 41 , 45 37 23 29 →= 45 42 , 51 48 18 19 →= 52 53 , 51 48 18 21 →= 52 54 , 51 48 23 24 →= 51 55 , 51 48 23 26 →= 51 48 , 51 48 23 27 →= 51 43 , 51 48 23 29 →= 51 44 , 13 30 39 3 →= 1 2 7 3 , 13 30 39 7 →= 1 2 7 7 , 13 30 39 9 →= 1 2 7 9 , 16 30 39 3 →= 17 2 7 3 , 16 30 39 7 →= 17 2 7 7 , 16 30 39 9 →= 17 2 7 9 , 13 30 40 56 →= 1 2 9 56 , 16 30 40 56 →= 17 2 9 56 , 23 29 39 3 →= 15 8 7 3 , 23 29 39 7 →= 15 8 7 7 , 23 29 39 9 →= 15 8 7 9 , 31 29 39 3 →= 32 8 7 3 , 31 29 39 7 →= 32 8 7 7 , 31 29 39 9 →= 32 8 7 9 , 23 29 40 56 →= 15 8 9 56 , 31 29 40 56 →= 32 8 9 56 , 28 42 39 3 →= 30 39 7 3 , 28 42 39 7 →= 30 39 7 7 , 28 42 39 9 →= 30 39 7 9 , 27 42 39 3 →= 29 39 7 3 , 27 42 39 7 →= 29 39 7 7 , 27 42 39 9 →= 29 39 7 9 , 41 42 39 3 →= 42 39 7 3 , 41 42 39 7 →= 42 39 7 7 , 41 42 39 9 →= 42 39 7 9 , 43 42 39 3 →= 44 39 7 3 , 43 42 39 7 →= 44 39 7 7 , 43 42 39 9 →= 44 39 7 9 , 45 42 39 3 →= 46 39 7 3 , 45 42 39 7 →= 46 39 7 7 , 45 42 39 9 →= 46 39 7 9 , 28 42 40 56 →= 30 39 9 56 , 27 42 40 56 →= 29 39 9 56 , 41 42 40 56 →= 42 39 9 56 , 43 42 40 56 →= 44 39 9 56 , 45 42 40 56 →= 46 39 9 56 , 4 5 8 3 →= 13 28 37 18 21 6 , 4 5 8 7 →= 13 28 37 18 21 8 , 4 5 8 9 →= 13 28 37 18 21 10 , 22 5 8 3 →= 16 28 37 18 21 6 , 22 5 8 7 →= 16 28 37 18 21 8 , 22 5 8 9 →= 16 28 37 18 21 10 , 18 21 8 3 →= 23 27 37 18 21 6 , 18 21 8 7 →= 23 27 37 18 21 8 , 18 21 8 9 →= 23 27 37 18 21 10 , 33 21 8 3 →= 31 27 37 18 21 6 , 33 21 8 7 →= 31 27 37 18 21 8 , 33 21 8 9 →= 31 27 37 18 21 10 , 14 15 8 3 →= 28 41 37 18 21 6 , 14 15 8 7 →= 28 41 37 18 21 8 , 14 15 8 9 →= 28 41 37 18 21 10 , 26 15 8 3 →= 27 41 37 18 21 6 , 26 15 8 7 →= 27 41 37 18 21 8 , 26 15 8 9 →= 27 41 37 18 21 10 , 37 15 8 3 →= 41 41 37 18 21 6 , 37 15 8 7 →= 41 41 37 18 21 8 , 37 15 8 9 →= 41 41 37 18 21 10 , 48 15 8 3 →= 43 41 37 18 21 6 , 48 15 8 7 →= 43 41 37 18 21 8 , 48 15 8 9 →= 43 41 37 18 21 10 , 49 15 8 3 →= 45 41 37 18 21 6 , 49 15 8 7 →= 45 41 37 18 21 8 , 49 15 8 9 →= 45 41 37 18 21 10 , 52 54 8 3 →= 51 43 37 18 21 6 , 52 54 8 7 →= 51 43 37 18 21 8 , 52 54 8 9 →= 51 43 37 18 21 10 } 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 23 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 24 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 25 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 26 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 27 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 28 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 29 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 30 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 31 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 32 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 33 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 34 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 35 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 36 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 37 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 38 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 39 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 40 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 41 