/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 { p ↦ 0, 0 ↦ 1, s ↦ 2, f ↦ 3, g ↦ 4, q ↦ 5, i ↦ 6 }, it remains to prove termination of the 8-rule system { 0 1 ⟶ 2 2 1 2 2 0 , 0 2 1 ⟶ 1 , 0 2 2 ⟶ 2 0 2 , 3 2 ⟶ 4 5 6 , 4 ⟶ 3 0 0 , 5 6 ⟶ 5 2 , 5 2 ⟶ 2 2 , 6 ⟶ 2 } Applying sparse tiling TRFC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (0,0) ↦ 0, (0,1) ↦ 1, (1,2) ↦ 2, (0,2) ↦ 3, (2,2) ↦ 4, (2,1) ↦ 5, (2,0) ↦ 6, (1,8) ↦ 7, (0,8) ↦ 8, (3,0) ↦ 9, (3,2) ↦ 10, (6,0) ↦ 11, (6,2) ↦ 12, (3,1) ↦ 13, (6,1) ↦ 14, (2,8) ↦ 15, (7,3) ↦ 16, (7,4) ↦ 17, (4,5) ↦ 18, (5,6) ↦ 19, (6,8) ↦ 20, (4,2) ↦ 21, (0,5) ↦ 22, (5,2) ↦ 23, (7,5) ↦ 24, (7,2) ↦ 25 }, it remains to prove termination of the 66-rule system { 0 1 2 ⟶ 3 4 5 2 4 6 3 , 0 1 7 ⟶ 3 4 5 2 4 6 8 , 6 1 2 ⟶ 4 4 5 2 4 6 3 , 6 1 7 ⟶ 4 4 5 2 4 6 8 , 9 1 2 ⟶ 10 4 5 2 4 6 3 , 9 1 7 ⟶ 10 4 5 2 4 6 8 , 11 1 2 ⟶ 12 4 5 2 4 6 3 , 11 1 7 ⟶ 12 4 5 2 4 6 8 , 0 3 5 2 ⟶ 1 2 , 0 3 5 7 ⟶ 1 7 , 6 3 5 2 ⟶ 5 2 , 6 3 5 7 ⟶ 5 7 , 9 3 5 2 ⟶ 13 2 , 9 3 5 7 ⟶ 13 7 , 11 3 5 2 ⟶ 14 2 , 11 3 5 7 ⟶ 14 7 , 0 3 4 6 ⟶ 3 6 3 6 , 0 3 4 5 ⟶ 3 6 3 5 , 0 3 4 4 ⟶ 3 6 3 4 , 0 3 4 15 ⟶ 3 6 3 15 , 6 3 4 6 ⟶ 4 6 3 6 , 6 3 4 5 ⟶ 4 6 3 5 , 6 3 4 4 ⟶ 4 6 3 4 , 6 3 4 15 ⟶ 4 6 3 15 , 9 3 4 6 ⟶ 10 6 3 6 , 9 3 4 5 ⟶ 10 6 3 5 , 9 3 4 4 ⟶ 10 6 3 4 , 9 3 4 15 ⟶ 10 6 3 15 , 11 3 4 6 ⟶ 12 6 3 6 , 11 3 4 5 ⟶ 12 6 3 5 , 11 3 4 4 ⟶ 12 6 3 4 , 11 3 4 15 ⟶ 12 6 3 15 , 16 10 6 ⟶ 17 18 19 11 , 16 10 5 ⟶ 17 18 19 14 , 16 10 4 ⟶ 17 18 19 12 , 16 10 15 ⟶ 17 18 19 20 , 17 21 ⟶ 16 9 0 3 , 17 18 ⟶ 16 9 0 22 , 22 19 11 ⟶ 22 23 6 , 22 19 14 ⟶ 22 23 5 , 22 19 12 ⟶ 22 23 4 , 22 19 20 ⟶ 22 23 15 , 18 19 11 ⟶ 18 23 6 , 18 19 14 ⟶ 18 23 5 , 18 19 12 ⟶ 18 23 4 , 18 19 20 ⟶ 18 23 15 , 24 19 11 ⟶ 24 23 6 , 24 19 14 ⟶ 24 23 5 , 24 19 12 ⟶ 24 23 4 , 24 19 20 ⟶ 24 23 15 , 22 23 6 ⟶ 3 4 6 , 22 23 5 ⟶ 3 4 5 , 22 23 4 ⟶ 3 4 4 , 22 23 15 ⟶ 3 4 15 , 18 23 6 ⟶ 21 4 6 , 18 23 5 ⟶ 21 4 5 , 18 23 4 ⟶ 21 4 4 , 18 23 15 ⟶ 21 4 15 , 24 23 6 ⟶ 25 4 6 , 24 23 5 ⟶ 25 4 5 , 24 23 4 ⟶ 25 4 4 , 24 23 15 ⟶ 25 4 15 , 19 11 ⟶ 23 6 , 19 14 ⟶ 23 5 , 19 12 ⟶ 23 4 , 19 20 ⟶ 23 15 } 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 1 ⎟ ⎜ 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 1 ⎟ ⎜ 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, 9 ↦ 7, 10 ↦ 8, 11 ↦ 9, 12 ↦ 10, 7 ↦ 11, 13 ↦ 12, 14 ↦ 13, 15 ↦ 14, 16 ↦ 15, 17 ↦ 16, 18 ↦ 17, 19 ↦ 18, 20 ↦ 19, 21 ↦ 20, 22 ↦ 21, 23 ↦ 22, 24 ↦ 23 }, it remains to prove termination of the 58-rule system { 0 1 2 ⟶ 3 4 5 2 4 6 3 , 6 1 2 ⟶ 4 4 5 2 4 6 3 , 7 1 2 ⟶ 8 4 5 2 4 6 3 , 9 1 2 ⟶ 10 4 5 2 4 6 3 , 0 3 5 2 ⟶ 1 2 , 0 3 5 11 ⟶ 1 11 , 6 3 5 2 ⟶ 5 2 , 6 3 5 11 ⟶ 5 11 , 7 3 5 2 ⟶ 12 2 , 7 3 5 11 ⟶ 12 11 , 9 3 5 2 ⟶ 13 2 , 9 3 5 11 ⟶ 13 11 , 0 3 4 6 ⟶ 3 6 3 6 , 0 3 4 5 ⟶ 3 6 3 5 , 0 3 4 4 ⟶ 3 6 3 4 , 0 3 4 14 ⟶ 3 6 3 14 , 6 3 4 6 ⟶ 4 6 3 6 , 6 3 4 5 ⟶ 4 6 3 5 , 6 3 4 4 ⟶ 4 6 3 4 , 6 3 4 14 ⟶ 4 6 3 14 , 7 3 4 6 ⟶ 8 6 3 6 , 7 3 4 5 ⟶ 8 6 3 5 , 7 3 4 4 ⟶ 8 6 3 4 , 7 3 4 14 ⟶ 8 6 3 14 , 9 3 4 6 ⟶ 10 6 3 6 , 9 3 4 5 ⟶ 10 6 3 5 , 9 3 4 4 ⟶ 10 6 3 4 , 9 3 4 14 ⟶ 10 6 3 14 , 15 8 6 ⟶ 16 17 18 9 , 15 8 5 ⟶ 16 17 18 13 , 15 8 4 ⟶ 16 17 18 10 , 15 8 14 ⟶ 16 17 18 19 , 16 20 ⟶ 15 7 0 3 , 16 17 ⟶ 15 7 0 21 , 21 18 9 ⟶ 21 22 6 , 21 18 13 ⟶ 21 22 5 , 21 18 10 ⟶ 21 22 4 , 21 18 19 ⟶ 21 22 14 , 17 18 9 ⟶ 17 22 6 , 17 18 13 ⟶ 17 22 5 , 17 18 10 ⟶ 17 22 4 , 17 18 19 ⟶ 17 22 14 , 23 18 9 ⟶ 23 22 6 , 23 18 13 ⟶ 23 22 5 , 23 18 10 ⟶ 23 22 4 , 23 18 19 ⟶ 23 22 14 , 21 22 6 ⟶ 3 4 6 , 21 22 5 ⟶ 3 4 5 , 21 22 4 ⟶ 3 4 4 , 21 22 14 ⟶ 3 4 14 , 17 22 6 ⟶ 20 4 6 , 17 22 5 ⟶ 20 4 5 , 17 22 4 ⟶ 20 4 4 , 17 22 14 ⟶ 20 4 14 , 18 9 ⟶ 22 6 , 18 13 ⟶ 22 5 , 18 10 ⟶ 22 4 , 18 19 ⟶ 22 14 } Applying sparse untiling TRFCU(2) after reversal [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { 2 ↦ 0, 1 ↦ 1, 0 ↦ 2, 7 ↦ 3, 5 ↦ 4, 6 ↦ 5, 4 ↦ 6, 3 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 18 ↦ 16, 17 ↦ 17, 16 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 50-rule system { 0 1 2 ⟶ 3 4 5 2 4 6 7 , 8 7 5 2 ⟶ 1 2 , 8 7 5 9 ⟶ 1 9 , 6 7 5 2 ⟶ 5 2 , 6 7 5 9 ⟶ 5 9 , 0 7 5 2 ⟶ 10 2 , 0 7 5 9 ⟶ 10 9 , 11 7 5 2 ⟶ 12 2 , 11 7 5 9 ⟶ 12 9 , 8 7 4 6 ⟶ 7 6 7 6 , 8 7 4 5 ⟶ 7 6 7 5 , 8 7 4 4 ⟶ 7 6 7 4 , 8 7 4 13 ⟶ 7 6 7 13 , 6 7 4 6 ⟶ 4 6 7 6 , 6 7 4 5 ⟶ 4 6 7 5 , 6 7 4 4 ⟶ 4 6 7 4 , 6 7 4 13 ⟶ 4 6 7 13 , 0 7 4 6 ⟶ 3 6 7 6 , 0 7 4 5 ⟶ 3 6 7 5 , 0 7 4 4 ⟶ 3 6 7 4 , 0 7 4 13 ⟶ 3 6 7 13 , 11 7 4 6 ⟶ 14 6 7 6 , 11 7 4 5 ⟶ 14 6 7 5 , 11 7 4 4 ⟶ 14 6 7 4 , 11 7 4 13 ⟶ 14 6 7 13 , 15 3 6 ⟶ 16 17 18 11 , 15 3 5 ⟶ 16 17 18 12 , 15 3 4 ⟶ 16 17 18 14 , 16 19 ⟶ 15 0 8 7 , 16 17 ⟶ 15 0 8 20 , 20 18 11 ⟶ 20 21 6 , 20 18 12 ⟶ 20 21 5 , 20 18 14 ⟶ 20 21 4 , 20 18 22 ⟶ 20 21 13 , 17 18 11 ⟶ 17 21 6 , 17 18 12 ⟶ 17 21 5 , 17 18 14 ⟶ 17 21 4 , 17 18 22 ⟶ 17 21 13 , 20 21 6 ⟶ 7 4 6 , 20 21 5 ⟶ 7 4 5 , 20 21 4 ⟶ 7 4 4 , 20 21 13 ⟶ 7 4 13 , 17 21 6 ⟶ 19 4 6 , 17 21 5 ⟶ 19 4 5 , 17 21 4 ⟶ 19 4 4 , 17 21 13 ⟶ 19 4 13 , 18 11 ⟶ 21 6 , 18 12 ⟶ 21 5 , 18 14 ⟶ 21 4 , 18 22 ⟶ 21 13 } 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 1 ⎟ ⎜ 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 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, 13 ↦ 13, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21 }, it remains to prove termination of the 47-rule system { 0 1 2 ⟶ 3 4 5 2 4 6 7 , 8 7 5 2 ⟶ 1 2 , 8 7 5 9 ⟶ 1 9 , 6 7 5 2 ⟶ 5 2 , 6 7 5 9 ⟶ 5 9 , 0 7 5 2 ⟶ 10 2 , 0 7 5 9 ⟶ 10 9 , 11 7 5 2 ⟶ 12 2 , 11 7 5 9 ⟶ 12 9 , 8 7 4 6 ⟶ 7 6 7 6 , 8 7 4 5 ⟶ 7 6 7 5 , 8 7 4 4 ⟶ 7 6 7 4 , 8 7 4 13 ⟶ 7 6 7 13 , 6 7 4 6 ⟶ 4 6 7 6 , 6 7 4 5 ⟶ 4 6 7 5 , 6 7 4 4 ⟶ 4 6 7 4 , 6 7 4 13 ⟶ 4 6 7 13 , 0 7 4 6 ⟶ 3 6 7 6 , 0 7 4 5 ⟶ 3 6 7 5 , 0 7 4 4 ⟶ 3 6 7 4 , 0 7 4 13 ⟶ 3 6 7 13 , 11 7 4 6 ⟶ 14 6 7 6 , 11 7 4 5 ⟶ 14 6 7 5 , 11 7 4 4 ⟶ 14 6 7 4 , 11 7 4 13 ⟶ 14 6 7 13 , 15 3 6 ⟶ 16 17 18 11 , 15 3 5 ⟶ 16 17 18 12 , 15 3 4 ⟶ 16 17 18 14 , 16 19 ⟶ 15 0 8 7 , 16 17 ⟶ 15 0 8 20 , 20 18 11 ⟶ 20 21 6 , 20 18 12 ⟶ 20 21 5 , 20 18 14 ⟶ 20 21 4 , 17 18 11 ⟶ 17 21 6 , 17 18 12 ⟶ 17 21 5 , 17 18 14 ⟶ 17 21 4 , 20 21 6 ⟶ 7 4 6 , 20 21 5 ⟶ 7 4 5 , 20 21 4 ⟶ 7 4 4 , 20 21 13 ⟶ 7 4 13 , 17 21 6 ⟶ 19 4 6 , 17 21 5 ⟶ 19 4 5 , 17 21 4 ⟶ 19 4 4 , 17 21 13 ⟶ 19 4 13 , 18 11 ⟶ 21 6 , 18 12 ⟶ 21 5 , 18 14 ⟶ 21 4 } Applying sparse untiling TRFCU(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, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21 }, it remains to prove termination of the 45-rule system { 0 1 2 ⟶ 3 4 5 2 4 6 7 , 8 7 5 2 ⟶ 1 2 , 8 7 5 9 ⟶ 1 9 , 6 7 5 2 ⟶ 5 2 , 6 7 5 9 ⟶ 5 9 , 0 7 5 2 ⟶ 10 2 , 0 7 5 9 ⟶ 10 9 , 11 7 5 2 ⟶ 12 2 , 11 7 5 9 ⟶ 12 9 , 8 7 4 6 ⟶ 7 6 7 6 , 8 7 4 5 ⟶ 7 6 7 5 , 8 7 4 4 ⟶ 7 6 7 4 , 8 7 4 13 ⟶ 7 6 7 13 , 6 7 4 6 ⟶ 4 6 7 6 , 6 7 4 5 ⟶ 4 6 7 5 , 6 7 4 4 ⟶ 4 6 7 4 , 6 7 4 13 ⟶ 4 6 7 13 , 0 7 4 6 ⟶ 3 6 7 6 , 0 7 4 5 ⟶ 3 6 7 5 , 0 7 4 4 ⟶ 3 6 7 4 , 0 7 4 13 ⟶ 3 6 7 13 , 11 7 4 6 ⟶ 14 6 7 6 , 11 7 4 5 ⟶ 14 6 7 5 , 11 7 4 4 ⟶ 14 6 7 4 , 11 7 4 13 ⟶ 14 6 7 13 , 15 3 6 ⟶ 16 17 18 11 , 15 3 5 ⟶ 16 17 18 12 , 15 3 4 ⟶ 16 17 18 14 , 16 19 ⟶ 15 0 8 7 , 16 17 ⟶ 15 0 8 20 , 20 18 11 ⟶ 20 21 6 , 20 18 12 ⟶ 20 21 5 , 20 18 14 ⟶ 20 21 4 , 17 18 11 ⟶ 17 21 6 , 17 18 12 ⟶ 17 21 5 , 17 18 