/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 1 2 , 0 1 ⟶ , 0 2 1 ⟶ 0 0 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 , 2 0 ⟶ , 2 1 0 ⟶ 0 0 0 } Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols. After renaming modulo the bijection { (0,↑) ↦ 0, (2,↑) ↦ 1, (2,↓) ↦ 2, (1,↓) ↦ 3, (0,↓) ↦ 4 }, it remains to prove termination of the 7-rule system { 0 ⟶ 1 2 , 0 ⟶ 1 , 1 3 4 ⟶ 0 4 4 , 1 3 4 ⟶ 0 4 , 4 →= 3 2 2 , 2 4 →= , 2 3 4 →= 4 4 4 } Applying sparse tiling TROC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (5,0) ↦ 0, (0,3) ↦ 1, (5,1) ↦ 2, (1,2) ↦ 3, (2,3) ↦ 4, (0,4) ↦ 5, (2,4) ↦ 6, (1,3) ↦ 7, (1,4) ↦ 8, (3,4) ↦ 9, (4,3) ↦ 10, (4,4) ↦ 11, (4,6) ↦ 12, (3,2) ↦ 13, (2,2) ↦ 14, (2,6) ↦ 15, (3,3) ↦ 16, (5,4) ↦ 17, (5,3) ↦ 18, (1,6) ↦ 19, (3,6) ↦ 20 }, it remains to prove termination of the 46-rule system { 0 1 ⟶ 2 3 4 , 0 5 ⟶ 2 3 6 , 0 1 ⟶ 2 7 , 0 5 ⟶ 2 8 , 2 7 9 10 ⟶ 0 5 11 10 , 2 7 9 11 ⟶ 0 5 11 11 , 2 7 9 12 ⟶ 0 5 11 12 , 2 7 9 10 ⟶ 0 5 10 , 2 7 9 11 ⟶ 0 5 11 , 2 7 9 12 ⟶ 0 5 12 , 5 10 →= 1 13 14 4 , 5 11 →= 1 13 14 6 , 5 12 →= 1 13 14 15 , 8 10 →= 7 13 14 4 , 8 11 →= 7 13 14 6 , 8 12 →= 7 13 14 15 , 6 10 →= 4 13 14 4 , 6 11 →= 4 13 14 6 , 6 12 →= 4 13 14 15 , 9 10 →= 16 13 14 4 , 9 11 →= 16 13 14 6 , 9 12 →= 16 13 14 15 , 11 10 →= 10 13 14 4 , 11 11 →= 10 13 14 6 , 11 12 →= 10 13 14 15 , 17 10 →= 18 13 14 4 , 17 11 →= 18 13 14 6 , 17 12 →= 18 13 14 15 , 3 6 10 →= 7 , 3 6 11 →= 8 , 3 6 12 →= 19 , 14 6 10 →= 4 , 14 6 11 →= 6 , 14 6 12 →= 15 , 13 6 10 →= 16 , 13 6 11 →= 9 , 13 6 12 →= 20 , 3 4 9 10 →= 8 11 11 10 , 3 4 9 11 →= 8 11 11 11 , 3 4 9 12 →= 8 11 11 12 , 14 4 9 10 →= 6 11 11 10 , 14 4 9 11 →= 6 11 11 11 , 14 4 9 12 →= 6 11 11 12 , 13 4 9 10 →= 9 11 11 10 , 13 4 9 11 →= 9 11 11 11 , 13 4 9 12 →= 9 11 11 12 } 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 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 ⎟ ⎝ ⎠ 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, 16 ↦ 15 }, it remains to prove termination of the 35-rule system { 0 1 ⟶ 2 3 4 , 0 5 ⟶ 2 3 6 , 0 1 ⟶ 2 7 , 0 5 ⟶ 2 8 , 2 7 9 10 ⟶ 0 5 11 10 , 2 7 9 11 ⟶ 0 5 11 11 , 2 7 9 12 ⟶ 0 5 11 12 , 2 7 9 10 ⟶ 0 5 10 , 2 7 9 11 ⟶ 0 5 11 , 2 7 9 12 ⟶ 0 5 12 , 5 10 →= 1 13 14 4 , 5 11 →= 1 13 14 6 , 8 10 →= 7 13 14 4 , 8 11 →= 7 13 14 6 , 6 10 →= 4 13 14 4 , 6 11 →= 4 13 14 6 , 9 10 →= 15 13 14 4 , 9 11 →= 15 13 14 6 , 11 10 →= 10 13 14 4 , 11 11 →= 10 13 14 6 , 3 6 10 →= 7 , 3 6 11 →= 8 , 14 6 10 →= 4 , 14 6 11 →= 6 , 13 6 10 →= 15 , 13 6 11 →= 9 , 3 4 9 10 →= 8 11 11 10 , 3 4 9 11 →= 8 11 11 11 , 3 4 9 12 →= 8 11 11 12 , 14 4 9 10 →= 6 11 11 10 , 14 4 9 11 →= 6 11 11 11 , 14 4 9 12 →= 6 11 11 12 , 13 4 9 10 →= 9 11 11 10 , 13 4 9 11 →= 9 11 11 11 , 13 4 9 12 →= 9 11 11 12 } Applying sparse tiling TROC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (16,0) ↦ 