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 42 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 43 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 44 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 45 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 46 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 47 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 48 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 49 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 50 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 51 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 52 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 53 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 54 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 55 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 56 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 7 ↦ 3, 4 ↦ 4, 5 ↦ 5, 8 ↦ 6, 9 ↦ 7, 10 ↦ 8, 12 ↦ 9, 13 ↦ 10, 14 ↦ 11, 15 ↦ 12, 6 ↦ 13, 11 ↦ 14, 16 ↦ 15, 17 ↦ 16, 18 ↦ 17, 21 ↦ 18, 22 ↦ 19, 23 ↦ 20, 26 ↦ 21, 27 ↦ 22, 28 ↦ 23, 29 ↦ 24, 30 ↦ 25, 31 ↦ 26, 32 ↦ 27, 19 ↦ 28, 33 ↦ 29, 24 ↦ 30, 34 ↦ 31, 35 ↦ 32, 37 ↦ 33, 38 ↦ 34, 39 ↦ 35, 40 ↦ 36, 41 ↦ 37, 42 ↦ 38, 43 ↦ 39, 44 ↦ 40, 45 ↦ 41, 46 ↦ 42, 48 ↦ 43, 49 ↦ 44, 51 ↦ 45, 52 ↦ 46, 54 ↦ 47, 3 ↦ 48, 56 ↦ 49 }, it remains to prove termination of the 132-rule system { 0 1 2 3 ⟶ 0 4 5 6 , 0 1 2 7 ⟶ 0 4 5 8 , 0 1 2 3 ⟶ 0 1 2 , 0 1 2 7 ⟶ 0 1 9 , 10 11 12 13 →= 1 14 , 10 11 12 6 →= 1 2 , 10 11 12 8 →= 1 9 , 15 11 12 13 →= 16 14 , 15 11 12 6 →= 16 2 , 15 11 12 8 →= 16 9 , 10 11 17 18 →= 4 5 , 15 11 17 18 →= 19 5 , 10 11 20 21 →= 10 11 , 10 11 20 22 →= 10 23 , 10 11 20 24 →= 10 25 , 15 11 20 21 →= 15 11 , 15 11 20 22 →= 15 23 , 15 11 20 24 →= 15 25 , 20 21 12 13 →= 12 13 , 20 21 12 6 →= 12 6 , 20 21 12 8 →= 12 8 , 26 21 12 13 →= 27 13 , 26 21 12 6 →= 27 6 , 26 21 12 8 →= 27 8 , 20 21 17 28 →= 17 28 , 20 21 17 18 →= 17 18 , 26 21 17 28 →= 29 28 , 26 21 17 18 →= 29 18 , 20 21 20 30 →= 20 30 , 20 21 20 21 →= 20 21 , 20 21 20 22 →= 20 22 , 20 21 20 24 →= 20 24 , 26 21 20 30 →= 26 30 , 26 21 20 21 →= 26 21 , 26 21 20 22 →= 26 22 , 26 21 20 24 →= 26 24 , 20 21 31 32 →= 31 32 , 23 33 12 13 →= 25 34 , 23 33 12 6 →= 25 35 , 23 33 12 8 →= 25 36 , 22 33 12 13 →= 24 34 , 22 33 12 6 →= 24 35 , 22 33 12 8 →= 24 36 , 37 33 12 13 →= 38 34 , 37 33 12 6 →= 38 35 , 37 33 12 8 →= 38 36 , 39 33 12 13 →= 40 34 , 39 33 12 6 →= 40 35 , 39 33 12 8 →= 40 36 , 41 33 12 13 →= 42 34 , 41 33 12 6 →= 42 35 , 41 33 12 8 →= 42 36 , 23 33 17 18 →= 11 12 , 22 33 17 18 →= 21 12 , 37 33 17 18 →= 33 12 , 39 33 17 18 →= 43 12 , 41 33 17 18 →= 44 12 , 23 33 20 21 →= 23 33 , 23 33 20 22 →= 23 37 , 23 33 20 24 →= 23 38 , 22 33 20 21 →= 22 33 , 22 33 20 22 →= 22 37 , 22 33 20 24 →= 22 38 , 37 33 20 21 →= 37 33 , 37 33 20 22 →= 37 37 , 37 33 20 24 →= 37 38 , 39 33 20 21 →= 39 33 , 39 33 20 22 →= 39 37 , 39 33 20 24 →= 39 38 , 41 33 20 21 →= 41 33 , 41 33 20 22 →= 41 37 , 41 33 20 24 →= 41 38 , 45 43 17 18 →= 46 47 , 45 43 20 21 →= 45 43 , 45 43 20 22 →= 45 39 , 45 43 20 24 →= 45 40 , 10 25 35 48 →= 1 2 3 48 , 10 25 35 3 →= 1 2 3 3 , 10 25 35 7 →= 1 2 3 7 , 15 25 35 48 →= 16 2 3 48 , 15 25 35 3 →= 16 