14 ⟶ 17 21 4 , 20 21 6 ⟶ 7 4 6 , 20 21 5 ⟶ 7 4 5 , 20 21 4 ⟶ 7 4 4 , 17 21 6 ⟶ 19 4 6 , 17 21 5 ⟶ 19 4 5 , 17 21 4 ⟶ 19 4 4 , 18 11 ⟶ 21 6 , 18 12 ⟶ 21 5 , 18 14 ⟶ 21 4 } Applying sparse untiling TRFCU(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, 14 ↦ 13, 15 ↦ 14, 16 ↦ 15, 17 ↦ 16, 18 ↦ 17, 19 ↦ 18, 20 ↦ 19, 21 ↦ 20 }, it remains to prove termination of the 41-rule system { 0 1 2 ⟶ 3 4 5 2 4 6 7 , 8 7 5 2 ⟶ 1 2 , 8 7 5 9 ⟶ 1 9 , 6 7 5 2 ⟶ 5 2 , 6 7 5 9 ⟶ 5 9 , 0 7 5 2 ⟶ 10 2 , 0 7 5 9 ⟶ 10 9 , 11 7 5 2 ⟶ 12 2 , 11 7 5 9 ⟶ 12 9 , 8 7 4 6 ⟶ 7 6 7 6 , 8 7 4 5 ⟶ 7 6 7 5 , 8 7 4 4 ⟶ 7 6 7 4 , 6 7 4 6 ⟶ 4 6 7 6 , 6 7 4 5 ⟶ 4 6 7 5 , 6 7 4 4 ⟶ 4 6 7 4 , 0 7 4 6 ⟶ 3 6 7 6 , 0 7 4 5 ⟶ 3 6 7 5 , 0 7 4 4 ⟶ 3 6 7 4 , 11 7 4 6 ⟶ 13 6 7 6 , 11 7 4 5 ⟶ 13 6 7 5 , 11 7 4 4 ⟶ 13 6 7 4 , 14 3 6 ⟶ 15 16 17 11 , 14 3 5 ⟶ 15 16 17 12 , 14 3 4 ⟶ 15 16 17 13 , 15 18 ⟶ 14 0 8 7 , 15 16 ⟶ 14 0 8 19 , 19 17 11 ⟶ 19 20 6 , 19 17 12 ⟶ 19 20 5 , 19 17 13 ⟶ 19 20 4 , 16 17 11 ⟶ 16 20 6 , 16 17 12 ⟶ 16 20 5 , 16 17 13 ⟶ 16 20 4 , 19 20 6 ⟶ 7 4 6 , 19 20 5 ⟶ 7 4 5 , 19 20 4 ⟶ 7 4 4 , 16 20 6 ⟶ 18 4 6 , 16 20 5 ⟶ 18 4 5 , 16 20 4 ⟶ 18 4 4 , 17 11 ⟶ 20 6 , 17 12 ⟶ 20 5 , 17 13 ⟶ 20 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 1 ⎟ ⎜ 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 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 0 ⎟ ⎜ 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, 11 ↦ 10, 12 ↦ 11, 13 ↦ 12, 14 ↦ 13, 15 ↦ 14, 16 ↦ 15, 17 ↦ 16, 18 ↦ 17, 19 ↦ 18, 20 ↦ 19 }, it remains to prove termination of the 39-rule system { 0 1 2 ⟶ 3 4 5 2 4 6 7 , 8 7 5 2 ⟶ 1 2 , 8 7 5 9 ⟶ 1 9 , 6 7 5 2 ⟶ 5 2 , 6 7 5 9 ⟶ 5 9 , 10 7 5 2 ⟶ 11 2 , 10 7 5 9 ⟶ 11 9 , 8 7 4 6 ⟶ 7 6 7 6 , 8 7 4 5 ⟶ 7 6 7 5 , 8 7 4 4 ⟶ 7 6 7 4 , 6 7 4 6 ⟶ 4 6 7 6 , 6 7 4 5 ⟶ 4 6 7 5 , 6 7 4 4 ⟶ 4 6 7 4 , 0 7 4 6 ⟶ 3 6 7 6 , 0 7 4 5 ⟶ 3 6 7 5 , 0 7 4 4 ⟶ 3 6 7 4 , 10 7 4 6 ⟶ 12 6 7 6 , 10 7 4 5 ⟶ 12 6 7 5 , 10 7 4 4 ⟶ 12 6 7 4 , 13 3 6 ⟶ 14 15 16 10 , 13 3 5 ⟶ 14 15 16 11 , 13 3 4 ⟶ 14 15 16 12 , 14 17 ⟶ 13 0 8 7 , 14 15 ⟶ 13 0 8 18 , 18 16 10 ⟶ 18 19 6 , 18 16 11 ⟶ 18 19 5 , 18 16 12 ⟶ 18 19 4 , 15 16 10 ⟶ 15 19 6 , 15 16 11 ⟶ 15 19 5 , 15 16 12 ⟶ 15 19 4 , 18 19 6 ⟶ 7 4 6 , 18 19 5 ⟶ 7 4 5 , 18 19 4 ⟶ 7 4 4 , 15 19 6 ⟶ 17 4 6 , 15 19 5 ⟶ 17 4 5 , 15 19 4 ⟶ 17 4 4 , 16 10 ⟶ 19 6 , 16 11 ⟶ 19 5 , 16 12 ⟶ 19 4 } Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols. After renaming modulo the bijection { (0,↑) ↦ 0, (1,↓) ↦ 1, (2,↓) ↦ 2, (6,↑) ↦ 3, (7,↓) ↦ 4, (8,↑) ↦ 5, (4,↓) ↦ 6, (6,↓) ↦ 7, (5,↓) ↦ 8, (10,↑) ↦ 9, (13,↑) ↦ 10, (3,↓) ↦ 11, (14,↑) ↦ 12, (15,↓) ↦ 13, (16,↓) ↦ 14, (10,↓) ↦ 15, (15,↑) ↦ 16, (16,↑) ↦ 17, (11,↓) ↦ 18, (12,↓) ↦ 19, (17,↓) ↦ 20, (0,↓) ↦ 21, (8,↓) ↦ 22, (18,↓) ↦ 23, (18,↑) ↦ 24, (19,↓) ↦ 25, (9,↓) ↦ 26, (13,↓) ↦ 27, (14,↓) ↦ 28 }, it remains to prove termination of the 78-rule system { 0 1 2 ⟶ 3 4 , 5 4 6 7 ⟶ 3 4 7 , 5 4 6 8 ⟶ 3 4 8 , 5 4 6 6 ⟶ 3 4 6 , 3 4 6 7 ⟶ 3 4 7 , 3 4 6 8 ⟶ 3 4 8 , 3 4 6 6 ⟶ 3 4 6 , 0 4 6 7 ⟶ 3 4 7 , 0 4 6 8 ⟶ 3 4 8 , 0 4 6 6 ⟶ 3 4 6 , 9 4 6 7 ⟶ 3 4 7 , 9 4 6 8 ⟶ 3 4 8 , 9 4 6 6 ⟶ 3 4 6 , 10 11 7 ⟶ 12 13 14 15 , 10 11 7 ⟶ 16 14 15 , 10 11 7 ⟶ 17 15 , 10 11 7 ⟶ 9 , 10 11 8 ⟶ 12 13 14 18 , 10 11 8 ⟶ 16 14 18 , 10 11 8 ⟶ 17 18 , 10 11 6 ⟶ 12 13 14 19 , 10 11 6 ⟶ 16 14 19 , 10 11 6 ⟶ 17 19 , 12 20 ⟶ 10 21 22 4 , 12 20 ⟶ 0 22 4 , 12 20 ⟶ 5 4 , 12 13 ⟶ 10 21 22 23 , 12 13 ⟶ 0 22 23 , 12 13 ⟶ 5 23 , 12 13 ⟶ 24 , 24 14 15 ⟶ 24 25 7 , 24 14 15 ⟶ 3 , 24 14 18 ⟶ 24 25 8 , 24 14 19 ⟶ 24 25 6 , 16 14 15 ⟶ 16 25 7 , 16 14 15 ⟶ 3 , 16 14 18 ⟶ 16 25 8 , 16 14 19 ⟶ 16 25 6 , 17 15 ⟶ 3 , 21 1 2 →= 11 6 8 2 6 7 4 , 22 4 8 2 →= 1 2 , 22 4 8 26 →= 1 26 , 7 4 8 2 →= 8 2 , 7 4 8 26 →= 8 26 , 15 4 8 2 →= 18 2 , 15 4 8 26 →= 18 26 , 22 4 6 7 →= 4 7 4 7 , 22 4 6 8 →= 4 7 4 8 , 22 4 6 6 →= 4 7 4 6 , 7 4 6 7 →= 6 7 4 7 , 7 4 6 8 →= 6 7 4 8 , 7 4 6 6 →= 6 7 4 6 , 21 4 6 7 →= 11 7 4 7 , 21 4 6 8 →= 11 7 4 8 , 21 4 6 6 →= 11 7 4 6 , 15 4 6 7 →= 19 7 4 7 , 15 4 6 8 →= 19 7 4 8 , 15 4 6 6 →= 19 7 4 6 , 27 11 7 →= 28 13 14 15 , 27 11 8 →= 28 13 14 18 , 27 11 6 →= 28 13 14 19 , 28 20 →= 27 21 22 4 , 28 13 →= 27 21 22 23 , 23 14 15 →= 23 25 7 , 23 14 18 →= 23 25 8 , 23 14 19 →= 23 25 6 , 13 14 15 →= 13 25 7 , 13 14 18 →= 13 25 8 , 13 14 19 →= 13 25 6 , 23 25 7 →= 4 6 7 , 23 25 8 →= 4 6 8 , 23 25 6 →= 4 6 6 , 13 25 7 →= 20 6 7 , 13 25 8 →= 20 6 8 , 13 25 6 →= 20 6 6 , 14 15 →= 25 7 , 14 18 →= 25 8 , 14 19 →= 25 6 } 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 1 ⎟ ⎜ 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 27 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 28 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 5 ↦ 0, 4 ↦ 1, 6 ↦ 2, 7 ↦ 3, 3 ↦ 4, 8 ↦ 5, 0 ↦ 6, 9 ↦ 7, 10 ↦ 8, 11 ↦ 9, 12 ↦ 10, 13 ↦ 11, 14 ↦ 12, 15 ↦ 13, 16 ↦ 14, 17 ↦ 15, 18 ↦ 16, 19 ↦ 17, 20 ↦ 18, 21 ↦ 19, 22 ↦ 20, 23 ↦ 21, 24 ↦ 22, 25 ↦ 23, 1 ↦ 24, 2 ↦ 25, 26 ↦ 26, 27 ↦ 27, 28 ↦ 28 }, it remains to prove termination of the 77-rule system { 0 1 2 3 ⟶ 4 1 3 , 0 1 2 5 ⟶ 4 1 5 , 0 1 2 2 ⟶ 4 1 2 , 4 1 2 3 ⟶ 4 1 3 , 4 1 2 5 ⟶ 4 1 5 , 4 1 2 2 ⟶ 4 1 2 , 6 1 2 3 ⟶ 4 1 3 , 6 1 2 5 ⟶ 4 1 5 , 6 1 2 2 ⟶ 4 1 2 , 7 1 2 3 ⟶ 4 1 3 , 7 1 2 5 ⟶ 4 1 5 , 7 1 2 2 ⟶ 4 1 2 , 8 9 3 ⟶ 10 11 12 13 , 8 9 3 ⟶ 14 12 13 , 8 9 3 ⟶ 15 13 , 8 9 3 ⟶ 7 , 8 9 5 ⟶ 10 11 12 16 , 8 9 5 ⟶ 14 12 16 , 8 9 5 ⟶ 15 16 , 8 9 2 ⟶ 10 11 12 17 , 8 9 2 ⟶ 14 12 17 , 8 9 2 ⟶ 15 17 , 10 18 ⟶ 8 19 20 1 , 10 18 ⟶ 6 20 1 , 10 18 ⟶ 0 1 , 10 11 ⟶ 8 19 20 21 , 10 11 ⟶ 6 20 21 , 10 11 ⟶ 0 21 , 10 11 ⟶ 22 , 22 12 13 ⟶ 22 23 3 , 22 12 13 ⟶ 4 , 22 12 16 ⟶ 22 23 5 , 22 12 17 ⟶ 22 23 2 , 14 12 13 ⟶ 14 23 3 , 14 12 13 ⟶ 4 , 14 12 16 ⟶ 14 23 5 , 14 12 17 ⟶ 14 23 2 , 15 13 ⟶ 4 , 19 24 25 →= 9 2 5 25 2 3 1 , 20 1 5 25 →= 24 25 , 20 1 5 26 →= 24 26 , 3 1 5 25 →= 5 25 , 3 1 5 26 →= 5 26 , 13 1 5 25 →= 16 25 , 13 1 5 26 →= 16 26 , 20 1 2 3 →= 1 3 1 3 , 20 1 2 5 →= 1 3 1 5 , 20 1 2 2 →= 1 3 1 2 , 3 1 2 3 →= 2 3 1 3 , 3 1 2 5 →= 2 3 1 5 , 3 1 2 2 →= 2 3 1 2 , 19 1 2 3 →= 9 3 1 3 , 19 1 2 5 →= 9 3 1 5 , 19 1 2 2 →= 9 3 1 2 , 13 1 2 3 →= 17 3 1 3 , 13 1 2 5 →= 17 3 1 5 , 13 1 2 2 →= 17 3 1 2 , 27 9 3 →= 28 11 12 13 , 27 9 5 →= 28 11 12 16 , 27 9 2 →= 28 11 12 17 , 28 18 →= 27 19 20 1 , 28 11 →= 27 19 20 21 , 21 12 13 →= 21 23 3 , 21 12 16 →= 21 23 5 , 21 12 17 →= 21 23 2 , 11 12 13 →= 11 23 3 , 11 12 16 →= 11 23 5 , 11 12 17 →= 11 23 2 , 21 23 3 →= 1 2 3 , 21 23 5 →= 1 2 5 , 21 23 2 →= 1 2 2 , 11 23 3 →= 18 2 3 , 11 23 5 →= 18 2 5 , 11 23 2 →= 18 2 2 , 12 13 →= 23 3 , 12 16 →= 23 5 , 12 17 →= 23 2 } 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 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 1 ⎟ ⎜ 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 ⎟ ⎝ ⎠ After renaming modulo the bijection { 4 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 5 ↦ 4, 8 ↦ 5, 9 ↦ 6, 10 ↦ 7, 11 ↦ 8, 12 ↦ 9, 13 ↦ 10, 16 ↦ 11, 17 ↦ 12, 18 ↦ 13, 19 ↦ 14, 20 ↦ 15, 21 ↦ 16, 22 ↦ 17, 23 ↦ 18, 14 ↦ 19, 24 ↦ 20, 25 ↦ 21, 26 ↦ 22, 27 ↦ 23, 28 ↦ 24 }, it remains to prove termination of the 53-rule system { 0 1 2 3 ⟶ 0 1 3 , 0 1 2 4 ⟶ 0 1 4 , 0 1 2 2 ⟶ 0 1 2 , 5 6 3 ⟶ 7 8 9 10 , 5 6 4 ⟶ 7 8 9 11 , 5 6 2 ⟶ 7 8 9 12 , 7 