0, (0,1) ↦ 1, (1,9) ↦ 2, (16,2) ↦ 3, (2,3) ↦ 4, (3,4) ↦ 5, (4,9) ↦ 6, (1,13) ↦ 7, (4,13) ↦ 8, (1,15) ↦ 9, (4,15) ↦ 10, (0,5) ↦ 11, (5,10) ↦ 12, (3,6) ↦ 13, (6,10) ↦ 14, (5,11) ↦ 15, (6,11) ↦ 16, (5,12) ↦ 17, (6,12) ↦ 18, (2,7) ↦ 19, (7,9) ↦ 20, (7,13) ↦ 21, (7,15) ↦ 22, (2,8) ↦ 23, (8,10) ↦ 24, (8,11) ↦ 25, (8,12) ↦ 26, (9,10) ↦ 27, (10,17) ↦ 28, (11,10) ↦ 29, (10,9) ↦ 30, (10,13) ↦ 31, (10,15) ↦ 32, (9,11) ↦ 33, (11,17) ↦ 34, (11,11) ↦ 35, (11,12) ↦ 36, (9,12) ↦ 37, (12,17) ↦ 38, (13,14) ↦ 39, (14,4) ↦ 40, (4,17) ↦ 41, (14,6) ↦ 42, (6,17) ↦ 43, (16,8) ↦ 44, (16,7) ↦ 45, (16,6) ↦ 46, (16,4) ↦ 47, (13,6) ↦ 48, (13,4) ↦ 49, (16,9) ↦ 50, (16,15) ↦ 51, (15,13) ↦ 52, (15,9) ↦ 53, (15,15) ↦ 54, (7,17) ↦ 55, (8,17) ↦ 56, (15,17) ↦ 57, (9,17) ↦ 58 }, it remains to prove termination of the 293-rule system { 0 1 2 ⟶ 3 4 5 6 , 0 1 7 ⟶ 3 4 5 8 , 0 1 9 ⟶ 3 4 5 10 , 0 11 12 ⟶ 3 4 13 14 , 0 11 15 ⟶ 3 4 13 16 , 0 11 17 ⟶ 3 4 13 18 , 0 1 2 ⟶ 3 19 20 , 0 1 7 ⟶ 3 19 21 , 0 1 9 ⟶ 3 19 22 , 0 11 12 ⟶ 3 23 24 , 0 11 15 ⟶ 3 23 25 , 0 11 17 ⟶ 3 23 26 , 3 19 20 27 28 ⟶ 0 11 15 29 28 , 3 19 20 27 30 ⟶ 0 11 15 29 30 , 3 19 20 27 31 ⟶ 0 11 15 29 31 , 3 19 20 27 32 ⟶ 0 11 15 29 32 , 3 19 20 33 34 ⟶ 0 11 15 35 34 , 3 19 20 33 29 ⟶ 0 11 15 35 29 , 3 19 20 33 35 ⟶ 0 11 15 35 35 , 3 19 20 33 36 ⟶ 0 11 15 35 36 , 3 19 20 37 38 ⟶ 0 11 15 36 38 , 3 19 20 27 28 ⟶ 0 11 12 28 , 3 19 20 27 30 ⟶ 0 11 12 30 , 3 19 20 27 31 ⟶ 0 11 12 31 , 3 19 20 27 32 ⟶ 0 11 12 32 , 3 19 20 33 34 ⟶ 0 11 15 34 , 3 19 20 33 29 ⟶ 0 11 15 29 , 3 19 20 33 35 ⟶ 0 11 15 35 , 3 19 20 33 36 ⟶ 0 11 15 36 , 3 19 20 37 38 ⟶ 0 11 17 38 , 11 12 28 →= 1 7 39 40 41 , 11 12 30 →= 1 7 39 40 6 , 11 12 31 →= 1 7 39 40 8 , 11 12 32 →= 1 7 39 40 10 , 11 15 34 →= 1 7 39 42 43 , 11 15 29 →= 1 7 39 42 14 , 11 15 35 →= 1 7 39 42 16 , 11 15 36 →= 1 7 39 42 18 , 44 24 28 →= 45 21 39 40 41 , 44 24 30 →= 45 21 39 40 6 , 44 24 31 →= 45 21 39 40 8 , 44 24 32 →= 45 21 39 40 10 , 23 24 28 →= 19 21 39 40 41 , 23 24 30 →= 19 21 39 40 6 , 23 24 31 →= 19 21 39 40 8 , 23 24 32 →= 19 21 39 40 10 , 44 25 34 →= 45 21 39 42 43 , 44 25 29 →= 45 21 39 42 14 , 44 25 35 →= 45 21 39 42 16 , 44 25 36 →= 45 21 39 42 18 , 23 25 34 →= 19 21 39 42 43 , 23 25 29 →= 19 21 39 42 14 , 23 25 35 →= 19 21 39 42 16 , 23 25 36 →= 19 21 39 42 18 , 46 14 28 →= 47 8 39 40 41 , 46 14 30 →= 47 8 39 40 6 , 46 14 31 →= 47 8 39 40 8 , 46 14 32 →= 47 8 39 40 10 , 13 14 28 →= 5 8 39 40 41 , 13 14 30 →= 5 8 39 40 6 , 13 14 31 →= 5 8 39 40 8 , 13 14 32 →= 5 8 39 40 10 , 48 14 28 →= 49 8 39 40 41 , 48 14 30 →= 49 8 39 40 6 , 48 14 31 →= 49 8 39 40 8 , 48 14 32 →= 49 8 39 40 10 , 42 14 28 →= 40 8 39 40 41 , 42 14 30 →= 40 8 39 40 6 , 42 14 31 →= 40 8 39 40 8 , 42 14 32 →= 40 8 39 40 10 , 46 16 34 →= 47 8 39 42 43 , 46 16 29 →= 47 8 39 42 14 , 46 16 35 →= 47 8 39 42 16 , 