2 3 3 , 15 25 35 7 →= 16 2 3 7 , 10 25 36 49 →= 1 2 7 49 , 15 25 36 49 →= 16 2 7 49 , 20 24 35 48 →= 12 6 3 48 , 20 24 35 3 →= 12 6 3 3 , 20 24 35 7 →= 12 6 3 7 , 26 24 35 48 →= 27 6 3 48 , 26 24 35 3 →= 27 6 3 3 , 26 24 35 7 →= 27 6 3 7 , 20 24 36 49 →= 12 6 7 49 , 26 24 36 49 →= 27 6 7 49 , 23 38 35 48 →= 25 35 3 48 , 23 38 35 3 →= 25 35 3 3 , 23 38 35 7 →= 25 35 3 7 , 22 38 35 48 →= 24 35 3 48 , 22 38 35 3 →= 24 35 3 3 , 22 38 35 7 →= 24 35 3 7 , 37 38 35 48 →= 38 35 3 48 , 37 38 35 3 →= 38 35 3 3 , 37 38 35 7 →= 38 35 3 7 , 39 38 35 48 →= 40 35 3 48 , 39 38 35 3 →= 40 35 3 3 , 39 38 35 7 →= 40 35 3 7 , 41 38 35 48 →= 42 35 3 48 , 41 38 35 3 →= 42 35 3 3 , 41 38 35 7 →= 42 35 3 7 , 23 38 36 49 →= 25 35 7 49 , 22 38 36 49 →= 24 35 7 49 , 37 38 36 49 →= 38 35 7 49 , 39 38 36 49 →= 40 35 7 49 , 41 38 36 49 →= 42 35 7 49 , 4 5 6 3 →= 10 23 33 17 18 6 , 4 5 6 7 →= 10 23 33 17 18 8 , 19 5 6 3 →= 15 23 33 17 18 6 , 19 5 6 7 →= 15 23 33 17 18 8 , 17 18 6 3 →= 20 22 33 17 18 6 , 17 18 6 7 →= 20 22 33 17 18 8 , 29 18 6 3 →= 26 22 33 17 18 6 , 29 18 6 7 →= 26 22 33 17 18 8 , 11 12 6 3 →= 23 37 33 17 18 6 , 11 12 6 7 →= 23 37 33 17 18 8 , 21 12 6 3 →= 22 37 33 17 18 6 , 21 12 6 7 →= 22 37 33 17 18 8 , 33 12 6 3 →= 37 37 33 17 18 6 , 33 12 6 7 →= 37 37 33 17 18 8 , 43 12 6 3 →= 39 37 33 17 18 6 , 43 12 6 7 →= 39 37 33 17 18 8 , 44 12 6 3 →= 41 37 33 17 18 6 , 44 12 6 7 →= 41 37 33 17 18 8 , 46 47 6 3 →= 45 39 33 17 18 6 , 46 47 6 7 →= 45 39 33 17 18 8 } 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 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 1 ⎟ ⎜ 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 ⎟ ⎝ ⎠ 26 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 27 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 28 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 29 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 30 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 31 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 32 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 33 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 34 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 35 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 36 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 37 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 38 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 39 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 40 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 41 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 42 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 43 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 44 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 45 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 46 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 47 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 48 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 49 ↦ ⎛ ⎞ ⎜ 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, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 15 ↦ 13, 16 ↦ 14, 17 ↦ 15, 18 ↦ 16, 19 ↦ 17, 20 ↦ 18, 21 ↦ 19, 22 ↦ 20, 23 ↦ 21, 24 ↦ 22, 25 ↦ 23, 