13 ⟶ 5 14 15 1 , 7 8 ⟶ 5 14 15 16 , 17 9 10 ⟶ 17 18 3 , 17 9 11 ⟶ 17 18 4 , 17 9 12 ⟶ 17 18 2 , 19 9 10 ⟶ 19 18 3 , 19 9 11 ⟶ 19 18 4 , 19 9 12 ⟶ 19 18 2 , 14 20 21 →= 6 2 4 21 2 3 1 , 15 1 4 21 →= 20 21 , 15 1 4 22 →= 20 22 , 3 1 4 21 →= 4 21 , 3 1 4 22 →= 4 22 , 10 1 4 21 →= 11 21 , 10 1 4 22 →= 11 22 , 15 1 2 3 →= 1 3 1 3 , 15 1 2 4 →= 1 3 1 4 , 15 1 2 2 →= 1 3 1 2 , 3 1 2 3 →= 2 3 1 3 , 3 1 2 4 →= 2 3 1 4 , 3 1 2 2 →= 2 3 1 2 , 14 1 2 3 →= 6 3 1 3 , 14 1 2 4 →= 6 3 1 4 , 14 1 2 2 →= 6 3 1 2 , 10 1 2 3 →= 12 3 1 3 , 10 1 2 4 →= 12 3 1 4 , 10 1 2 2 →= 12 3 1 2 , 23 6 3 →= 24 8 9 10 , 23 6 4 →= 24 8 9 11 , 23 6 2 →= 24 8 9 12 , 24 13 →= 23 14 15 1 , 24 8 →= 23 14 15 16 , 16 9 10 →= 16 18 3 , 16 9 11 →= 16 18 4 , 16 9 12 →= 16 18 2 , 8 9 10 →= 8 18 3 , 8 9 11 →= 8 18 4 , 8 9 12 →= 8 18 2 , 16 18 3 →= 1 2 3 , 16 18 4 →= 1 2 4 , 16 18 2 →= 1 2 2 , 8 18 3 →= 13 2 3 , 8 18 4 →= 13 2 4 , 8 18 2 →= 13 2 2 , 9 10 →= 18 3 , 9 11 →= 18 4 , 9 12 →= 18 2 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 4: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 1 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 0 1 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 23 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎝ ⎠ 24 ↦ ⎛ ⎞ ⎜ 1 0 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 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, 14 ↦ 14, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22, 23 ↦ 23, 24 ↦ 24 }, it remains to prove termination of the 52-rule system { 0 1 2 3 ⟶ 0 1 3 , 0 1 2 4 ⟶ 0 1 4 , 0 1 2 2 ⟶ 0 1 2 , 5 6 3 ⟶ 7 8 9 10 , 5 6 4 ⟶ 7 8 9 11 , 5 6 2 ⟶ 7 8 9 12 , 7 13 ⟶ 5 14 15 1 , 7 8 ⟶ 5 14 15 16 , 17 9 11 ⟶ 17 18 4 , 17 9 12 ⟶ 17 18 2 , 19 9 10 ⟶ 19 18 3 , 19 9 11 ⟶ 19 18 4 , 19 9 12 ⟶ 19 18 2 , 14 20 21 →= 6 2 4 21 2 3 1 , 15 1 4 21 →= 20 21 , 15 1 4 22 →= 20 22 , 3 1 4 21 →= 4 21 , 3 1 4 22 →= 4 22 , 10 1 4 21 →= 11 21 , 10 1 4 22 →= 11 22 , 15 1 2 3 →= 1 3 1 3 , 15 1 2 4 →= 1 3 1 4 , 15 1 2 2 →= 1 3 1 2 , 3 1 2 3 →= 2 3 1 3 , 3 1 2 4 →= 2 3 1 4 , 3 1 2 2 →= 2 3 1 2 , 14 1 2 3 →= 6 3 1 3 , 14 1 2 4 →= 6 3 1 4 , 14 1 2 2 →= 6 3 1 2 , 10 1 2 3 →= 12 3 1 3 , 10 1 2 4 →= 12 3 1 4 , 10 1 2 2 →= 12 3 1 2 , 23 6 3 →= 24 8 9 10 , 23 6 4 →= 24 8 9 11 , 23 6 2 →= 24 8 9 12 , 24 13 →= 23 14 15 1 , 24 8 →= 23 14 15 16 , 16 9 10 →= 16 18 3 , 16 9 11 →= 16 18 4 , 16 9 12 →= 16 18 2 , 8 9 10 →= 8 18 3 , 8 9 11 →= 8 18 4 , 8 9 12 →= 8 18 2 , 16 18 3 →= 1 2 3 , 16 18 4 →= 1 2 4 , 16 18 2 →= 1 2 2 , 8 18 3 →= 13 2 3 , 8 18 4 →= 13 2 4 , 8 18 2 →= 13 2 2 , 9 10 →= 18 3 , 9 11 →= 18 4 , 9 12 →= 18 2 } Applying sparse untiling TROCU(2) after reversal [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { 3 ↦ 0, 2 ↦ 1, 1 ↦ 2, 0 ↦ 3, 4 ↦ 4, 6 ↦ 5, 5 ↦ 6, 10 ↦ 7, 9 ↦ 8, 8 ↦ 9, 7 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13, 15 ↦ 14, 14 ↦ 15, 16 ↦ 16, 18 ↦ 17, 17 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 22 ↦ 22 }, it remains to prove termination of the 47-rule system { 0 1 2 3 ⟶ 0 1 3 , 0 1 2 4 ⟶ 0 1 4 , 0 1 2 2 ⟶ 0 1 2 , 5 6 3 ⟶ 7 8 9 10 , 5 6 4 ⟶ 7 8 9 11 , 5 6 2 ⟶ 7 8 9 12 , 7 13 ⟶ 5 14 15 1 , 7 8 ⟶ 5 14 15 16 , 14 17 18 →= 6 2 4 18 2 3 1 , 15 1 4 18 →= 17 18 , 15 1 4 19 →= 17 19 , 3 1 4 18 →= 4 18 , 3 1 4 19 →= 4 19 , 10 1 4 18 →= 11 18 , 10 1 4 19 →= 11 19 , 15 1 2 3 →= 1 3 1 3 , 15 1 2 4 →= 1 3 1 4 , 15 1 2 2 →= 1 3 1 2 , 3 1 2 3 →= 2 3 1 3 , 3 1 2 4 →= 2 3 1 4 , 3 1 2 2 →= 2 3 1 2 , 14 1 2 3 →= 6 3 1 3 , 14 1 2 4 →= 6 3 1 4 , 14 1 2 2 →= 6 3 1 2 , 10 1 2 3 →= 12 3 1 3 , 10 1 2 4 →= 12 3 1 4 , 10 1 2 2 →= 12 3 1 2 , 20 6 3 →= 21 8 9 10 , 20 6 4 →= 21 8 9 11 , 20 6 2 →= 21 8 9 12 , 21 13 →= 20 14 15 1 , 21 8 →= 20 14 15 16 , 16 9 10 →= 16 22 3 , 16 9 11 →= 16 22 4 , 16 9 12 →= 16 22 2 , 8 9 10 →= 8 22 3 , 8 9 11 →= 8 22 4 , 8 9 12 →= 8 22 2 , 16 22 3 →= 1 2 3 , 16 22 4 →= 1 2 4 , 16 22 2 →= 1 2 2 , 8 22 3 →= 13 2 3 , 8 22 4 →= 13 2 4 , 8 22 2 →= 13 2 2 , 9 10 →= 22 3 , 9 11 →= 22 4 , 9 12 →= 22 2 } Applying sparse tiling TROC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (23,0) ↦ 0, (0,1) ↦ 1, (1,2) ↦ 2, (2,3) ↦ 3, (3,1) ↦ 4, (1,3) ↦ 5, (3,24) ↦ 6, (2,4) ↦ 7, (4,18) ↦ 8, (1,4) ↦ 9, (4,19) ↦ 10, (4,24) ↦ 11, (2,2) ↦ 12, (2,24) ↦ 13, (23,5) ↦ 14, (5,6) ↦ 15, (6,3) ↦ 16, (23,7) ↦ 17, (7,8) ↦ 18, (8,9) ↦ 19, (9,10) ↦ 20, (10,1) ↦ 21, (10,24) ↦ 22, (6,4) ↦ 23, (9,11) ↦ 24, (11,18) ↦ 25, (11,19) ↦ 26, (11,24) ↦ 27, (6,2) ↦ 28, (9,12) ↦ 29, (12,2) ↦ 30, (12,3) ↦ 31, (12,4) ↦ 32, (12,24) ↦ 33, (7,13) ↦ 34, (13,2) ↦ 35, (5,14) ↦ 36, (14,15) ↦ 37, (15,1) ↦ 38, (8,22) ↦ 39, (15,16) ↦ 40, (16,22) ↦ 41, (16,9) ↦ 42, (20,14) ↦ 43, (14,17) ↦ 44, (17,18) ↦ 45, (18,2) ↦ 46, (20,6) ↦ 47, (18,24) ↦ 48, (1,24) ↦ 49, (19,24) ↦ 50, (17,19) ↦ 51, (22,3) ↦ 52, (22,4) ↦ 53, (14,1) ↦ 54, (22,2) ↦ 55, (23,20) ↦ 56, (23,21) ↦ 57, (21,8) ↦ 58, (21,13) ↦ 59, (23,16) ↦ 60, (23,8) ↦ 61, (23,1) ↦ 62, (23,13) ↦ 63 }, it remains to prove termination of the 247-rule system { 0 1 2 3 4 ⟶ 0 1 5 4 , 0 1 2 3 6 ⟶ 0 1 5 6 , 0 1 2 7 8 ⟶ 0 1 9 8 , 0 1 2 7 10 ⟶ 0 1 9 10 , 0 1 2 7 11 ⟶ 0 1 9 11 , 0 1 2 12 12 ⟶ 0 1 2 12 , 0 1 2 12 3 ⟶ 0 1 2 3 , 0 1 2 12 7 ⟶ 0 1 2 7 , 0 1 2 12 13 ⟶ 0 1 2 13 , 14 15 16 4 ⟶ 17 18 19 20 21 , 14 15 16 6 ⟶ 17 18 19 20 22 , 14 15 23 8 ⟶ 17 18 19 24 25 , 14 15 23 10 ⟶ 17 18 19 24 26 , 14 15 23 11 ⟶ 17 18 19 24 27 , 14 15 28 12 ⟶ 17 18 19 29 30 , 14 15 28 3 ⟶ 17 18 19 29 31 , 14 15 28 7 ⟶ 17 18 19 29 32 , 14 15 28 13 ⟶ 17 18 19 29 33 , 17 34 35 ⟶ 14 36 37 38 2 , 17 18 39 ⟶ 14 36 37 40 41 , 17 18 19 ⟶ 14 36 37 40 42 , 43 44 45 46 →= 47 28 7 8 46 3 4 2 , 43 44 45 48 →= 47 28 7 8 46 3 4 49 , 36 44 45 46 →= 15 28 7 8 46 3 4 2 , 36 44 45 48 →= 15 28 7 8 46 3 4 49 , 37 38 9 8 46 →= 44 45 46 , 37 38 9 8 48 →= 44 45 48 , 37 38 9 10 50 →= 44 51 50 , 5 4 9 8 46 →= 9 8 46 , 5 4 9 8 48 →= 9 8 48 , 3 4 9 8 46 →= 7 8 46 , 3 4 9 8 48 →= 7 8 48 , 16 4 9 8 46 →= 23 8 46 , 16 4 9 8 48 →= 23 8 48 , 52 4 9 8 46 →= 53 8 46 , 52 4 9 8 48 →= 53 8 48 , 31 4 9 8 46 →= 32 8 46 , 31 4 9 8 48 →= 32 8 48 , 5 4 9 10 50 →= 9 10 50 , 3 4 9 10 50 →= 7 10 50 , 16 4 9 10 50 →= 23 10 50 , 52 4 9 10 50 →= 53 10 50 , 31 4 9 10 50 →= 32 10 50 , 20 21 9 8 46 →= 24 25 46 , 20 21 9 8 48 →= 24 25 48 , 20 21 9 10 50 →= 24 26 50 , 37 38 2 3 4 →= 54 5 4 5 4 , 37 38 2 3 6 →= 54 5 4 5 6 , 37 38 2 7 8 →= 54 5 4 9 8 , 37 38 2 7 10 →= 54 5 4 9 10 , 37 38 2 7 11 →= 54 5 4 9 11 , 37 38 2 12 12 →= 54 5 4 2 12 , 37 38 2 12 3 →= 54 5 4 2 3 , 37 38 2 12 7 →= 54 5 4 2 7 , 37 38 2 12 13 →= 54 5 4 2 13 , 5 4 2 3 4 →= 2 3 4 5 4 , 5 4 2 3 6 →= 2 3 4 5 6 , 3 4 2 3 4 →= 12 3 4 5 4 , 3 4 2 3 6 →= 12 3 4 5 6 , 16 4 2 3 4 →= 28 3 4 5 4 , 16 4 2 3 6 →= 28 3 4 5 6 , 52 4 2 3 4 →= 55 3 4 5 4 , 52 4 2 3 6 →= 55 3 4 5 6 , 31 4 2 3 4 →= 30 3 4 5 4 , 31 4 2 3 6 →= 30 3 4 5 6 , 5 4 2 7 8 →= 2 3 4 9 8 , 5 4 2 7 10 →= 2 3 4 9 10 , 5 4 2 7 11 →= 2 3 4 9 11 , 3 4 2 7 8 →= 12 3 4 9 8 , 3 4 2 7 10 →= 12 3 4 9 10 , 3 4 2 7 11 →= 12 3 4 9 11 , 16 4 2 7 8 →= 28 3 4 9 8 , 16 4 2 7 10 →= 28 3 4 9 10 , 16 4 2 7 11 →= 28 3 4 9 11 , 52 4 2 7 8 →= 55 3 4 9 8 , 52 4 2 7 10 →= 55 3 4 9 10 , 52 4 2 7 11 →= 55 3 4 9 11 , 31 4 2 7 8 →= 30 3 4 9 8 , 31 4 2 7 10 →= 