46 16 36 →= 47 8 39 42 18 , 13 16 34 →= 5 8 39 42 43 , 13 16 29 →= 5 8 39 42 14 , 13 16 35 →= 5 8 39 42 16 , 13 16 36 →= 5 8 39 42 18 , 48 16 34 →= 49 8 39 42 43 , 48 16 29 →= 49 8 39 42 14 , 48 16 35 →= 49 8 39 42 16 , 48 16 36 →= 49 8 39 42 18 , 42 16 34 →= 40 8 39 42 43 , 42 16 29 →= 40 8 39 42 14 , 42 16 35 →= 40 8 39 42 16 , 42 16 36 →= 40 8 39 42 18 , 50 27 28 →= 51 52 39 40 41 , 50 27 30 →= 51 52 39 40 6 , 50 27 31 →= 51 52 39 40 8 , 50 27 32 →= 51 52 39 40 10 , 2 27 28 →= 9 52 39 40 41 , 2 27 30 →= 9 52 39 40 6 , 2 27 31 →= 9 52 39 40 8 , 2 27 32 →= 9 52 39 40 10 , 6 27 28 →= 10 52 39 40 41 , 6 27 30 →= 10 52 39 40 6 , 6 27 31 →= 10 52 39 40 8 , 6 27 32 →= 10 52 39 40 10 , 20 27 28 →= 22 52 39 40 41 , 20 27 30 →= 22 52 39 40 6 , 20 27 31 →= 22 52 39 40 8 , 20 27 32 →= 22 52 39 40 10 , 30 27 28 →= 32 52 39 40 41 , 30 27 30 →= 32 52 39 40 6 , 30 27 31 →= 32 52 39 40 8 , 30 27 32 →= 32 52 39 40 10 , 53 27 28 →= 54 52 39 40 41 , 53 27 30 →= 54 52 39 40 6 , 53 27 31 →= 54 52 39 40 8 , 53 27 32 →= 54 52 39 40 10 , 50 33 34 →= 51 52 39 42 43 , 50 33 29 →= 51 52 39 42 14 , 50 33 35 →= 51 52 39 42 16 , 50 33 36 →= 51 52 39 42 18 , 2 33 34 →= 9 52 39 42 43 , 2 33 29 →= 9 52 39 42 14 , 2 33 35 →= 9 52 39 42 16 , 2 33 36 →= 9 52 39 42 18 , 6 33 34 →= 10 52 39 42 43 , 6 33 29 →= 10 52 39 42 14 , 6 33 35 →= 10 52 39 42 16 , 6 33 36 →= 10 52 39 42 18 , 20 33 34 →= 22 52 39 42 43 , 20 33 29 →= 22 52 39 42 14 , 20 33 35 →= 22 52 39 42 16 , 20 33 36 →= 22 52 39 42 18 , 30 33 34 →= 32 52 39 42 43 , 30 33 29 →= 32 52 39 42 14 , 30 33 35 →= 32 52 39 42 16 , 30 33 36 →= 32 52 39 42 18 , 53 33 34 →= 54 52 39 42 43 , 53 33 29 →= 54 52 39 42 14 , 53 33 35 →= 54 52 39 42 16 , 53 33 36 →= 54 52 39 42 18 , 15 29 28 →= 12 31 39 40 41 , 15 29 30 →= 12 31 39 40 6 , 15 29 31 →= 12 31 39 40 8 , 15 29 32 →= 12 31 39 40 10 , 16 29 28 →= 14 31 39 40 41 , 16 29 30 →= 14 31 39 40 6 , 16 29 31 →= 14 31 39 40 8 , 16 29 32 →= 14 31 39 40 10 , 25 29 28 →= 24 31 39 40 41 , 25 29 30 →= 24 31 39 40 6 , 25 29 31 →= 24 31 39 40 8 , 25 29 32 →= 24 31 39 40 10 , 33 29 28 →= 27 31 39 40 41 , 33 29 30 →= 27 31 39 40 6 , 33 29 31 →= 27 31 39 40 8 , 33 29 32 →= 27 31 39 40 10 , 35 29 28 →= 29 31 39 40 41 , 35 29 30 →= 29 31 39 40 6 , 35 29 31 →= 29 31 39 40 8 , 35 29 32 →= 29 31 39 40 10 , 15 35 34 →= 12 31 39 42 43 , 15 35 29 →= 12 31 39 42 14 , 15 35 35 →= 12 31 39 42 16 , 15 35 36 →= 12 31 39 42 18 , 16 35 34 →= 14 31 39 42 43 , 16 35 29 →= 14 31 39 42 14 , 16 35 35 →= 14 31 39 42 16 , 16 35 36 →= 14 31 39 42 18 , 25 35 34 →= 24 31 39 42 43 , 25 35 29 →= 24 31 39 42 14 , 25 35 35 →= 24 31 39 42 16 , 25 35 36 →= 24 31 39 42 18 , 33 35 34 →= 27 31 39 42 43 , 33 35 29 →= 27 31 39 42 14 , 33 35 35 →= 27 31 39 42 16 , 33 35 36 →= 27 31 39 42 18 , 35 35 34 →= 29 31 39 42 43 , 35 35 29 →= 