13 ↦ 24, 26 ↦ 25, 27 ↦ 26, 28 ↦ 27, 29 ↦ 28, 30 ↦ 29, 31 ↦ 30, 32 ↦ 31, 33 ↦ 32, 35 ↦ 33, 36 ↦ 34, 37 ↦ 35, 38 ↦ 36, 39 ↦ 37, 40 ↦ 38, 41 ↦ 39, 42 ↦ 40, 43 ↦ 41, 44 ↦ 42, 45 ↦ 43, 46 ↦ 44, 47 ↦ 45, 48 ↦ 46, 49 ↦ 47 }, it remains to prove termination of the 125-rule system { 0 1 2 3 ⟶ 0 4 5 6 , 0 1 2 7 ⟶ 0 4 5 8 , 0 1 2 3 ⟶ 0 1 2 , 0 1 2 7 ⟶ 0 1 9 , 10 11 12 6 →= 1 2 , 10 11 12 8 →= 1 9 , 13 11 12 6 →= 14 2 , 13 11 12 8 →= 14 9 , 10 11 15 16 →= 4 5 , 13 11 15 16 →= 17 5 , 10 11 18 19 →= 10 11 , 10 11 18 20 →= 10 21 , 10 11 18 22 →= 10 23 , 13 11 18 19 →= 13 11 , 13 11 18 20 →= 13 21 , 13 11 18 22 →= 13 23 , 18 19 12 24 →= 12 24 , 18 19 12 6 →= 12 6 , 18 19 12 8 →= 12 8 , 25 19 12 24 →= 26 24 , 25 19 12 6 →= 26 6 , 25 19 12 8 →= 26 8 , 18 19 15 27 →= 15 27 , 18 19 15 16 →= 15 16 , 25 19 15 27 →= 28 27 , 25 19 15 16 →= 28 16 , 18 19 18 29 →= 18 29 , 18 19 18 19 →= 18 19 , 18 19 18 20 →= 18 20 , 18 19 18 22 →= 18 22 , 25 19 18 29 →= 25 29 , 25 19 18 19 →= 25 19 , 25 19 18 20 →= 25 20 , 25 19 18 22 →= 25 22 , 18 19 30 31 →= 30 31 , 21 32 12 6 →= 23 33 , 21 32 12 8 →= 23 34 , 20 32 12 6 →= 22 33 , 20 32 12 8 →= 22 34 , 35 32 12 6 →= 36 33 , 35 32 12 8 →= 36 34 , 37 32 12 6 →= 38 33 , 37 32 12 8 →= 38 34 , 39 32 12 6 →= 40 33 , 39 32 12 8 →= 40 34 , 21 32 15 16 →= 11 12 , 20 32 15 16 →= 19 12 , 35 32 15 16 →= 32 12 , 37 32 15 16 →= 41 12 , 39 32 15 16 →= 42 12 , 21 32 18 19 →= 21 32 , 21 32 18 20 →= 21 35 , 21 32 18 22 →= 21 36 , 20 32 18 19 →= 20 32 , 20 32 18 20 →= 20 35 , 20 32 18 22 →= 20 36 , 35 32 18 19 →= 35 32 , 35 32 18 20 →= 35 35 , 35 32 18 22 →= 35 36 , 37 32 18 19 →= 37 32 , 37 32 18 20 →= 37 35 , 37 32 18 22 →= 37 36 , 39 32 18 19 →= 39 32 , 39 32 18 20 →= 39 35 , 39 32 18 22 →= 39 36 , 43 41 15 16 →= 44 45 , 43 41 18 19 →= 43 41 , 43 41 18 20 →= 43 37 , 43 41 18 22 →= 43 38 , 10 23 33 46 →= 1 2 3 46 , 10 23 33 3 →= 1 2 3 3 , 10 23 33 7 →= 1 2 3 7 , 13 23 33 46 →= 14 2 3 46 , 13 23 33 3 →= 14 2 3 3 , 13 23 33 7 →= 14 2 3 7 , 10 23 34 47 →= 1 2 7 47 , 13 23 34 47 →= 14 2 7 47 , 18 22 33 46 →= 12 6 3 46 , 18 22 33 3 →= 12 6 3 3 , 18 22 33 7 →= 12 6 3 7 , 25 22 33 46 →= 26 6 3 46 , 25 22 33 3 →= 26 6 3 3 , 25 22 33 7 →= 26 6 3 7 , 18 22 34 47 →= 12 6 7 47 , 25 22 34 47 →= 26 6 7 47 , 21 36 33 46 →= 23 33 3 46 , 21 36 33 3 →= 23 33 3 3 , 21 36 33 7 →= 23 33 3 7 , 20 36 33 46 →= 22 33 3 46 , 20 36 33 3 →= 22 33 3 3 , 20 36 33 7 →= 22 33 3 7 , 35 36 33 46 →= 36 33 3 46 , 35 36 33 3 →= 36 33 3 3 , 35 36 33 7 →= 36 33 3 7 , 37 36 33 46 →= 38 33 3 46 , 37 36 33 3 →= 38 33 3 3 , 37 36 33 7 →= 38 33 3 7 , 39 36 33 46 →= 40 33 3 46 , 39 36 33 3 →= 40 33 3 3 , 39 36 33 7 →= 40 33 3 7 , 21 36 34 47 →= 23 33 7 47 , 20 36 34 47 →= 22 33 7 47 , 35 36 34 47 →= 36 33 7 47 , 37 36 34 47 →= 38 33 7 47 , 39 36 34 47 →= 40 33 7 47 , 4 5 6 3 →= 10 21 32 15 16 6 , 4 5 6 7 →= 10 21 32 15 16 8 , 17 5 6 3 →= 13 21 32 15 16 6 , 17 5 6 7 →= 13 21 32 15 16 8 , 15 16 6 3 →= 18 20 32 15 16 6 , 15 16 6 7 →= 18 20 32 15 16 8 , 28 16 6 3 →= 25 20 32 15 16 6 , 28 16 6 7 →= 25 20 32 15 16 8 , 11 12 6 3 →= 21 35 32 15 16 6 , 11 12 6 7 →= 21 35 32 15 16 8 , 19 12 6 3 →= 20 35 32 15 16 6 , 19 12 6 7 →= 20 35 32 15 16 8 , 32 12 6 3 →= 35 35 32 15 