30 3 4 9 10 , 31 4 2 7 11 →= 30 3 4 9 11 , 5 4 2 12 12 →= 2 3 4 2 12 , 5 4 2 12 3 →= 2 3 4 2 3 , 5 4 2 12 7 →= 2 3 4 2 7 , 5 4 2 12 13 →= 2 3 4 2 13 , 3 4 2 12 12 →= 12 3 4 2 12 , 3 4 2 12 3 →= 12 3 4 2 3 , 3 4 2 12 7 →= 12 3 4 2 7 , 3 4 2 12 13 →= 12 3 4 2 13 , 16 4 2 12 12 →= 28 3 4 2 12 , 16 4 2 12 3 →= 28 3 4 2 3 , 16 4 2 12 7 →= 28 3 4 2 7 , 16 4 2 12 13 →= 28 3 4 2 13 , 52 4 2 12 12 →= 55 3 4 2 12 , 52 4 2 12 3 →= 55 3 4 2 3 , 52 4 2 12 7 →= 55 3 4 2 7 , 52 4 2 12 13 →= 55 3 4 2 13 , 31 4 2 12 12 →= 30 3 4 2 12 , 31 4 2 12 3 →= 30 3 4 2 3 , 31 4 2 12 7 →= 30 3 4 2 7 , 31 4 2 12 13 →= 30 3 4 2 13 , 43 54 2 3 4 →= 47 16 4 5 4 , 43 54 2 3 6 →= 47 16 4 5 6 , 36 54 2 3 4 →= 15 16 4 5 4 , 36 54 2 3 6 →= 15 16 4 5 6 , 43 54 2 7 8 →= 47 16 4 9 8 , 43 54 2 7 10 →= 47 16 4 9 10 , 43 54 2 7 11 →= 47 16 4 9 11 , 36 54 2 7 8 →= 15 16 4 9 8 , 36 54 2 7 10 →= 15 16 4 9 10 , 36 54 2 7 11 →= 15 16 4 9 11 , 43 54 2 12 12 →= 47 16 4 2 12 , 43 54 2 12 3 →= 47 16 4 2 3 , 43 54 2 12 7 →= 47 16 4 2 7 , 43 54 2 12 13 →= 47 16 4 2 13 , 36 54 2 12 12 →= 15 16 4 2 12 , 36 54 2 12 3 →= 15 16 4 2 3 , 36 54 2 12 7 →= 15 16 4 2 7 , 36 54 2 12 13 →= 15 16 4 2 13 , 20 21 2 3 4 →= 29 31 4 5 4 , 20 21 2 3 6 →= 29 31 4 5 6 , 20 21 2 7 8 →= 29 31 4 9 8 , 20 21 2 7 10 →= 29 31 4 9 10 , 20 21 2 7 11 →= 29 31 4 9 11 , 20 21 2 12 12 →= 29 31 4 2 12 , 20 21 2 12 3 →= 29 31 4 2 3 , 20 21 2 12 7 →= 29 31 4 2 7 , 20 21 2 12 13 →= 29 31 4 2 13 , 56 47 16 4 →= 57 58 19 20 21 , 56 47 16 6 →= 57 58 19 20 22 , 56 47 23 8 →= 57 58 19 24 25 , 56 47 23 10 →= 57 58 19 24 26 , 56 47 23 11 →= 57 58 19 24 27 , 56 47 28 12 →= 57 58 19 29 30 , 56 47 28 3 →= 57 58 19 29 31 , 56 47 28 7 →= 57 58 19 29 32 , 56 47 28 13 →= 57 58 19 29 33 , 57 59 35 →= 56 43 37 38 2 , 57 58 39 →= 56 43 37 40 41 , 57 58 19 →= 56 43 37 40 42 , 60 42 20 21 →= 60 41 52 4 , 60 42 20 22 →= 60 41 52 6 , 40 42 20 21 →= 40 41 52 4 , 40 42 20 22 →= 40 41 52 6 , 60 42 24 25 →= 60 41 53 8 , 60 42 24 26 →= 60 41 53 10 , 60 42 24 27 →= 60 41 53 11 , 40 42 24 25 →= 40 41 53 8 , 40 42 24 26 →= 40 41 53 10 , 40 42 24 27 →= 40 41 53 11 , 60 42 29 30 →= 60 41 55 12 , 60 42 29 31 →= 60 41 55 3 , 60 42 29 32 →= 60 41 55 7 , 60 42 29 33 →= 60 41 55 13 , 40 42 29 30 →= 40 41 55 12 , 40 42 29 31 →= 40 41 55 3 , 40 42 29 32 →= 40 41 55 7 , 40 42 29 33 →= 40 41 55 13 , 58 19 20 21 →= 58 39 52 4 , 58 19 20 22 →= 58 39 52 6 , 18 19 20 21 →= 18 39 52 4 , 18 19 20 22 →= 18 39 52 6 , 61 19 20 21 →= 61 39 52 4 , 61 19 20 22 →= 61 39 52 6 , 58 19 24 25 →= 58 39 53 8 , 58 19 24 26 →= 58 39 53 10 , 58 19 24 27 →= 58 39 53 11 , 18 19 24 25 →= 18 39 53 8 , 18 19 24 26 →= 18 39 53 10 , 18 19 24 27 →= 18 39 53 11 , 61 19 24 25 →= 61 39 53 8 , 61 19 24 26 →= 61 39 53 10 , 61 19 24 27 →= 61 39 53 11 , 58 19 29 30 →= 58 39 55 12 , 58 19 29 31 →= 58 39 55 3 , 58 19 29 32 →= 58 39 55 7 , 58 19 29 33 →= 58 39 55 13 , 18 19 29 30 →= 18 39 55 12 , 18 19 29 31 →= 18 39 55 3 , 18 19 29 32 →= 18 39 55 7 , 18 19 29 33 →= 18 39 55 13 , 61 19 29 30 →= 61 39 55 12 , 61 19 29 31 →= 61 39 55 3 , 61 19 29 32 →= 61 39 55 7 , 61 19 29 33 →= 61 39 55 13 , 60 41 52 4 →= 62 2 3 4 , 60 41 52 6 →= 62 2 3 6 , 40 41 52 4 →= 38 2 3 4 , 40 41 52 6 →= 38 2 3 6 , 60 41 53 8 →= 62 2 7 8 , 60 41 53 10 →= 62 2 7 10 , 60 41 53 11 →= 62 2 7 11 , 40 41 53 8 →= 38 2 7 8 , 40 41 53 10 →= 38 2 7 10 , 40 41 53 11 →= 38 2 7 11 , 60 41 55 12 →= 62 2 12 12 , 60 41 55 3 →= 62 2 12 3 , 60 41 55 7 →= 62 2 12 7 , 60 41 55 13 →= 62 2 12 13 , 40 41 55 12 →= 38 2 12 12 , 40 41 55 3 →= 38 2 12 3 , 40 41 55 7 →= 38 2 12 7 , 40 41 55 13 →= 38 2 12 13 , 58 39 52 4 →= 59 35 3 4 , 58 39 52 6 →= 59 35 3 6 , 18 39 52 4 →= 34 35 3 4 , 18 39 52 6 →= 34 35 3 6 , 61 39 52 4 →= 63 35 3 4 , 61 39 52 6 →= 63 35 3 6 , 58 39 53 8 →= 59 35 7 8 , 58 39 53 10 →= 59 35 7 10 , 58 39 53 11 →= 59 35 7 11 , 18 39 53 8 →= 34 35 7 8 , 18 39 53 10 →= 34 35 7 10 , 18 39 53 11 →= 34 35 7 11 , 61 39 53 8 →= 63 35 7 8 , 61 39 53 10 →= 63 35 7 10 , 61 39 53 11 →= 63 35 7 11 , 58 39 55 12 →= 59 35 12 12 , 58 39 55 3 →= 59 35 12 3 , 58 39 55 7 →= 59 35 12 7 , 58 39 55 13 →= 59 35 12 13 , 18 39 55 12 →= 34 35 12 12 , 18 39 55 3 →= 34 35 12 3 , 18 39 55 7 →= 34 35 12 7 , 18 39 55 13 →= 34 35 12 13 , 61 39 55 12 →= 63 35 12 12 , 61 39 55 3 →= 63 35 12 3 , 61 39 55 7 →= 63 35 12 7 , 61 39 55 13 →= 63 35 12 13 , 42 20 21 →= 41 52 4 , 42 20 22 →= 41 52 6 , 19 20 21 →= 39 52 4 , 19 20 22 →= 39 52 6 , 42 24 25 →= 41 53 8 , 42 24 26 →= 41 53 10 , 42 24 27 →= 41 53 11 , 19 24 25 →= 39 53 8 , 19 24 26 →= 39 53 10 , 19 24 27 →= 39 53 11 , 42 29 30 →= 41 55 12 , 42 29 31 →= 41 55 3 , 42 29 32 →= 41 55 7 , 42 29 33 →= 41 55 13 , 19 29 30 →= 39 55 12 , 19 29 31 →= 39 55 3 , 19 29 32 →= 39 55 7 , 19 29 33 →= 39 55 13 } 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 ⎟ ⎝ ⎠ 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 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 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 ⎟ ⎝ ⎠ 59 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 60 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 61 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 62 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 63 ↦ ⎛ ⎞ ⎜ 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, 41 ↦ 41, 42 ↦ 42, 43 ↦ 43, 44 ↦ 44, 45 ↦ 45, 46 ↦ 46, 47 ↦ 47, 48 ↦ 48, 50 ↦ 49, 51 ↦ 50, 52 ↦ 51, 53 ↦ 52, 54 ↦ 53, 55 ↦ 54, 56 ↦ 55, 57 ↦ 56, 58 ↦ 57, 59 ↦ 58, 60 ↦ 59, 61 ↦ 60 }, it remains to prove termination of the 227-rule system { 0 1 2 3 4 ⟶ 0 1 5 4 , 0 1 2 3 6 ⟶ 0 1 5 6 , 0 1 2 7 8 ⟶ 0 1 9 8 , 0 1 2 7 10 ⟶ 0 1 9 10 , 0 1 2 7 11 ⟶ 0 1 9 11 , 0 1 2 12 12 ⟶ 0 1 2 12 , 0 1 2 12 3 ⟶ 0 1 2 3 , 0 1 2 12 7 ⟶ 0 1 2 7 , 0 1 2 12 13 ⟶ 0 1 2 13 , 14 15 16 4 ⟶ 17 18 19 20 21 , 14 15 16 6 ⟶ 17 18 19 20 22 , 14 15 23 8 ⟶ 17 18 19 24 25 , 14 15 23 10 ⟶ 17 18 19 24 26 , 14 15 23 11 ⟶ 17 18 19 24 27 , 14 15 28 12 ⟶ 17 18 19 29 30 , 14 15 28 3 ⟶ 17 18 19 29 31 , 14 15 28 7 ⟶ 17 18 19 29 32 , 14 15 28 13 ⟶ 17 18 19 29 33 , 17 34 35 ⟶ 14 36 37 38 2 , 17 18 39 ⟶ 14 36 37 40 41 , 17 18 19 ⟶ 14 36 37 40 42 , 43 44 45 46 →= 47 28 7 8 46 3 4 2 , 36 44 45 46 →= 15 28 7 8 46 3 4 2 , 37 38 9 8 46 →= 44 45 46 , 37 38 9 8 48 →= 44 45 48 , 37 38 9 10 49 →= 44 50 49 , 5 4 9 8 46 →= 9 8 46 , 5 4 9 8 48 →= 9 8 48 , 3 4 9 8 46 →= 7 8 46 , 3 4 9 8 48 →= 7 8 48 , 16 4 9 8 46 →= 23 8 46 , 16 4 9 8 48 →= 23 8 48 , 51 4 9 8 46 →= 52 8 46 , 51 4 9 8 48 →= 52 8 48 , 31 4 9 8 46 →= 32 8 46 , 31 4 9 8 48 →= 32 8 48 , 5 4 9 10 49 →= 9 10 49 , 3 4 9 10 49 →= 7 10 49 , 16 4 9 10 49 →= 23 10 49 , 51 4 9 10 49 →= 52 10 49 , 31 4 9 10 49 →= 32 10 49 , 20 21 9 8 46 →= 24 25 46 , 20 21 9 8 48 →= 24 25 48 , 20 21 9 10 49 →= 24 26 49 , 37 38 2 3 4 →= 53 5 4 5 4 , 37 38 2 3 6 →= 53 5 4 5 6 , 37 38 2 7 8 →= 53 5 4 9 8 , 37 38 2 7 10 →= 53 5 4 9 10 , 37 38 2 7 11 →= 53 5 4 9 11 , 37 38 2 12 12 →= 53 5 4 2 12 , 37 38 2 12 3 →= 53 5 4 2 3 , 37 38 2 12 7 →= 53 5 4 2 7 , 37 38 2 12 13 →= 53 5 4 2 13 , 5 4 2 3 4 →= 2 3 4 5 4 , 5 4 2 3 6 →= 2 3 4 5 6 , 3 4 2 3 4 →= 12 3 4 5 4 , 3 4 2 3 6 →= 12 3 4 5 6 , 16 4 2 3 4 →= 28 3 4 5 4 , 