29 31 39 42 14 , 35 35 35 →= 29 31 39 42 16 , 35 35 36 →= 29 31 39 42 18 , 4 13 14 28 →= 19 55 , 4 13 14 30 →= 19 20 , 4 13 14 31 →= 19 21 , 4 13 14 32 →= 19 22 , 4 13 16 34 →= 23 56 , 4 13 16 29 →= 23 24 , 4 13 16 35 →= 23 25 , 4 13 16 36 →= 23 26 , 39 42 14 28 →= 49 41 , 39 42 14 30 →= 49 6 , 39 42 14 31 →= 49 8 , 39 42 14 32 →= 49 10 , 39 42 16 34 →= 48 43 , 39 42 16 29 →= 48 14 , 39 42 16 35 →= 48 16 , 39 42 16 36 →= 48 18 , 7 48 14 28 →= 9 57 , 7 48 14 30 →= 9 53 , 7 48 14 31 →= 9 52 , 7 48 14 32 →= 9 54 , 8 48 14 28 →= 10 57 , 8 48 14 30 →= 10 53 , 8 48 14 31 →= 10 52 , 8 48 14 32 →= 10 54 , 21 48 14 28 →= 22 57 , 21 48 14 30 →= 22 53 , 21 48 14 31 →= 22 52 , 21 48 14 32 →= 22 54 , 31 48 14 28 →= 32 57 , 31 48 14 30 →= 32 53 , 31 48 14 31 →= 32 52 , 31 48 14 32 →= 32 54 , 52 48 14 28 →= 54 57 , 52 48 14 30 →= 54 53 , 52 48 14 31 →= 54 52 , 52 48 14 32 →= 54 54 , 7 48 16 34 →= 2 58 , 7 48 16 29 →= 2 27 , 7 48 16 35 →= 2 33 , 7 48 16 36 →= 2 37 , 8 48 16 34 →= 6 58 , 8 48 16 29 →= 6 27 , 8 48 16 35 →= 6 33 , 8 48 16 36 →= 6 37 , 21 48 16 34 →= 20 58 , 21 48 16 29 →= 20 27 , 21 48 16 35 →= 20 33 , 21 48 16 36 →= 20 37 , 31 48 16 34 →= 30 58 , 31 48 16 29 →= 30 27 , 31 48 16 35 →= 30 33 , 31 48 16 36 →= 30 37 , 52 48 16 34 →= 53 58 , 52 48 16 29 →= 53 27 , 52 48 16 35 →= 53 33 , 52 48 16 36 →= 53 37 , 4 5 6 27 28 →= 23 25 35 29 28 , 4 5 6 27 30 →= 23 25 35 29 30 , 4 5 6 27 31 →= 23 25 35 29 31 , 4 5 6 27 32 →= 23 25 35 29 32 , 4 5 6 33 34 →= 23 25 35 35 34 , 4 5 6 33 29 →= 23 25 35 35 29 , 4 5 6 33 35 →= 23 25 35 35 35 , 4 5 6 33 36 →= 23 25 35 35 36 , 4 5 6 37 38 →= 23 25 35 36 38 , 39 40 6 27 28 →= 48 16 35 29 28 , 39 40 6 27 30 →= 48 16 35 29 30 , 39 40 6 27 31 →= 48 16 35 29 31 , 39 40 6 27 32 →= 48 16 35 29 32 , 39 40 6 33 34 →= 48 16 35 35 34 , 39 40 6 33 29 →= 48 16 35 35 29 , 39 40 6 33 35 →= 48 16 35 35 35 , 39 40 6 33 36 →= 48 16 35 35 36 , 39 40 6 37 38 →= 48 16 35 36 38 , 7 49 6 27 28 →= 2 33 35 29 28 , 7 49 6 27 30 →= 2 33 35 29 30 , 7 49 6 27 31 →= 2 33 35 29 31 , 7 49 6 27 32 →= 2 33 35 29 32 , 8 49 6 27 28 →= 6 33 35 29 28 , 8 49 6 27 30 →= 6 33 35 29 30 , 8 49 6 27 31 →= 6 33 35 29 31 , 8 49 6 27 32 →= 6 33 35 29 32 , 21 49 6 27 28 →= 20 33 35 29 28 , 21 49 6 27 30 →= 20 33 35 29 30 , 21 49 6 27 31 →= 20 33 35 29 31 , 21 49 6 27 32 →= 20 33 35 29 32 , 31 49 6 27 28 →= 30 33 35 29 28 , 31 49 6 27 30 →= 30 33 35 29 30 , 31 49 6 27 31 →= 30 33 35 29 31 , 31 49 6 27 32 →= 30 33 35 29 32 , 52 49 6 27 28 →= 53 33 35 29 28 , 52 49 6 27 30 →= 53 33 35 29 30 , 52 49 6 27 31 →= 53 33 35 29 31 , 52 49 6 27 32 →= 53 33 35 29 32 , 7 49 6 33 34 →= 2 33 35 35 34 , 7 49 6 33 29 →= 2 33 35 35 29 , 7 49 6 33 35 →= 2 33 35 35 35 , 7 49 6 33 36 →= 2 33 35 35 36 , 8 49 6 33 34 →= 6 33 35 35 34 , 8 49 6 33 29 →= 6 33 35 35 29 , 8 49 6 33 35 →= 6 