16 6 , 32 12 6 7 →= 35 35 32 15 16 8 , 41 12 6 3 →= 37 35 32 15 16 6 , 41 12 6 7 →= 37 35 32 15 16 8 , 42 12 6 3 →= 39 35 32 15 16 6 , 42 12 6 7 →= 39 35 32 15 16 8 , 44 45 6 3 →= 43 37 32 15 16 6 , 44 45 6 7 →= 43 37 32 15 16 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, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13, 14 ↦ 14, 18 ↦ 15, 19 ↦ 16, 24 ↦ 17, 25 ↦ 18, 26 ↦ 19, 21 ↦ 20, 32 ↦ 21, 23 ↦ 22, 33 ↦ 23, 34 ↦ 24, 20 ↦ 25, 22 ↦ 26, 35 ↦ 27, 36 ↦ 28, 37 ↦ 29, 38 ↦ 30, 39 ↦ 31, 40 ↦ 32, 15 ↦ 33, 16 ↦ 34, 41 ↦ 35, 42 ↦ 36, 46 ↦ 37, 47 ↦ 38, 17 ↦ 39, 28 ↦ 40, 44 ↦ 41, 45 ↦ 42, 43 ↦ 43 }, it remains to prove termination of the 100-rule system { 0 1 2 3 ⟶ 0 4 5 6 , 0 1 2 7 ⟶ 0 4 5 8 , 0 1 2 3 ⟶ 0 1 2 , 0 1 2 7 ⟶ 0 1 9 , 10 11 12 6 →= 1 2 , 10 11 12 8 →= 1 9 , 13 11 12 6 →= 14 2 , 13 11 12 8 →= 14 9 , 15 16 12 17 →= 12 17 , 15 16 12 6 →= 12 6 , 15 16 12 8 →= 12 8 , 18 16 12 17 →= 19 17 , 18 16 12 6 →= 19 6 , 18 16 12 8 →= 19 8 , 20 21 12 6 →= 22 23 , 20 21 12 8 →= 22 24 , 25 21 12 6 →= 26 23 , 25 21 12 8 →= 26 24 , 27 21 12 6 →= 28 23 , 27 21 12 8 →= 28 24 , 29 21 12 6 →= 30 23 , 29 21 12 8 →= 30 24 , 31 21 12 6 →= 32 23 , 31 21 12 8 →= 32 24 , 20 21 33 34 →= 11 12 , 25 21 33 34 →= 16 12 , 27 21 33 34 →= 21 12 , 29 21 33 34 →= 35 12 , 31 21 33 34 →= 36 12 , 20 21 15 16 →= 20 21 , 20 21 15 25 →= 20 27 , 20 21 15 26 →= 20 28 , 25 21 15 16 →= 25 21 , 25 21 15 25 →= 25 27 , 25 21 15 26 →= 25 28 , 27 21 15 16 →= 27 21 , 27 21 15 25 →= 27 27 , 27 21 15 26 →= 27 28 , 29 21 15 16 →= 29 21 , 29 21 15 25 →= 29 27 , 29 21 15 26 →= 29 28 , 31 21 15 16 →= 31 21 , 31 21 15 25 →= 31 27 , 31 21 15 26 →= 31 28 , 10 22 23 37 →= 1 2 3 37 , 10 22 23 3 →= 1 2 3 3 , 10 22 23 7 →= 1 2 3 7 , 13 22 23 37 →= 14 2 3 37 , 13 22 23 3 →= 14 2 3 3 , 13 22 23 7 →= 14 2 3 7 , 10 22 24 38 →= 1 2 7 38 , 13 22 24 38 →= 14 2 7 38 , 15 26 23 37 →= 12 6 3 37 , 15 26 23 3 →= 12 6 3 3 , 15 26 23 7 →= 12 6 3 7 , 18 26 23 37 →= 19 6 3 37 , 18 26 23 3 →= 19 6 3 3 , 18 26 23 7 →= 19 6 3 7 , 15 26 24 38 →= 12 6 7 38 , 18 26 24 38 →= 19 6 7 38 , 20 28 23 37 →= 22 23 3 37 , 20 28 23 3 →= 22 23 3 3 , 20 28 23 7 →= 22 23 3 7 , 25 28 23 37 →= 26 23 3 37 , 25 28 23 3 →= 26 23 3 3 , 25 28 23 7 →= 26 23 3 7 , 27 28 23 37 →= 28 23 3 37 , 27 28 23 3 →= 28 23 3 3 , 27 28 23 7 →= 28 23 3 7 , 29 28 23 37 →= 30 23 3 37 , 29 28 23 3 →= 30 23 3 3 , 29 28 23 7 →= 30 23 3 7 , 31 28 23 37 →= 32 23 3 37 , 31 28 23 3 →= 32 23 3 3 , 31 28 23 7 →= 32 23 3 7 , 20 28 24 38 →= 22 23 7 38 , 25 28 24 38 →= 26 23 7 38 , 27 28 24 38 →= 28 23 7 38 , 29 28 24 38 →= 30 23 7 38 , 31 28 24 38 →= 32 23 7 38 , 4 5 6 3 →= 10 20 21 33 34 6 , 4 5 6 7 →= 10 20 21 33 34 8 , 39 5 6 3 →= 13 20 21 33 34 6 , 39 5 6 7 →= 13 20 21 33 34 8 , 33 34 6 3 →= 15 25 21 33 34 6 , 33 34 6 7 →= 15 25 21 33 34 8 , 40 34 6 3 →= 18 25 21 33 34 6 , 40 34 6 7 →= 18 25 21 33 34 8 , 11 12 6 3 →= 20 27 21 33 34 6 , 11 12 6 7 →= 20 27 21 33 34 8 , 16 12 6 3 →= 25 27 21 33 34 6 , 16 12 6 7 →= 25 27 21 33 34 8 , 21 12 6 3 →= 27 27 21 33 34 6 , 21 12 6 7 →= 27 27 21 33 34 8 , 35 12 6 3 →= 29 27 21 33 34 6 , 35 12 6 7 →= 29 27 21 33 34 8 , 36 12 6 3 →= 31 27 21 33 34 6 , 36 12 6 7 →= 31 27 21 33 34 8 , 41 42 6 3 →= 