16 4 2 3 6 →= 28 3 4 5 6 , 51 4 2 3 4 →= 54 3 4 5 4 , 51 4 2 3 6 →= 54 3 4 5 6 , 31 4 2 3 4 →= 30 3 4 5 4 , 31 4 2 3 6 →= 30 3 4 5 6 , 5 4 2 7 8 →= 2 3 4 9 8 , 5 4 2 7 10 →= 2 3 4 9 10 , 5 4 2 7 11 →= 2 3 4 9 11 , 3 4 2 7 8 →= 12 3 4 9 8 , 3 4 2 7 10 →= 12 3 4 9 10 , 3 4 2 7 11 →= 12 3 4 9 11 , 16 4 2 7 8 →= 28 3 4 9 8 , 16 4 2 7 10 →= 28 3 4 9 10 , 16 4 2 7 11 →= 28 3 4 9 11 , 51 4 2 7 8 →= 54 3 4 9 8 , 51 4 2 7 10 →= 54 3 4 9 10 , 51 4 2 7 11 →= 54 3 4 9 11 , 31 4 2 7 8 →= 30 3 4 9 8 , 31 4 2 7 10 →= 30 3 4 9 10 , 31 4 2 7 11 →= 30 3 4 9 11 , 5 4 2 12 12 →= 2 3 4 2 12 , 5 4 2 12 3 →= 2 3 4 2 3 , 5 4 2 12 7 →= 2 3 4 2 7 , 5 4 2 12 13 →= 2 3 4 2 13 , 3 4 2 12 12 →= 12 3 4 2 12 , 3 4 2 12 3 →= 12 3 4 2 3 , 3 4 2 12 7 →= 12 3 4 2 7 , 3 4 2 12 13 →= 12 3 4 2 13 , 16 4 2 12 12 →= 28 3 4 2 12 , 16 4 2 12 3 →= 28 3 4 2 3 , 16 4 2 12 7 →= 28 3 4 2 7 , 16 4 2 12 13 →= 28 3 4 2 13 , 51 4 2 12 12 →= 54 3 4 2 12 , 51 4 2 12 3 →= 54 3 4 2 3 , 51 4 2 12 7 →= 54 3 4 2 7 , 51 4 2 12 13 →= 54 3 4 2 13 , 31 4 2 12 12 →= 30 3 4 2 12 , 31 4 2 12 3 →= 30 3 4 2 3 , 31 4 2 12 7 →= 30 3 4 2 7 , 31 4 2 12 13 →= 30 3 4 2 13 , 43 53 2 3 4 →= 47 16 4 5 4 , 43 53 2 3 6 →= 47 16 4 5 6 , 36 53 2 3 4 →= 15 16 4 5 4 , 36 53 2 3 6 →= 15 16 4 5 6 , 43 53 2 7 8 →= 47 16 4 9 8 , 43 53 2 7 10 →= 47 16 4 9 10 , 43 53 2 7 11 →= 47 16 4 9 11 , 36 53 2 7 8 →= 15 16 4 9 8 , 36 53 2 7 10 →= 15 16 4 9 10 , 36 53 2 7 11 →= 15 16 4 9 11 , 43 53 2 12 12 →= 47 16 4 2 12 , 43 53 2 12 3 →= 47 16 4 2 3 , 43 53 2 12 7 →= 47 16 4 2 7 , 43 53 2 12 13 →= 47 16 4 2 13 , 36 53 2 12 12 →= 15 16 4 2 12 , 36 53 2 12 3 →= 15 16 4 2 3 , 36 53 2 12 7 →= 15 16 4 2 7 , 36 53 2 12 13 →= 15 16 4 2 13 , 20 21 2 3 4 →= 29 31 4 5 4 , 20 21 2 3 6 →= 29 31 4 5 6 , 20 21 2 7 8 →= 29 31 4 9 8 , 20 21 2 7 10 →= 29 31 4 9 10 , 20 21 2 7 11 →= 29 31 4 9 11 , 20 21 2 12 12 →= 29 31 4 2 12 , 20 21 2 12 3 →= 29 31 4 2 3 , 20 21 2 12 7 →= 29 31 4 2 7 , 20 21 2 12 13 →= 29 31 4 2 13 , 55 47 16 4 →= 56 57 19 20 21 , 55 47 16 6 →= 56 57 19 20 22 , 55 47 23 8 →= 56 57 19 24 25 , 55 47 23 10 →= 56 57 19 24 26 , 55 47 23 11 →= 56 57 19 24 27 , 55 47 28 12 →= 56 57 19 29 30 , 55 47 28 3 →= 56 57 19 29 31 , 55 47 28 7 →= 56 57 19 29 32 , 55 47 28 13 →= 56 57 19 29 33 , 56 58 35 →= 55 43 37 38 2 , 56 57 39 →= 55 43 37 40 41 , 56 57 19 →= 55 43 37 40 42 , 59 42 20 21 →= 59 41 51 4 , 59 42 20 22 →= 59 41 51 6 , 40 42 20 21 →= 40 41 51 4 , 40 42 20 22 →= 40 41 51 6 , 59 42 24 25 →= 59 41 52 8 , 59 42 24 26 →= 59 41 52 10 , 59 42 24 27 →= 59 41 52 11 , 40 42 24 25 →= 40 41 52 8 , 40 42 24 26 →= 40 41 52 10 , 40 42 24 27 →= 40 41 52 11 , 59 42 29 30 →= 59 41 54 12 , 59 42 29 31 →= 59 41 54 3 , 59 42 29 32 →= 59 41 54 7 , 59 42 29 33 →= 59 41 54 13 , 40 42 29 30 →= 40 41 54 12 , 40 42 29 31 →= 40 41 54 3 , 40 42 29 32 →= 40 41 54 7 , 40 42 29 33 →= 40 41 54 13 , 57 19 20 21 →= 57 39 51 4 , 57 19 20 22 →= 57 39 51 6 , 18 19 20 21 →= 18 39 51 4 , 18 19 20 22 →= 18 39 51 6 , 60 19 20 21 →= 60 39 51 4 , 60 19 20 22 →= 60 39 51 6 , 57 19 24 25 →= 57 39 52 8 , 57 19 24 26 →= 57 39 52 10 , 57 19 24 27 →= 57 39 52 11 , 18 19 24 25 →= 18 39 52 8 , 18 19 24 26 →= 18 39 52 10 , 18 19 24 27 →= 18 39 52 11 , 60 19 24 25 →= 60 39 52 8 , 60 19 24 26 →= 60 39 52 10 , 60 19 24 27 →= 60 39 52 11 , 57 19 29 30 →= 57 39 54 12 , 57 19 29 31 →= 57 39 54 3 , 57 19 29 32 →= 57 39 54 7 , 57 19 29 33 →= 57 39 54 13 , 18 19 29 30 →= 18 39 54 12 , 18 19 29 31 →= 18 39 54 3 , 18 19 29 32 →= 18 39 54 7 , 18 19 29 33 →= 18 39 54 13 , 60 19 29 30 →= 60 39 54 12 , 60 19 29 31 →= 60 39 54 3 , 60 19 29 32 →= 60 39 54 7 , 60 19 29 33 →= 60 39 54 13 , 40 41 51 4 →= 38 2 3 4 , 40 41 51 6 →= 38 2 3 6 , 40 41 52 8 →= 38 2 7 8 , 40 41 52 10 →= 38 2 7 10 , 40 41 52 11 →= 38 2 7 11 , 40 41 54 12 →= 38 2 12 12 , 40 41 54 3 →= 38 2 12 3 , 40 41 54 7 →= 38 2 12 7 , 40 41 54 13 →= 38 2 12 13 , 57 39 51 4 →= 58 35 3 4 , 57 39 51 6 →= 58 35 3 6 , 18 39 51 4 →= 34 35 3 4 , 18 39 51 6 →= 34 35 3 6 , 57 39 52 8 →= 58 35 7 8 , 57 39 52 10 →= 58 35 7 10 , 57 39 52 11 →= 58 35 7 11 , 18 39 52 8 →= 34 35 7 8 , 18 39 52 10 →= 34 35 7 10 , 18 39 52 11 →= 34 35 7 11 , 57 39 54 12 →= 58 35 12 12 , 57 39 54 3 →= 58 35 12 3 , 57 39 54 7 →= 58 35 12 7 , 57 39 54 13 →= 58 35 12 13 , 18 39 54 12 →= 34 35 12 12 , 18 39 54 3 →= 34 35 12 3 , 18 39 54 7 →= 34 35 12 7 , 18 39 54 13 →= 34 35 12 13 , 42 20 21 →= 41 51 4 , 42 20 22 →= 41 51 6 , 19 20 21 →= 39 51 4 , 19 20 22 →= 39 51 6 , 42 24 25 →= 41 52 8 , 42 24 26 →= 41 52 10 , 42 24 27 →= 41 52 11 , 19 24 25 →= 39 52 8 , 19 24 26 →= 39 52 10 , 19 24 27 →= 39 52 11 , 42 29 30 →= 41 54 12 , 42 29 31 →= 41 54 3 , 42 29 32 →= 41 54 7 , 42 29 33 →= 41 54 13 , 19 29 30 →= 39 54 12 , 19 29 31 →= 39 54 3 , 19 29 32 →= 39 54 7 , 19 29 33 →= 39 54 13 } 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, 15 ↦ 15, 16 ↦ 16, 17 ↦ 17, 18 ↦ 18, 19 ↦ 19, 20 ↦ 20, 21 ↦ 21, 23 ↦ 22, 24 ↦ 23, 25 ↦ 24, 26 ↦ 25, 28 ↦ 26, 29 ↦ 27, 30 ↦ 28, 31 ↦ 29, 32 ↦ 30, 34 ↦ 31, 35 ↦ 32, 36 ↦ 33, 37 ↦ 34, 38 ↦ 35, 39 ↦ 36, 40 ↦ 37, 41 ↦ 38, 42 ↦ 39, 46 ↦ 40, 48 ↦ 41, 51 ↦ 42, 52 ↦ 43, 49 ↦ 44, 53 ↦ 45, 54 ↦ 46, 43 ↦ 47, 47 ↦ 48, 55 ↦ 49, 56 ↦ 50, 57 ↦ 51, 58 ↦ 52, 22 ↦ 53, 27 ↦ 54, 33 ↦ 55 }, it remains to prove termination of the 198-rule system { 0 1 2 3 4 ⟶ 0 1 5 4 , 0 1 2 3 6 ⟶ 0 1 5 6 , 0 1 2 7 8 ⟶ 0 1 9 8 , 0 1 2 7 10 ⟶ 0 1 9 10 , 0 1 2 7 11 ⟶ 0 1 9 11 , 0 1 2 12 12 ⟶ 0 1 2 12 , 0 1 2 12 3 ⟶ 0 1 2 3 , 0 1 2 12 7 ⟶ 0 1 2 7 , 0 1 2 12 13 ⟶ 0 1 2 13 , 14 15 16 4 ⟶ 17 18 19 20 21 , 14 15 22 8 ⟶ 17 18 19 23 24 , 14 15 22 10 ⟶ 17 18 19 23 25 , 14 15 26 12 ⟶ 17 18 19 27 28 , 14 15 26 3 ⟶ 17 18 19 27 29 , 14 15 26 7 ⟶ 17 18 19 27 30 , 17 31 32 ⟶ 14 33 34 35 2 , 17 18 36 ⟶ 14 33 34 37 38 , 17 18 19 ⟶ 14 33 34 37 39 , 5 4 9 8 40 →= 9 8 40 , 5 4 9 8 41 →= 9 8 41 , 3 4 9 8 40 →= 7 8 40 , 3 4 9 8 41 →= 7 8 41 , 16 4 9 8 40 →= 22 8 40 , 16 4 9 8 41 →= 22 8 41 , 42 4 9 8 40 →= 43 8 40 , 42 4 9 8 41 →= 43 8 41 , 29 4 9 8 40 →= 30 8 40 , 29 4 9 8 41 →= 30 8 41 , 5 4 9 10 44 →= 9 10 44 , 3 4 9 10 44 →= 7 10 44 , 16 4 9 10 44 →= 22 10 44 , 42 4 9 10 44 →= 43 10 44 , 29 4 9 10 44 →= 30 10 44 , 20 21 9 8 40 →= 23 24 40 , 20 21 9 8 41 →= 23 24 41 , 20 21 9 10 44 →= 23 25 44 , 34 35 2 3 4 →= 45 5 4 5 4 , 34 35 2 3 6 →= 45 5 4 5 6 , 34 35 2 7 8 →= 45 5 4 9 8 , 34 35 2 7 10 →= 45 5 4 9 10 , 34 35 2 7 11 →= 45 5 4 9 11 , 34 35 2 12 12 →= 45 5 4 2 12 , 34 35 2 12 3 →= 45 5 4 2 3 , 34 35 2 12 7 →= 45 5 4 2 7 , 34 35 2 12 13 →= 45 5 4 2 13 , 5 4 2 3 4 →= 2 3 4 5 4 , 5 4 2 3 6 →= 2 3 4 5 6 , 3 4 2 3 4 →= 12 3 4 5 4 , 3 4 2 3 6 →= 12 3 4 5 6 , 16 4 2 3 4 →= 26 3 4 5 4 , 16 4 2 3 6 →= 26 3 4 5 6 , 42 4 2 3 4 →= 46 3 4 5 4 , 42 4 2 3 6 →= 46 3 4 5 6 , 29 4 2 3 4 →= 28 3 4 5 4 , 29 4 2 3 6 →= 28 3 4 5 6 , 5 4 2 7 8 →= 2 3 4 9 8 , 5 4 2 7 10 →= 2 3 4 9 10 , 5 4 2 7 11 →= 2 3 4 9 11 , 3 4 2 7 8 →= 12 3 4 9 8 , 3 4 2 7 10 →= 12 3 4 9 10 , 3 4 2 7 11 →= 12 3 4 9 11 , 16 4 2 7 8 →= 26 3 4 9 8 , 16 4 2 7 10 →= 26 3 4 9 10 , 16 4 2 7 11 →= 26 3 4 9 11 , 42 4 2 7 8 →= 46 3 4 9 8 , 42 4 2 7 10 →= 46 3 4 9 10 , 42 4 2 7 11 →= 46 3 4 9 11 , 29 4 2 7 8 →= 28 3 4 9 8 , 29 4 2 7 