33 35 35 35 , 8 49 6 33 36 →= 6 33 35 35 36 , 21 49 6 33 34 →= 20 33 35 35 34 , 21 49 6 33 29 →= 20 33 35 35 29 , 21 49 6 33 35 →= 20 33 35 35 35 , 21 49 6 33 36 →= 20 33 35 35 36 , 31 49 6 33 34 →= 30 33 35 35 34 , 31 49 6 33 29 →= 30 33 35 35 29 , 31 49 6 33 35 →= 30 33 35 35 35 , 31 49 6 33 36 →= 30 33 35 35 36 , 52 49 6 33 34 →= 53 33 35 35 34 , 52 49 6 33 29 →= 53 33 35 35 29 , 52 49 6 33 35 →= 53 33 35 35 35 , 52 49 6 33 36 →= 53 33 35 35 36 , 7 49 6 37 38 →= 2 33 35 36 38 , 8 49 6 37 38 →= 6 33 35 36 38 , 21 49 6 37 38 →= 20 33 35 36 38 , 31 49 6 37 38 →= 30 33 35 36 38 , 52 49 6 37 38 →= 53 33 35 36 38 } 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 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 ⎟ ⎝ ⎠ 26 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 27 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 28 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 36 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 37 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 38 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 45 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 46 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 47 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 48 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 49 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 50 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 57 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 58 ↦ ⎛ ⎞ ⎜ 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, 39 ↦ 39, 40 ↦ 40, 42 ↦ 41, 48 ↦ 42, 49 ↦ 43, 52 ↦ 44, 53 ↦ 45, 54 ↦ 46 }, it remains to prove termination of the 225-rule system { 0 1 2 ⟶ 3 4 5 6 , 0 1 7 ⟶ 3 4 5 8 , 0 1 9 ⟶ 3 4 5 10 , 0 11 12 ⟶ 3 4 13 14 , 0 11 15 ⟶ 3 4 13 16 , 0 11 17 ⟶ 3 4 13 18 , 0 1 2 ⟶ 3 19 20 , 0 1 7 ⟶ 3 19 21 , 0 1 9 ⟶ 3 19 22 , 0 11 12 ⟶ 3 23 24 , 0 11 15 ⟶ 3 23 25 , 0 11 17 ⟶ 3 23 26 , 3 19 20 27 28 ⟶ 0 11 15 29 28 , 3 19 20 27 30 ⟶ 0 11 15 29 30 , 3 19 20 27 31 ⟶ 0 11 15 29 31 , 3 19 20 27 32 ⟶ 0 11 15 29 32 , 3 19 20 33 34 ⟶ 0 11 15 35 34 , 3 19 20 33 29 ⟶ 0 11 15 35 29 , 3 19 20 33 35 ⟶ 0 11 15 35 35 , 3 19 20 33 36 ⟶ 0 11 15 35 36 , 3 19 20 37 38 ⟶ 0 11 15 36 38 , 3 19 20 27 28 ⟶ 0 11 12 28 , 3 19 20 27 30 ⟶ 0 11 12 30 , 3 19 20 27 31 ⟶ 0 11 12 31 , 3 19 20 27 32 ⟶ 0 11 12 32 , 3 19 20 33 34 ⟶ 0 11 15 34 , 3 19 20 33 29 ⟶ 0 11 15 29 , 3 19 20 33 35 ⟶ 0 11 15 35 , 3 19 20 33 36 ⟶ 0 11 15 36 , 3 19 20 37 38 ⟶ 0 11 17 38 , 11 12 30 →= 1 7 39 40 6 , 11 12 31 →= 1 7 39 40 8 , 11 12 32 →= 1 7 39 40 10 , 11 15 29 →= 1 7 39 41 14 , 11 15 35 →= 1 7 39 41 16 , 11 15 36 →= 1 7 39 41 18 , 23 24 30 →= 19 21 39 40 6 , 23 24 31 →= 19 21 39 40 8 , 23 24 32 →= 19 21 39 40 10 , 23 25 29 →= 19 21 39 41 14 , 23 25 35 →= 19 21 39 41 16 , 23 25 36 →= 19 21 39 41 18 , 13 14 