43 29 21 33 34 6 , 41 42 6 7 →= 43 29 21 33 34 8 } 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 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 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 ⎟ ⎝ ⎠ 24 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 25 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 26 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 27 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 28 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 29 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 30 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 31 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 32 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 33 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 34 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 35 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 36 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 37 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 38 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 39 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 40 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 41 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 42 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 43 ↦ ⎛ ⎞ ⎜ 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, 21 ↦ 21, 22 ↦ 22, 23 ↦ 23, 24 ↦ 24, 25 ↦ 25, 26 ↦ 26, 27 ↦ 27, 28 ↦ 28, 29 ↦ 29, 30 ↦ 30, 31 ↦ 31, 32 ↦ 32, 33 ↦ 33, 34 ↦ 34, 35 ↦ 35, 36 ↦ 36, 37 ↦ 37, 38 ↦ 38 }, it remains to prove termination of the 94-rule system { 0 1 2 3 ⟶ 0 4 5 6 , 0 1 2 7 ⟶ 0 4 5 8 , 0 1 2 3 ⟶ 0 1 2 , 0 1 2 7 ⟶ 0 1 9 , 10 11 12 6 →= 1 2 , 10 11 12 8 →= 1 9 , 13 11 12 6 →= 14 2 , 13 11 12 8 →= 14 9 , 15 16 12 17 →= 12 17 , 15 16 12 6 →= 12 6 , 15 16 12 8 →= 12 8 , 18 16 12 17 →= 19 17 , 18 16 12 6 →= 19 6 , 18 16 12 8 →= 19 8 , 20 21 12 6 →= 22 23 , 20 21 12 8 →= 22 24 , 25 21 12 6 →= 26 23 , 25 21 12 8 →= 26 24 , 27 21 12 6 →= 28 23 , 27 21 12 8 →= 28 24 , 29 21 12 6 →= 30 23 , 29 21 12 8 →= 30 24 , 31 21 12 6 →= 32 23 , 31 21 12 8 →= 32 24 , 20 21 33 34 →= 11 12 , 25 21 33 34 →= 16 12 , 27 21 33 34 →= 21 12 , 29 21 33 34 →= 35 12 , 31 21 33 34 →= 36 12 , 20 21 15 16 →= 20 21 , 20 21 15 25 →= 20 27 , 20 21 15 26 →= 20 28 , 25 21 15 16 →= 25 21 , 25 21 15 25 →= 25 27 , 25 21 15 26 →= 25 28 , 27 21 15 16 →= 27 21 , 27 21 15 25 →= 27 27 , 27 21 15 26 →= 27 28 , 29 21 15 16 →= 29 21 , 29 21 15 25 →= 29 27 , 29 21 15 26 →= 29 28 , 31 21 15 16 →= 31 21 , 31 21 15 25 →= 31 27 , 31 21 15 26 →= 31 28 , 10 22 23 37 →= 1 2 3 37 , 10 22 23 3 →= 1 2 3 3 , 10 22 23 7 →= 1 2 3 7 , 13 22 23 37 →= 14 2 3 37 , 13 22 23 3 →= 14 2 3 3 , 13 22 23 7 →= 14 2 3 7 , 10 22 24 38 →= 1 2 7 38 , 13 22 24 38 →= 14 2 7 38 , 15 26 23 37 →= 12 6 3 37 , 15 26 23 3 →= 12 6 3 3 , 15 26 23 7 →= 12 6 3 7 , 18 26 23 37 →= 19 6 3 37 , 18 26 23 3 →= 19 6 3 3 , 18 26 23 7 →= 19 6 3 7 , 15 26 24 38 →= 12 6 7 38 , 18 26 24 38 →= 19 6 7 38 , 20 28 23 37 →= 22 23 3 37 , 20 28 23 3 →= 22 23 3 3 , 20 28 23 7 →= 22 23 3 7 , 25 28 23 37 →= 26 23 3 37 , 25 28 23 3 →= 26 23 3 3 , 25 28 23 7 →= 26 23 3 7 , 27 28 23 37 →= 28 23 3 37 , 27 28 23 3 →= 