10 →= 28 3 4 9 10 , 29 4 2 7 11 →= 28 3 4 9 11 , 5 4 2 12 12 →= 2 3 4 2 12 , 5 4 2 12 3 →= 2 3 4 2 3 , 5 4 2 12 7 →= 2 3 4 2 7 , 5 4 2 12 13 →= 2 3 4 2 13 , 3 4 2 12 12 →= 12 3 4 2 12 , 3 4 2 12 3 →= 12 3 4 2 3 , 3 4 2 12 7 →= 12 3 4 2 7 , 3 4 2 12 13 →= 12 3 4 2 13 , 16 4 2 12 12 →= 26 3 4 2 12 , 16 4 2 12 3 →= 26 3 4 2 3 , 16 4 2 12 7 →= 26 3 4 2 7 , 16 4 2 12 13 →= 26 3 4 2 13 , 42 4 2 12 12 →= 46 3 4 2 12 , 42 4 2 12 3 →= 46 3 4 2 3 , 42 4 2 12 7 →= 46 3 4 2 7 , 42 4 2 12 13 →= 46 3 4 2 13 , 29 4 2 12 12 →= 28 3 4 2 12 , 29 4 2 12 3 →= 28 3 4 2 3 , 29 4 2 12 7 →= 28 3 4 2 7 , 29 4 2 12 13 →= 28 3 4 2 13 , 47 45 2 3 4 →= 48 16 4 5 4 , 47 45 2 3 6 →= 48 16 4 5 6 , 33 45 2 3 4 →= 15 16 4 5 4 , 33 45 2 3 6 →= 15 16 4 5 6 , 47 45 2 7 8 →= 48 16 4 9 8 , 47 45 2 7 10 →= 48 16 4 9 10 , 47 45 2 7 11 →= 48 16 4 9 11 , 33 45 2 7 8 →= 15 16 4 9 8 , 33 45 2 7 10 →= 15 16 4 9 10 , 33 45 2 7 11 →= 15 16 4 9 11 , 47 45 2 12 12 →= 48 16 4 2 12 , 47 45 2 12 3 →= 48 16 4 2 3 , 47 45 2 12 7 →= 48 16 4 2 7 , 47 45 2 12 13 →= 48 16 4 2 13 , 33 45 2 12 12 →= 15 16 4 2 12 , 33 45 2 12 3 →= 15 16 4 2 3 , 33 45 2 12 7 →= 15 16 4 2 7 , 33 45 2 12 13 →= 15 16 4 2 13 , 20 21 2 3 4 →= 27 29 4 5 4 , 20 21 2 3 6 →= 27 29 4 5 6 , 20 21 2 7 8 →= 27 29 4 9 8 , 20 21 2 7 10 →= 27 29 4 9 10 , 20 21 2 7 11 →= 27 29 4 9 11 , 20 21 2 12 12 →= 27 29 4 2 12 , 20 21 2 12 3 →= 27 29 4 2 3 , 20 21 2 12 7 →= 27 29 4 2 7 , 20 21 2 12 13 →= 27 29 4 2 13 , 49 48 16 4 →= 50 51 19 20 21 , 49 48 22 8 →= 50 51 19 23 24 , 49 48 22 10 →= 50 51 19 23 25 , 49 48 26 12 →= 50 51 19 27 28 , 49 48 26 3 →= 50 51 19 27 29 , 49 48 26 7 →= 50 51 19 27 30 , 50 52 32 →= 49 47 34 35 2 , 50 51 36 →= 49 47 34 37 38 , 50 51 19 →= 49 47 34 37 39 , 37 39 20 21 →= 37 38 42 4 , 37 39 20 53 →= 37 38 42 6 , 37 39 23 24 →= 37 38 43 8 , 37 39 23 25 →= 37 38 43 10 , 37 39 23 54 →= 37 38 43 11 , 37 39 27 28 →= 37 38 46 12 , 37 39 27 29 →= 37 38 46 3 , 37 39 27 30 →= 37 38 46 7 , 37 39 27 55 →= 37 38 46 13 , 51 19 20 21 →= 51 36 42 4 , 51 19 20 53 →= 51 36 42 6 , 18 19 20 21 →= 18 36 42 4 , 18 19 20 53 →= 18 36 42 6 , 51 19 23 24 →= 51 36 43 8 , 51 19 23 25 →= 51 36 43 10 , 51 19 23 54 →= 51 36 43 11 , 18 19 23 24 →= 18 36 43 8 , 18 19 23 25 →= 18 36 43 10 , 18 19 23 54 →= 18 36 43 11 , 51 19 27 28 →= 51 36 46 12 , 51 19 27 29 →= 51 36 46 3 , 51 19 27 30 →= 51 36 46 7 , 51 19 27 55 →= 51 36 46 13 , 18 19 27 28 →= 18 36 46 12 , 18 19 27 29 →= 18 36 46 3 , 18 19 27 30 →= 18 36 46 7 , 18 19 27 55 →= 18 36 46 13 , 37 38 42 4 →= 35 2 3 4 , 37 38 42 6 →= 35 2 3 6 , 37 38 43 8 →= 35 2 7 8 , 37 38 43 10 →= 35 2 7 10 , 37 38 43 11 →= 35 2 7 11 , 37 38 46 12 →= 35 2 12 12 , 37 38 46 3 →= 35 2 12 3 , 37 38 46 7 →= 35 2 12 7 , 37 38 46 13 →= 35 2 12 13 , 51 36 42 4 →= 52 32 3 4 , 51 36 42 6 →= 52 32 3 6 , 18 36 42 4 →= 31 32 3 4 , 18 36 42 6 →= 31 32 3 6 , 51 36 43 8 →= 52 32 7 8 , 51 36 43 10 →= 52 32 7 10 , 51 36 43 11 →= 52 32 7 11 , 18 36 43 8 →= 31 32 7 8 , 18 36 43 10 →= 31 32 7 10 , 18 36 43 11 →= 31 32 7 11 , 51 36 46 12 →= 52 32 12 12 , 51 36 46 3 →= 52 32 12 3 , 51 36 46 7 →= 52 32 12 7 , 51 36 46 13 →= 52 32 12 13 , 18 36 46 12 →= 31 32 12 12 , 18 36 46 3 →= 31 32 12 3 , 18 36 46 7 →= 31 32 12 7 , 18 36 46 13 →= 31 32 12 13 , 39 20 21 →= 38 42 4 , 39 20 53 →= 38 42 6 , 19 20 21 →= 36 42 4 , 19 20 53 →= 36 42 6 , 39 23 24 →= 38 43 8 , 39 23 25 →= 38 43 10 , 39 23 54 →= 38 43 11 , 19 23 24 →= 36 43 8 , 19 23 25 →= 36 43 10 , 19 23 54 →= 36 43 11 , 39 27 28 →= 38 46 12 , 39 27 29 →= 38 46 3 , 39 27 30 →= 38 46 7 , 39 27 55 →= 38 46 13 , 19 27 28 →= 36 46 12 , 19 27 29 →= 36 46 3 , 19 27 30 →= 36 46 7 , 19 27 55 →= 36 46 13 } 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 ⎟ ⎝ ⎠ 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 38 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 39 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 40 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 41 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 42 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 43 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 44 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 54 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 55 ↦ ⎛ ⎞ ⎜ 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, 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, 41 ↦ 41, 42 ↦ 42, 43 ↦ 43, 44 ↦ 44, 45 ↦ 45, 46 ↦ 46, 47 ↦ 47, 48 ↦ 48, 49 ↦ 49, 50 ↦ 50, 51 ↦ 51, 52 ↦ 52 }, it remains to prove termination of the 183-rule system { 0 1 2 3 4 ⟶ 0 1 5 4 , 0 1 2 3 6 ⟶ 0 1 5 6 , 0 1 2 7 8 ⟶ 0 1 9 8 , 0 1 2 7 10 ⟶ 0 1 9 10 , 0 1 2 7 11 ⟶ 0 1 9 11 , 0 1 2 12 12 ⟶ 0 1 2 12 , 0 1 2 12 3 ⟶ 0 1 2 3 , 0 1 2 12 7 ⟶ 0 1 2 7 , 0 1 2 12 13 ⟶ 0 1 2 13 , 14 15 16 4 ⟶ 17 18 19 20 21 , 14 15 22 8 ⟶ 17 18 19 23 24 , 14 15 22 10 ⟶ 17 18 19 23 25 , 14 15 26 12 ⟶ 17 18 19 27 28 , 14 15 26 3 ⟶ 17 18 19 27 29 , 14 15 26 7 ⟶ 17 18 19 27 30 , 17 31 32 ⟶ 14 33 34 35 2 , 17 18 36 ⟶ 14 33 34 37 38 , 17 18 19 ⟶ 14 33 34 37 39 , 5 4 9 8 40 →= 9 8 40 , 5 4 9 8 41 →= 9 8 41 , 3 4 9 8 40 →= 7 8 40 , 3 4 9 8 41 →= 7 8 41 , 16 4 9 8 40 →= 22 8 40 , 16 4 9 8 41 →= 22 8 41 , 42 4 9 8 40 →= 43 8 40 , 42 4 9 8 41 →= 43 8 41 , 29 4 9 8 40 →= 30 8 40 , 29 4 9 8 41 →= 30 8 41 , 5 4 9 10 44 →= 9 10 44 , 3 4 9 10 44 →= 7 10 44 , 16 4 9 10 44 →= 22 10 44 , 42 4 9 10 44 →= 43 10 44 , 29 4 9 10 44 →= 30 10 44 , 20 21 9 8 40 →= 23 24 40 , 20 21 9 8 41 →= 23 24 41 , 20 21 9 10 44 →= 23 25 44 , 34 35 2 3 4 →= 45 5 4 5 4 , 34 35 2 3 6 →= 45 5 4 5 6 , 34 35 2 7 8 →= 45 5 4 9 8 , 34 35 2 7 10 →= 45 5 4 9 10 , 34 35 2 7 11 →= 45 5 4 9 11 , 34 35 2 12 12 →= 45 5 4 2 12 , 34 35 2 12 3 →= 45 5 4 2 3 , 34 35 2 12 7 →= 45 5 4 2 7 , 34 35 2 12 13 →= 45 5 4 2 13 , 5 4 2 3 4 →= 2 3 4 5 4 , 5 4 2 3 6 →= 2 3 4 5 6 , 3 4 2 3 4 →= 12 3 4 5 4 , 3 4 2 3 6 →= 12 3 4 5 6 , 16 4 2 3 4 →= 26 3 4 5 4 , 16 4 2 3 6 →= 26 3 4 5 6 , 42 4 2 3 4 →= 46 3 4 5 4 , 42 4 2 3 6 →= 46 3 4 5 6 , 29 4 2 3 4 →= 28 3 4 5 4 , 29 4 2 3 6 →= 28 3 4 5 6 , 5 4 2 7 8 →= 2 3 4 9 8 , 5 4 2 7 10 →= 2 3 4 9 10 , 5 4 2 7 11 →= 2 3 4 9 11 , 3 4 2 7 8 →= 12 3 4 9 8 , 3 4 2 7 10 →= 12 3 4 9 10 , 3 4 2 7 11 →= 12 3 4 9 11 , 16 4 2 7 8 →= 26 3 4 9 8 , 16 4 2 7 10 →= 26 3 4 9 10 , 16 4 2 7 11 →= 26 3 4 9 11 , 42 4 2 7 8 →= 46 3 4 9 8 , 42 4 2 7 10 →= 46 3 4 9 10 , 42 4 2 7 11 →= 46 3 4 9 11 , 29 4 2 7 8 →= 28 3 4 9 8 , 29 4 2 7 10 →= 28 3 4 9 10 , 29 4 2 7 11 →= 28 3 4 9 11 , 5 4 2 12 12 →= 2 3 4 2 12 , 5 4 2 12 3 →= 2 3 4 2 3 , 5 4 2 12 7 →= 2 3 4 2 7 , 5 4 2 12 13 →= 2 3 4 2 13 , 3 4 2 12 12 →= 12 3 4 2 12 , 3 4 2 12 3 →= 12 3 4 2 3 , 3 4 2 12 7 →= 12 3 4 2 7 , 3 4 2 12 13 →= 12 3 4 2 13 , 16 4 2 12 12 →= 26 3 4 2 12 , 16 4 2 12 3 →= 26 3 4 2 3 , 16 4 2 12 7 →= 26 3 4 2 7 , 16 4 2 12 13 →= 26 3 4 2 13 , 42 4 2 12 12 →= 46 3 4 2 12 , 42 4 2 12 3 →= 46 3 4 2 3 , 42 4 2 12 7 →= 46 3 4 2 7 , 42 4 2 12 13 →= 46 3 4 2 13 , 29 4 2 12 12 →= 28 3 4 2 12 , 29 4 2 12 3 →= 28 3 4 2 3 , 29 4 2 12 7 →= 28 3 4 2 7 , 29 4 2 12 13 →= 28 3 4 2 13 , 47 45 2 3 4 →= 48 16 4 5 4 , 47 45 2 3 6 →= 48 16 4 5 6 , 33 45 2 3 4 →= 15 16 4 5 4 , 33 45 2 3 6 →= 15 16 4 5 6 , 47 45 2 7 8 →= 48 16 4 9 8 , 47 45 2 7 10 →= 48 16 4 9 10 , 47 45 2 7 11 →= 48 16 4 9 11 , 33 45 2 7 8 →= 15 16 4 9 8 , 33 45 2 7 10 →= 15 16 4 9 10 , 33 45 2 7 11 →= 15 16 4 9 11 , 47 45 2 12 12 →= 48 16 4 2 12 , 47 45 2 12 3 →= 48 16 4 2 3 , 47 45 2 12 7 →= 48 16 4 2 7 , 47 45 2 12 13 →= 48 16 4 2 13 , 33 45 2 12 12 →= 15 16 4 2 12 , 33 45 2 12 3 →= 15 16 4 2 3 , 33 45 2 12 7 →= 15 16 4 2 7 , 33 45 2 12 13 →= 15 16 4 2 13 , 20 21 2 3 4 →= 27 29 4 5 4 , 20 21 2 3 6 →= 27 29 4 5 6 , 20 21 2 7 8 →= 27 29 4 9 8 , 20 21 2 7 10 →= 27 29 4 9 10 , 20 21 2 7 11 →= 27 29 4 9 11 , 20 21 2 12 12 →= 27 29 4 2 12 , 20 21 2 12 3 →= 27 29 4 2 3 , 20 21 2 12 7 →= 27 29 4 2 7 , 20 21 2 12 13 →= 27 29 4 2 13 , 49 48 16 4 →= 50 51 19 20 21 , 49 48 22 8 →= 50 51 19 23 24 , 49 48 22 10 →= 50 51 19 23 25 , 49 48 26 12 →= 50 51 19 27 28 , 49 48 26 3 →= 50 51 19 27 29 , 49 48 26 7 →= 50 51 19 27 30 , 50 52 32 →= 49 47 34 35 2 , 50 51 36 →= 49 47 34 37 38 , 50 51 19 →= 49 47 34 37 39 , 37 39 20 21 →= 37 38 42 4 , 37 39 23 24 →= 37 38 43 8 , 37 39 23 25 →= 37 38 43 10 , 37 39 27 28 →= 37 38 46 12 , 37 39 27 29 →= 37 38 46 3 , 37 39 27 30 →= 37 38 46 7 , 51 19 20 21 →= 51 36 42 4 , 18 19 20 21 →= 18 36 42 4 , 51 19 23 24 →= 51 36 43 8 , 51 19 23 25 →= 51 36 43 10 , 18 19 23 24 →= 18 36 43 8 , 18 19 23 25 →= 18 36 43 10 , 51 19 27 28 →= 51 36 46 12 , 51 19 27 29 →= 51 36 46 3 , 51 19 27 30 →= 51 36 46 7 , 18 19 27 28 →= 18 36 46 12 , 18 19 27 29 →= 18 36 46 3 , 18 19 27 30 →= 18 36 46 7 , 37 38 42 4 →= 35 2 3 4 , 37 38 42 6 →= 35 2 3 6 , 37 38 43 8 →= 35 2 7 8 , 37 38 43 10 →= 35 2 7 10 , 37 38 43 11 →= 35 2 7 11 , 37 38 46 12 →= 35 2 12 12 , 37 38 46 3 →= 35 2 12 3 , 37 38 46 7 →= 35 2 12 7 , 37 38 46 13 →= 35 2 12 13 , 51 36 42 4 →= 52 32 3 4 , 51 36 42 6 →= 52 32 3 6 , 18 36 42 4 →= 31 32 3 4 , 18 36 42 6 →= 31 32 3 6 , 51 36 43 8 →= 52 32 7 8 , 51 36 43 10 →= 52 32 7 10 , 51 36 43 11 →= 52 32 7 11 , 18 36 43 8 →= 31 32 7 8 , 18 36 43 10 →= 31 32 7 10 , 18 36 43 11 →= 31 32 7 11 , 51 36 46 12 →= 52 32 12 12 , 51 36 46 3 →= 52 32 12 3 , 51 36 46 7 →= 52 32 12 7 , 51 36 46 13 →= 52 32 12 13 , 18 36 46 12 →= 31 32 12 12 , 18 36 46 3 →= 31 32 12 3 , 18 36 46 7 →= 31 32 12 7 , 18 36 46 13 →= 31 32 12 13 , 39 20 21 →= 38 42 4 , 19 20 21 →= 36 42 4 , 39 23 24 →= 38 43 8 , 39 23 25 →= 38 43 10 , 19 23 24 →= 36 43 8 , 19 23 25 →= 36 43 10 , 39 27 28 →= 38 46 12 , 39 27 29 →= 38 46 3 , 39 27 30 →= 38 46 7 , 19 27 28 →= 36 46 12 , 19 27 29 →= 36 46 3 , 19 27 30 →= 36 46 7 } 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, 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, 41 ↦ 41, 42 ↦ 42, 43 ↦ 43, 44 ↦ 44, 45 ↦ 45, 46 ↦ 46, 47 ↦ 47, 48 ↦ 48, 49 ↦ 49, 50 ↦ 50, 51 ↦ 51, 52 ↦ 52 }, it remains to prove termination of the 174-rule system { 0 1 2 3 4 ⟶ 0 1 5 4 , 0 1 2 3 6 ⟶ 0 1 5 6 , 0 1 2 7 8 ⟶ 0 1 9 8 , 0 1 2 7 10 ⟶ 0 1 9 10 , 0 1 2 7 11 ⟶ 0 1 9 11 , 0 1 2 12 12 ⟶ 0 1 2 12 , 0 1 2 12 3 ⟶ 0 1 2 3 , 0 1 2 12 7 ⟶ 0 1 2 7 , 0 1 2 12 13 ⟶ 0 1 2 13 , 14 15 16 4 ⟶ 17 18 19 20 21 , 14 15 22 8 ⟶ 17 18 19 23 24 , 14 15 22 10 ⟶ 17 18 19 23 25 , 14 15 26 12 ⟶ 17 18 19 27 28 , 14 15 26 3 ⟶ 17 18 19 27 29 , 14 15 26 7 ⟶ 17 18 19 27 30 , 17 31 32 ⟶ 14 33 34 35 2 , 17 18 36 ⟶ 14 33 34 37 38 , 17 18 19 ⟶ 14 33 34 37 39 , 5 4 9 8 40 →= 9 8 40 , 5 4 9 8 41 →= 9 8 41 , 3 4 9 8 40 →= 7 8 40 , 3 4 9 8 41 →= 7 8 41 , 16 4 9 8 40 →= 22 8 40 , 16 4 9 8 41 →= 22 8 41 , 42 4 9 8 40 →= 43 8 40 , 42 4 9 8 41 →= 43 8 41 , 29 4 9 8 40 →= 30 8 40 , 29 4 9 8 41 →= 30 8 41 , 5 4 9 10 44 →= 9 10 44 , 3 4 9 10 44 →= 7 10 44 , 16 4 9 10 44 →= 22 10 44 , 42 4 9 10 44 →= 43 10 44 , 29 4 9 10 44 →= 30 10 44 , 20 21 9 8 40 →= 23 24 40 , 20 21 9 8 41 →= 23 24 41 , 20 21 9 10 44 →= 23 25 44 , 34 35 2 3 4 →= 45 5 4 5 4 , 34 35 2 3 6 →= 45 5 4 5 6 , 34 35 2 7 8 →= 45 5 4 9 8 , 34 35 2 7 10 →= 45 5 4 9 10 , 34 35 2 7 11 →= 45 5 4 9 11 , 34 35 2 12 12 →= 45 5 4 2 12 , 34 35 2 12 3 →= 45 5 4 2 3 , 34 35 2 12 7 →= 45 5 4 2 7 , 34 35 2 12 13 →= 45 5 4 2 13 , 5 4 2 3 4 →= 2 3 4 5 4 , 5 4 2 3 6 →= 2 3 4 5 6 , 3 4 2 3 4 →= 12 3 4 5 4 , 3 4 2 3 6 →= 12 3 4 5 6 , 16 4 2 3 4 →= 26 3 4 5 4 , 16 4 2 3 6 →= 26 3 4 5 6 , 42 4 2 3 4 →= 46 3 4 5 4 , 42 4 2 3 6 →= 46 3 4 5 6 , 29 4 2 3 4 →= 28 3 4 5 4 , 29 4 2 3 6 →= 28 3 4 5 6 , 5 4 2 7 8 →= 2 3 4 9 8 , 5 4 2 7 10 →= 2 3 4 9 10 , 5 4 2 7 11 →= 2 3 4 9 11 , 3 4 2 7 8 →= 12 3 4 9 8 , 3 4 2 7 10 →= 12 3 4 9 10 , 3 4 2 7 11 →= 12 3 4 9 11 , 16 4 2 7 8 →= 26 3 4 9 8 , 16 4 2 7 10 →= 26 3 4 9 10 , 16 4 2 7 11 →= 26 3 4 9 11 , 42 4 2 7 8 →= 46 3 4 9 8 , 42 4 2 7 10 →= 46 3 4 9 10 , 42 4 2 7 11 →= 46 3 4 9 11 , 29 4 2 7 8 →= 28 3 4 9 8 , 29 4 2 7 10 →= 28 3 4 9 10 , 29 4 2 7 11 →= 28 3 4 9 11 , 5 4 2 12 12 →= 2 3 4 2 12 , 5 4 2 12 3 →= 2 3 4 2 3 , 5 4 2 12 7 →= 2 3 4 2 7 , 5 4 2 12 13 →= 2 3 4 2 13 , 3 4 2 12 12 →= 12 3 4 2 12 , 3 4 2 12 3 →= 12 3 4 2 3 , 3 4 2 12 7 →= 12 3 4 2 7 , 3 4 2 12 13 →= 12 3 4 2 13 , 16 4 2 12 12 →= 26 3 4 2 12 , 16 4 2 12 3 →= 26 3 4 2 3 , 16 4 2 12 7 →= 26 3 4 2 7 , 16 4 2 12 13 →= 26 3 4 2 13 , 42 4 2 12 12 →= 46 3 4 2 12 , 42 4 2 12 3 →= 46 3 4 2 3 , 42 4 2 12 7 →= 46 3 4 2 7 , 42 4 2 12 13 →= 46 3 4 2 13 , 29 4 2 12 12 →= 28 3 4 2 12 , 29 4 2 12 3 →= 28 3 4 2 3 , 29 4 2 12 7 →= 28 3 4 2 7 , 29 4 2 12 13 →= 28 3 4 2 13 , 47 45 2 3 4 →= 48 16 4 5 4 , 47 45 2 3 6 →= 48 16 4 5 6 , 33 45 2 3 4 →= 15 16 4 5 4 , 33 45 2 3 6 →= 15 16 4 5 6 , 47 45 2 7 8 →= 48 16 4 9 8 , 47 45 2 7 10 →= 48 16 4 9 10 , 47 45 2 7 11 →= 48 16 4 9 11 , 33 45 2 7 8 →= 15 16 4 9 8 , 33 45 2 7 10 →= 15 16 4 9 10 , 33 45 2 7 11 →= 15 16 4 9 11 , 47 45 2 12 12 →= 48 16 4 2 12 , 47 45 2 12 3 →= 48 16 4 2 3 , 47 45 2 12 7 →= 48 16 4 2 7 , 47 45 2 12 13 →= 48 16 4 2 13 , 33 45 2 12 12 →= 15 16 4 2 12 , 33 45 2 12 3 →= 15 16 4 2 3 , 33 45 2 12 7 →= 15 16 4 2 7 , 33 45 2 12 13 →= 15 16 4 2 13 , 20 21 2 3 4 →= 27 29 4 5 4 , 20 21 2 3 6 →= 27 29 4 5 6 , 20 21 2 7 8 →= 27 29 4 9 8 , 20 21 2 7 10 →= 27 29 4 9 10 , 20 21 2 7 11 →= 27 29 4 9 11 , 20 21 2 12 12 →= 27 29 4 2 12 , 20 21 2 12 3 →= 27 29 4 2 3 , 20 21 2 12 7 →= 27 29 4 2 7 , 20 21 2 12 13 →= 27 29 4 2 13 , 49 48 16 4 →= 50 51 19 20 21 , 49 48 22 8 →= 50 51 19 23 24 , 49 48 22 10 →= 50 51 19 23 25 , 49 48 26 12 →= 50 51 19 27 28 , 49 48 26 3 →= 50 51 19 27 29 , 49 48 26 7 →= 50 51 19 27 30 , 50 52 32 →= 49 47 34 35 2 , 50 51 36 →= 49 47 34 37 38 , 50 51 19 →= 49 47 34 37 39 , 37 39 20 21 →= 37 38 42 4 , 37 39 23 24 →= 37 38 43 8 , 37 39 23 25 →= 37 38 43 10 , 37 39 27 28 →= 37 38 46 12 , 37 39 27 29 →= 37 38 46 3 , 37 39 27 30 →= 37 38 46 7 , 51 19 20 21 →= 51 36 42 4 , 18 19 20 21 →= 18 36 42 4 , 51 19 23 24 →= 51 36 43 8 , 51 19 23 25 →= 51 36 43 10 , 18 19 23 24 →= 18 36 43 8 , 18 19 23 25 →= 18 36 43 10 , 51 19 27 28 →= 51 36 46 12 , 51 19 27 29 →= 51 36 46 3 , 51 19 27 30 →= 51 36 46 7 , 18 19 27 28 →= 18 36 46 12 , 18 19 27 29 →= 18 36 46 3 , 18 19 27 30 →= 18 36 46 7 , 37 38 42 4 →= 35 2 3 4 , 37 38 43 8 →= 35 2 7 8 , 37 38 43 10 →= 35 2 7 10 , 37 38 46 12 →= 35 2 12 12 , 37 38 46 3 →= 35 2 12 3 , 37 38 46 7 →= 35 2 12 7 , 51 36 42 4 →= 52 32 3 4 , 18 36 42 4 →= 31 32 3 4 , 51 36 43 