30 →= 5 8 39 40 6 , 13 14 31 →= 5 8 39 40 8 , 13 14 32 →= 5 8 39 40 10 , 42 14 30 →= 43 8 39 40 6 , 42 14 31 →= 43 8 39 40 8 , 42 14 32 →= 43 8 39 40 10 , 41 14 30 →= 40 8 39 40 6 , 41 14 31 →= 40 8 39 40 8 , 41 14 32 →= 40 8 39 40 10 , 13 16 29 →= 5 8 39 41 14 , 13 16 35 →= 5 8 39 41 16 , 13 16 36 →= 5 8 39 41 18 , 42 16 29 →= 43 8 39 41 14 , 42 16 35 →= 43 8 39 41 16 , 42 16 36 →= 43 8 39 41 18 , 41 16 29 →= 40 8 39 41 14 , 41 16 35 →= 40 8 39 41 16 , 41 16 36 →= 40 8 39 41 18 , 2 27 30 →= 9 44 39 40 6 , 2 27 31 →= 9 44 39 40 8 , 2 27 32 →= 9 44 39 40 10 , 6 27 30 →= 10 44 39 40 6 , 6 27 31 →= 10 44 39 40 8 , 6 27 32 →= 10 44 39 40 10 , 20 27 30 →= 22 44 39 40 6 , 20 27 31 →= 22 44 39 40 8 , 20 27 32 →= 22 44 39 40 10 , 30 27 30 →= 32 44 39 40 6 , 30 27 31 →= 32 44 39 40 8 , 30 27 32 →= 32 44 39 40 10 , 45 27 30 →= 46 44 39 40 6 , 45 27 31 →= 46 44 39 40 8 , 45 27 32 →= 46 44 39 40 10 , 2 33 29 →= 9 44 39 41 14 , 2 33 35 →= 9 44 39 41 16 , 2 33 36 →= 9 44 39 41 18 , 6 33 29 →= 10 44 39 41 14 , 6 33 35 →= 10 44 39 41 16 , 6 33 36 →= 10 44 39 41 18 , 20 33 29 →= 22 44 39 41 14 , 20 33 35 →= 22 44 39 41 16 , 20 33 36 →= 22 44 39 41 18 , 30 33 29 →= 32 44 39 41 14 , 30 33 35 →= 32 44 39 41 16 , 30 33 36 →= 32 44 39 41 18 , 45 33 29 →= 46 44 39 41 14 , 45 33 35 →= 46 44 39 41 16 , 45 33 36 →= 46 44 39 41 18 , 15 29 30 →= 12 31 39 40 6 , 15 29 31 →= 12 31 39 40 8 , 15 29 32 →= 12 31 39 40 10 , 16 29 30 →= 14 31 39 40 6 , 16 29 31 →= 14 31 39 40 8 , 16 29 32 →= 14 31 39 40 10 , 25 29 30 →= 24 31 39 40 6 , 25 29 31 →= 24 31 39 40 8 , 25 29 32 →= 24 31 39 40 10 , 33 29 30 →= 27 31 39 40 6 , 33 29 31 →= 27 31 39 40 8 , 33 29 32 →= 27 31 39 40 10 , 35 29 30 →= 29 31 39 40 6 , 35 29 31 →= 29 31 39 40 8 , 35 29 32 →= 29 31 39 40 10 , 15 35 29 →= 12 31 39 41 14 , 15 35 35 →= 12 31 39 41 16 , 15 35 36 →= 12 31 39 41 18 , 16 35 29 →= 14 31 39 41 14 , 16 35 35 →= 14 31 39 41 16 , 16 35 36 →= 14 31 39 41 18 , 25 35 29 →= 24 31 39 41 14 , 25 35 35 →= 24 31 39 41 16 , 25 35 36 →= 24 31 39 41 18 , 33 35 29 →= 27 31 39 41 14 , 33 35 35 →= 27 31 39 41 16 , 33 35 36 →= 27 31 39 41 18 , 35 35 29 →= 29 31 39 41 14 , 35 35 35 →= 29 31 39 41 16 , 35 35 36 →= 29 31 39 41 18 , 4 13 14 30 →= 19 20 , 4 13 14 31 →= 19 21 , 4 13 14 32 →= 19 22 , 4 13 16 29 →= 23 24 , 4 13 16 35 →= 23 25 , 4 13 16 36 →= 23 26 , 39 41 14 30 →= 43 6 , 39 41 14 31 →= 43 8 , 39 41 14 32 →= 43 10 , 39 41 16 29 →= 42 14 , 39 41 16 35 →= 42 16 , 39 41 16 36 →= 42 18 , 7 42 14 30 →= 9 45 , 7 42 14 31 →= 9 44 , 7 42 14 32 →= 9 46 , 8 42 14 30 →= 10 45 , 8 42 14 31 →= 10 44 , 8 42 14 32 →= 10 46 , 21 42 14 30 →= 22 45 , 21 42 14 31 →= 22 44 , 21 42 14 32 →= 22 46 , 31 42 14 30 →= 32 45 , 31 42 14 31 →= 32 44 , 31 42 14 32 →= 32 46 , 44 42 14 30 →= 46 45 , 44 42 14 31 →= 46 44 , 44 42 14 32 →= 46 46 , 7 42 16 29 →= 2 27 , 7 42 16 35 →= 2 33 , 7 42 16 36 →= 2 37 , 8 42 16 29 →= 6 27 , 8 42 16 35 →= 6 33 , 8 42 16 36 →= 6 37 , 21 42 16 29 →= 20 27 , 21 42 16 35 →= 20 33 , 21 42 16 36 →= 20 37 , 31 42 16 29 →= 30 27 , 31 42 16 35 →= 30 33 , 31 42 16 36 →= 30 37 , 44 42 16 29 →= 45 27 , 44 42 16 35 →= 45 33 , 44 42 16 36 →= 45 37 , 4 5 6 27 28 →= 23 25 35 29 28 , 4 5 6 27 30 →= 23 25 35 29 30 , 4 5 6 27 31 →= 23 25 35 29 31 , 4 5 6 27 32 →= 23 25 35 29 32 , 4 5 6 33 34 →= 23 25 35 35 34 , 4 5 6 33 29 →= 23 25 35 35 29 , 4 5 6 33 35 →= 23 25 35 35 35 , 4 5 6 33 36 →= 23 25 35 35 36 , 4 5 6 37 38 →= 23 25 35 36 38 , 39 40 6 27 28 →= 42 16 35 29 28 , 39 40 6 27 30 →= 42 16 35 29 30 , 39 40 6 27 31 →= 42 16 35 29 31 , 39 40 6 27 32 →= 42 16 35 29 32 , 39 40 6 33 34 →= 42 16 35 35 34 , 39 40 6 33 29 →= 42 16 35 35 29 , 39 40 6 33 35 →= 42 16 35 35 35 , 39 40 6 33 36 →= 42 16 35 35 36 , 39 40 6 37 38 →= 42 16 35 36 38 , 7 43 6 27 28 →= 2 33 35 29 28 , 7 43 6 27 30 →= 2 33 35 29 30 , 7 43 6 27 31 →= 2 33 35 29 31 , 7 43 6 27 32 →= 2 33 35 29 32 , 8 43 6 27 28 →= 6 33 35 29 28 , 8 43 6 27 30 →= 6 33 35 29 30 , 8 43 6 27 31 →= 6 33 35 29 31 , 8 43 6 27 32 →= 6 33 35 29 32 , 21 43 6 27 28 →= 20 33 35 29 28 , 21 43 6 27 30 →= 20 33 35 29 30 , 21 43 6 27 31 →= 20 33 35 29 31 , 21 43 6 27 32 →= 20 33 35 29 32 , 31 43 6 27 28 →= 30 33 35 29 28 , 31 43 6 27 30 →= 30 33 35 29 30 , 31 43 6 27 31 →= 30 33 35 29 31 , 31 43 6 27 32 →= 30 33 35 29 32 , 44 43 6 27 28 →= 45 33 35 29 28 , 44 43 6 27 30 →= 45 33 35 29 30 , 44 43 6 27 31 →= 45 33 35 29 31 , 44 43 6 27 32 →= 45 33 35 29 32 , 7 43 6 33 34 →= 2 33 35 35 34 , 7 43 6 33 29 →= 2 33 35 35 29 , 7 43 6 33 35 →= 2 33 35 35 35 , 7 43 6 33 36 →= 2 33 35 35 36 , 8 43 6 33 34 →= 6 33 35 35 34 , 8 43 6 33 29 →= 6 33 35 35 29 , 8 43 6 33 35 →= 6 33 35 35 35 , 8 43 6 33 36 →= 6 33 35 35 36 , 21 43 6 33 34 →= 20 33 35 35 34 , 21 43 6 33 29 →= 20 33 35 35 29 , 21 43 6 33 35 →= 20 33 35 35 35 , 21 43 6 33 36 →= 20 33 35 35 36 , 31 43 6 33 34 →= 30 33 35 35 34 , 31 43 6 33 29 →= 30 33 35 35 29 , 31 43 6 33 35 →= 30 33 35 35 35 , 31 43 6 33 36 →= 30 33 35 35 36 , 44 43 6 33 34 →= 45 33 35 35 34 , 44 43 6 33 29 →= 45 33 35 35 29 , 44 43 6 33 35 →= 45 33 35 35 35 , 44 43 6 33 36 →= 45 33 35 35 36 , 7 43 6 37 38 →= 2 33 35 36 38 , 8 43 6 37 38 →= 6 33 35 36 38 , 21 43 6 37 38 →= 20 33 35 36 38 , 31 43 6 37 38 →= 30 33 35 36 38 , 44 43 6 37 38 →= 45 33 35 36 38 } 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 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 26 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 27 ↦ ⎛ ⎞ ⎜ 1 5 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 28 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 29 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 30 ↦ ⎛ ⎞ ⎜ 1 7 