28 23 3 3 , 27 28 23 7 →= 28 23 3 7 , 29 28 23 37 →= 30 23 3 37 , 29 28 23 3 →= 30 23 3 3 , 29 28 23 7 →= 30 23 3 7 , 31 28 23 37 →= 32 23 3 37 , 31 28 23 3 →= 32 23 3 3 , 31 28 23 7 →= 32 23 3 7 , 20 28 24 38 →= 22 23 7 38 , 25 28 24 38 →= 26 23 7 38 , 27 28 24 38 →= 28 23 7 38 , 29 28 24 38 →= 30 23 7 38 , 31 28 24 38 →= 32 23 7 38 , 4 5 6 3 →= 10 20 21 33 34 6 , 4 5 6 7 →= 10 20 21 33 34 8 , 33 34 6 3 →= 15 25 21 33 34 6 , 33 34 6 7 →= 15 25 21 33 34 8 , 11 12 6 3 →= 20 27 21 33 34 6 , 11 12 6 7 →= 20 27 21 33 34 8 , 16 12 6 3 →= 25 27 21 33 34 6 , 16 12 6 7 →= 25 27 21 33 34 8 , 21 12 6 3 →= 27 27 21 33 34 6 , 21 12 6 7 →= 27 27 21 33 34 8 , 35 12 6 3 →= 29 27 21 33 34 6 , 35 12 6 7 →= 29 27 21 33 34 8 , 36 12 6 3 →= 31 27 21 33 34 6 , 36 12 6 7 →= 31 27 21 33 34 8 } 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 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 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 ⎟ ⎝ ⎠ 26 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 27 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 28 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 29 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 30 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 31 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 32 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 33 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 34 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 35 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 36 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 37 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 38 ↦ ⎛ ⎞ ⎜ 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, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 15 ↦ 13, 16 ↦ 14, 17 ↦ 15, 20 ↦ 16, 21 ↦ 17, 22 ↦ 18, 23 ↦ 19, 24 ↦ 20, 25 ↦ 21, 26 ↦ 22, 27 ↦ 23, 28 ↦ 24, 29 ↦ 25, 30 ↦ 26, 31 ↦ 27, 32 ↦ 28, 33 ↦ 29, 34 ↦ 30, 35 ↦ 31, 36 ↦ 32, 37 ↦ 33, 38 ↦ 34 }, it remains to prove termination of the 81-rule system { 0 1 2 3 ⟶ 0 4 5 6 , 0 1 2 7 ⟶ 0 4 5 8 , 0 1 2 3 ⟶ 0 1 2 , 0 1 2 7 ⟶ 0 1 9 , 10 11 12 6 →= 1 2 , 10 11 12 8 →= 1 9 , 13 14 12 15 →= 12 15 , 13 14 12 6 →= 12 6 , 13 14 12 8 →= 12 8 , 16 17 12 6 →= 18 19 , 16 17 12 8 →= 18 20 , 21 17 12 6 →= 22 19 , 21 17 12 8 →= 22 20 , 23 17 12 6 →= 24 19 , 23 17 12 8 →= 24 20 , 25 17 12 6 →= 26 19 , 25 17 12 8 →= 26 20 , 27 17 12 6 →= 28 19 , 27 17 12 8 →= 28 20 , 16 17 29 30 →= 11 12 , 21 17 29 30 →= 14 12 , 23 17 29 30 →= 17 12 , 25 17 29 30 →= 31 12 , 27 17 29 30 →= 32 12 , 16 17 13 14 →= 16 17 , 16 17 13 21 →= 16 23 , 16 17 13 22 →= 16 24 , 21 17 13 14 →= 21 17 , 21 17 13 21 →= 21 23 , 21 17 13 22 →= 21 24 , 23 17 13 14 →= 23 17 , 23 17 13 21 →= 23 23 , 23 17 13 22 →= 23 24 , 25 17 13 14 →= 25 17 , 25 17 13 21 →= 25 23 , 25 17 13 22 →= 25 24 , 27 17 13 14 →= 27 17 , 27 17 13 21 →= 27 23 , 27 17 13 22 →= 27 24 , 10 18 19 33 →= 1 2 3 33 , 10 18 19 3 →= 1 2 3 3 , 10 18 19 7 →= 1 2 3 7 , 10 18 20 34 →= 1 2 7 34 , 13 22 19 33 →= 12 6 3 33 , 13 22 19 3 →= 12 6 3 3 , 13 22 19 7 →= 12 6 3 7 , 13 22 20 34 →= 12 6 7 34 , 16 24 19 33 →= 18 19 3 33 , 16 24 19 3 →= 18 19 3 3 , 16 24 19 7 →= 18 19 3 7 , 21 24 19 33 →= 22 19 3 33 , 21 24 19 3 →= 22 19 3 3 , 21 24 19 7 →= 22 19 3 7 , 23 24 19 33 →= 24 19 3 33 , 23 24 19 3 →= 24 19 3 3 , 23 24 19 7 →= 24 19 3 7 , 25 24 19 33 →= 26 19 3 33 , 25 24 19 3 →= 26 19 3 3 , 25 24 19 7 →= 26 19 3 7 , 27 24 19 33 →= 28 19 3 33 , 27 24 19 3 →= 28 19 3 3 , 27 24 19 7 →= 28 19 3 7 , 16 24 20 34 →= 18 19 7 34 , 21 24 20 34 →= 22 19 7 34 , 23 24 20 34 →= 24 19 7 34 , 25 24 20 34 →= 26 19 7 34 , 27 24 20 34 →= 28 19 7 34 , 4 5 6 3 →= 10 16 17 29 30 6 , 4 5 6 7 →= 10 16 17 29 30 8 , 29 30 6 3 →= 13 21 17 29 30 6 , 29 30 6 7 →= 13 21 17 29 30 8 , 11 12 6 3 →= 16 23 17 29 30 6 , 11 12 6 7 →= 16 23 17 29 30 8 , 14 12 6 3 →= 21 23 17 29 30 6 , 14 12 6 7 →= 21 23 17 29 30 8 , 17 12 6 3 →= 23 23 17 29 30 6 , 17 12 6 7 →= 23 23 17 29 30 8 , 31 12 6 3 →= 25 23 17 29 30 6 , 31 12 6 7 →= 25 23 17 29 30 8 , 32 12 6 3 →= 27 23 17 29 30 6 , 32 12 6 7 →= 27 23 17 29 30 8 } 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 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 23 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 24 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 25 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 26 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 27 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 28 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 29 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 30 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 31 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 32 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 33 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 34 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 13 ↦ 0, 14 ↦ 1, 12 ↦ 2, 15 ↦ 3, 6 ↦ 4, 8 ↦ 5, 21 ↦ 6, 17 ↦ 7, 22 ↦ 8, 19 ↦ 9, 20 ↦ 10, 23 ↦ 11, 24 ↦ 12, 16 ↦ 13, 29 ↦ 14, 30 ↦ 15, 11 ↦ 16, 25 ↦ 17, 31 ↦ 18, 27 ↦ 19, 32 ↦ 20, 33 ↦ 21, 3 ↦ 22, 7 ↦ 23, 34 ↦ 24 }, it remains to prove termination of the 51-rule system { 0 1 2 3 →= 2 3 , 0 1 2 4 →= 2 4 , 0 1 2 5 →= 2 5 , 6 7 2 4 →= 8 9 , 6 7 2 5 →= 8 10 , 11 7 2 4 →= 12 9 , 11 7 2 5 →= 12 10 , 13 7 14 15 →= 16 2 , 6 7 14 15 →= 1 2 , 11 7 14 15 →= 7 2 , 17 7 14 15 →= 18 2 , 19 7 14 15 →= 20 2 , 13 7 0 1 →= 13 7 , 13 7 0 6 →= 13 11 , 13 7 0 8 →= 13 12 , 6 7 0 1 →= 6 7 , 6 7 0 6 →= 6 11 , 6 7 0 8 →= 6 12 , 11 7 0 1 →= 11 7 , 11 7 0 6 →= 11 11 , 11 7 0 8 →= 11 12 , 17 7 0 1 →= 17 7 , 17 7 0 6 →= 17 11 , 17 7 0 8 →= 17 12 , 19 7 0 1 →= 19 7 , 19 7 0 6 →= 19 11 , 19 7 0 8 →= 19 12 , 0 8 9 21 →= 2 4 22 21 , 0 8 9 22 →= 2 4 22 22 , 0 8 9 23 →= 2 4 22 23 , 0 8 10 24 →= 2 4 23 24 , 6 12 9 21 →= 8 9 22 21 , 6 12 9 22 →= 8 9 22 22 , 6 12 9 23 →= 8 9 22 23 , 11 12 9 21 →= 12 9 22 21 , 11 12 9 22 →= 12 9 22 22 , 11 12 9 23 →= 12 9 22 23 , 6 12 10 24 →= 8 9 23 24 , 11 12 10 24 →= 12 9 23 24 , 14 15 4 22 →= 0 6 7 14 15 4 , 14 15 4 23 →= 0 6 7 14 15 5 , 16 2 4 22 →= 13 11 7 14 15 4 , 16 2 4 23 →= 13 11 7 14 15 5 , 1 2 4 22 →= 6 11 7 14 15 4 , 1 2 4 23 →= 6 11 7 14 15 5 , 7 2 4 22 →= 11 11 7 14 15 4 , 7 2 4 23 →= 11 11 7 14 15 5 , 18 2 4 22 →= 17 11 7 14 15 4 , 18 2 4 23 →= 17 11 7 14 15 5 , 20 2 4 22 →= 19 11 7 14 15 4 , 20 2 4 23 →= 19 11 7 14 15 5 } The system is trivially terminating.