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 31 ↦ ⎛ ⎞ ⎜ 1 6 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 32 ↦ ⎛ ⎞ ⎜ 1 14 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 33 ↦ ⎛ ⎞ ⎜ 1 5 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 34 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 35 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 36 ↦ ⎛ ⎞ ⎜ 1 4 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 37 ↦ ⎛ ⎞ ⎜ 1 6 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 38 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 39 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 40 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 41 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 42 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 43 ↦ ⎛ ⎞ ⎜ 1 4 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 44 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 45 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 46 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 42 ↦ 0, 14 ↦ 1, 30 ↦ 2, 43 ↦ 3, 8 ↦ 4, 39 ↦ 5, 40 ↦ 6, 6 ↦ 7, 31 ↦ 8, 41 ↦ 9, 16 ↦ 10, 29 ↦ 11, 35 ↦ 12, 33 ↦ 13, 27 ↦ 14, 36 ↦ 15, 37 ↦ 16, 28 ↦ 17, 32 ↦ 18, 34 ↦ 19, 38 ↦ 20 }, it remains to prove termination of the 57-rule system { 0 1 2 →= 3 4 5 6 7 , 0 1 8 →= 3 4 5 6 4 , 9 1 2 →= 6 4 5 6 7 , 9 1 8 →= 6 4 5 6 4 , 0 10 11 →= 3 4 5 9 1 , 0 10 12 →= 3 4 5 9 10 , 9 10 11 →= 6 4 5 9 1 , 9 10 12 →= 6 4 5 9 10 , 10 11 2 →= 1 8 5 6 7 , 10 11 8 →= 1 8 5 6 4 , 13 11 2 →= 14 8 5 6 7 , 13 11 8 →= 14 8 5 6 4 , 12 11 2 →= 11 8 5 6 7 , 12 11 8 →= 11 8 5 6 4 , 10 12 11 →= 1 8 5 9 1 , 10 12 12 →= 1 8 5 9 10 , 13 12 11 →= 14 8 5 9 1 , 13 12 12 →= 14 8 5 9 10 , 12 12 11 →= 11 8 5 9 1 , 12 12 12 →= 11 8 5 9 10 , 5 9 1 2 →= 3 7 , 5 9 1 8 →= 3 4 , 5 9 10 11 →= 0 1 , 5 9 10 12 →= 0 10 , 4 0 10 11 →= 7 14 , 4 0 10 12 →= 7 13 , 4 0 10 15 →= 7 16 , 8 0 10 11 →= 2 14 , 8 0 10 12 →= 2 13 , 8 0 10 15 →= 2 16 , 5 6 7 14 17 →= 0 10 12 11 17 , 5 6 7 14 2 →= 0 10 12 11 2 , 5 6 7 14 8 →= 0 10 12 11 8 , 5 6 7 14 18 →= 0 10 12 11 18 , 5 6 7 13 19 →= 0 10 12 12 19 , 5 6 7 13 11 →= 0 10 12 12 11 , 5 6 7 13 12 →= 0 10 12 12 12 , 5 6 7 13 15 →= 0 10 12 12 15 , 5 6 7 16 20 →= 0 10 12 15 20 , 4 3 7 14 17 →= 7 13 12 11 17 , 4 3 7 14 2 →= 7 13 12 11 2 , 4 3 7 14 8 →= 7 13 12 11 8 , 4 3 7 14 18 →= 7 13 12 11 18 , 8 3 7 14 17 →= 2 13 12 11 17 , 8 3 7 14 2 →= 2 13 12 11 2 , 8 3 7 14 8 →= 2 13 12 11 8 , 8 3 7 14 18 →= 2 13 12 11 18 , 4 3 7 13 19 →= 7 13 12 12 19 , 4 3 7 13 11 →= 7 13 12 12 11 , 4 3 7 13 12 →= 7 13 12 12 12 , 4 3 7 13 15 →= 7 13 12 12 15 , 8 3 7 13 19 →= 2 13 12 12 19 , 8 3 7 13 11 →= 2 13 12 12 11 , 8 3 7 13 12 →= 2 13 12 12 12 , 8 3 7 13 15 →= 2 13 12 12 15 , 4 3 7 16 20 →= 7 13 12 15 20 , 8 3 7 16 20 →= 2 13 12 15 20 } The system is trivially terminating.