8 →= 52 32 7 8 , 51 36 43 10 →= 52 32 7 10 , 18 36 43 8 →= 31 32 7 8 , 18 36 43 10 →= 31 32 7 10 , 51 36 46 12 →= 52 32 12 12 , 51 36 46 3 →= 52 32 12 3 , 51 36 46 7 →= 52 32 12 7 , 18 36 46 12 →= 31 32 12 12 , 18 36 46 3 →= 31 32 12 3 , 18 36 46 7 →= 31 32 12 7 , 39 20 21 →= 38 42 4 , 19 20 21 →= 36 42 4 , 39 23 24 →= 38 43 8 , 39 23 25 →= 38 43 10 , 19 23 24 →= 36 43 8 , 19 23 25 →= 36 43 10 , 39 27 28 →= 38 46 12 , 39 27 29 →= 38 46 3 , 39 27 30 →= 38 46 7 , 19 27 28 →= 36 46 12 , 19 27 29 →= 36 46 3 , 19 27 30 →= 36 46 7 } 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, 7 ↦ 6, 8 ↦ 7, 9 ↦ 8, 10 ↦ 9, 12 ↦ 10, 14 ↦ 11, 15 ↦ 12, 16 ↦ 13, 17 ↦ 14, 18 ↦ 15, 19 ↦ 16, 20 ↦ 17, 21 ↦ 18, 22 ↦ 19, 23 ↦ 20, 24 ↦ 21, 25 ↦ 22, 26 ↦ 23, 27 ↦ 24, 28 ↦ 25, 29 ↦ 26, 30 ↦ 27, 31 ↦ 28, 32 ↦ 29, 33 ↦ 30, 34 ↦ 31, 35 ↦ 32, 36 ↦ 33, 37 ↦ 34, 38 ↦ 35, 39 ↦ 36, 40 ↦ 37, 41 ↦ 38, 42 ↦ 39, 43 ↦ 40, 44 ↦ 41, 45 ↦ 42, 46 ↦ 43, 47 ↦ 44, 48 ↦ 45, 49 ↦ 46, 50 ↦ 47, 51 ↦ 48, 52 ↦ 49 }, it remains to prove termination of the 144-rule system { 0 1 2 3 4 ⟶ 0 1 5 4 , 0 1 2 6 7 ⟶ 0 1 8 7 , 0 1 2 6 9 ⟶ 0 1 8 9 , 0 1 2 10 10 ⟶ 0 1 2 10 , 0 1 2 10 3 ⟶ 0 1 2 3 , 0 1 2 10 6 ⟶ 0 1 2 6 , 11 12 13 4 ⟶ 14 15 16 17 18 , 11 12 19 7 ⟶ 14 15 16 20 21 , 11 12 19 9 ⟶ 14 15 16 20 22 , 11 12 23 10 ⟶ 14 15 16 24 25 , 11 12 23 3 ⟶ 14 15 16 24 26 , 11 12 23 6 ⟶ 14 15 16 24 27 , 14 28 29 ⟶ 11 30 31 32 2 , 14 15 33 ⟶ 11 30 31 34 35 , 14 15 16 ⟶ 11 30 31 34 36 , 5 4 8 7 37 →= 8 7 37 , 5 4 8 7 38 →= 8 7 38 , 3 4 8 7 37 →= 6 7 37 , 3 4 8 7 38 →= 6 7 38 , 13 4 8 7 37 →= 19 7 37 , 13 4 8 7 38 →= 19 7 38 , 39 4 8 7 37 →= 40 7 37 , 39 4 8 7 38 →= 40 7 38 , 26 4 8 7 37 →= 27 7 37 , 26 4 8 7 38 →= 27 7 38 , 5 4 8 9 41 →= 8 9 41 , 3 4 8 9 41 →= 6 9 41 , 13 4 8 9 41 →= 19 9 41 , 39 4 8 9 41 →= 40 9 41 , 26 4 8 9 41 →= 27 9 41 , 17 18 8 7 37 →= 20 21 37 , 17 18 8 7 38 →= 20 21 38 , 17 18 8 9 41 →= 20 22 41 , 31 32 2 3 4 →= 42 5 4 5 4 , 31 32 2 6 7 →= 42 5 4 8 7 , 31 32 2 6 9 →= 42 5 4 8 9 , 31 32 2 10 10 →= 42 5 4 2 10 , 31 32 2 10 3 →= 42 5 4 2 3 , 31 32 2 10 6 →= 42 5 4 2 6 , 5 4 2 3 4 →= 2 3 4 5 4 , 3 4 2 3 4 →= 10 3 4 5 4 , 13 4 2 3 4 →= 23 3 4 5 4 , 39 4 2 3 4 →= 43 3 4 5 4 , 26 4 2 3 4 →= 25 3 4 5 4 , 5 4 2 6 7 →= 2 3 4 8 7 , 5 4 2 6 9 →= 2 3 4 8 9 , 3 4 2 6 7 →= 10 3 4 8 7 , 3 4 2 6 9 →= 10 3 4 8 9 , 13 4 2 6 7 →= 23 3 4 8 7 , 13 4 2 6 9 →= 23 3 4 8 9 , 39 4 2 6 7 →= 43 3 4 8 7 , 39 4 2 6 9 →= 43 3 4 8 9 , 26 4 2 6 7 →= 25 3 4 8 7 , 26 4 2 6 9 →= 25 3 4 8 9 , 5 4 2 10 10 →= 2 3 4 2 10 , 5 4 2 10 3 →= 2 3 4 2 3 , 5 4 2 10 6 →= 2 3 4 2 6 , 3 4 2 10 10 →= 10 3 4 2 10 , 3 4 2 10 3 →= 10 3 4 2 3 , 3 4 2 10 6 →= 10 3 4 2 6 , 13 4 2 10 10 →= 23 3 4 2 10 , 13 4 2 10 3 →= 23 3 4 2 3 , 13 4 2 10 6 →= 23 3 4 2 6 , 39 4 2 10 10 →= 43 3 4 2 10 , 39 4 2 10 3 →= 43 3 4 2 3 , 39 4 2 10 6 →= 43 3 4 2 6 , 26 4 2 10 10 →= 25 3 4 2 10 , 26 4 2 10 3 →= 25 3 4 2 3 , 26 4 2 10 6 →= 25 3 4 2 6 , 44 42 2 3 4 →= 45 13 4 5 4 , 30 42 2 3 4 →= 12 13 4 5 4 , 44 42 2 6 7 →= 45 13 4 8 7 , 44 42 2 6 9 →= 45 13 4 8 9 , 30 42 2 6 7 →= 12 13 4 8 7 , 30 42 2 6 9 →= 12 13 4 8 9 , 44 42 2 10 10 →= 45 13 4 2 10 , 44 42 2 10 3 →= 45 13 4 2 3 , 44 42 2 10 6 →= 45 13 4 2 6 , 30 42 2 10 10 →= 12 13 4 2 10 , 30 42 2 10 3 →= 12 13 4 2 3 , 30 42 2 10 6 →= 12 13 4 2 6 , 17 18 2 3 4 →= 24 26 4 5 4 , 17 18 2 6 7 →= 24 26 4 8 7 , 17 18 2 6 9 →= 24 26 4 8 9 , 17 18 2 10 10 →= 24 26 4 2 10 , 17 18 2 10 3 →= 24 26 4 2 3 , 17 18 2 10 6 →= 24 26 4 2 6 , 46 45 13 4 →= 47 48 16 17 18 , 46 45 19 7 →= 47 48 16 20 21 , 46 45 19 9 →= 47 48 16 20 22 , 46 45 23 10 →= 47 48 16 24 25 , 46 45 23 3 →= 47 48 16 24 26 , 46 45 23 6 →= 47 48 16 24 27 , 47 49 29 →= 46 44 31 32 2 , 47 48 33 →= 46 44 31 34 35 , 47 48 16 →= 46 44 31 34 36 , 34 36 17 18 →= 34 35 39 4 , 34 36 20 21 →= 34 35 40 7 , 34 36 20 22 →= 34 35 40 9 , 34 36 24 25 →= 34 35 43 10 , 34 36 24 26 →= 34 35 43 3 , 34 36 24 27 →= 34 35 43 6 , 48 16 17 18 →= 48 33 39 4 , 15 16 17 18 →= 15 33 39 4 , 48 16 20 21 →= 48 33 40 7 , 48 16 20 22 →= 48 33 40 9 , 15 16 20 21 →= 15 33 40 7 , 15 16 20 22 →= 15 33 40 9 , 48 16 24 25 →= 48 33 43 10 , 48 16 24 26 →= 48 33 43 3 , 48 16 24 27 →= 48 33 43 6 , 15 16 24 25 →= 15 33 43 10 , 15 16 24 26 →= 15 33 43 3 , 15 16 24 27 →= 15 33 43 6 , 34 35 39 4 →= 32 2 3 4 , 34 35 40 7 →= 32 2 6 7 , 34 35 40 9 →= 32 2 6 9 , 34 35 43 10 →= 32 2 10 10 , 34 35 43 3 →= 32 2 10 3 , 34 35 43 6 →= 32 2 10 6 , 48 33 39 4 →= 49 29 3 4 , 15 33 39 4 →= 28 29 3 4 , 48 33 40 7 →= 49 29 6 7 , 48 33 40 9 →= 49 29 6 9 , 15 33 40 7 →= 28 29 6 7 , 15 33 40 9 →= 28 29 6 9 , 48 33 43 10 →= 49 29 10 10 , 48 33 43 3 →= 49 29 10 3 , 48 33 43 6 →= 49 29 10 6 , 15 33 43 10 →= 28 29 10 10 , 15 33 43 3 →= 28 29 10 3 , 15 33 43 6 →= 28 29 10 6 , 36 17 18 →= 35 39 4 , 16 17 18 →= 33 39 4 , 36 20 21 →= 35 40 7 , 36 20 22 →= 35 40 9 , 16 20 21 →= 33 40 7 , 16 20 22 →= 33 40 9 , 36 24 25 →= 35 43 10 , 36 24 26 →= 35 43 3 , 36 24 27 →= 35 43 6 , 16 24 25 →= 33 43 10 , 16 24 26 →= 33 43 3 , 16 24 27 →= 33 43 6 } 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 12 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 13 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 12 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 17 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 15 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 16 ↦ ⎛ ⎞ ⎜ 1 14 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 17 ↦ ⎛ ⎞ ⎜ 1 5 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 18 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 19 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 20 ↦ ⎛ ⎞ ⎜ 1 3 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 21 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 22 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 23 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 24 ↦ ⎛ ⎞ ⎜ 1 13 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 25 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 26 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 27 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 28 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 29 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 30 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 31 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 32 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 33 ↦ ⎛ ⎞ ⎜ 1 14 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 34 ↦ ⎛ ⎞ ⎜ 1 13 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 35 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 36 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 37 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 38 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 39 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 40 ↦ ⎛ ⎞ ⎜ 1 2 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 41 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 42 ↦ ⎛ ⎞ ⎜ 1 9 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 43 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 44 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 45 ↦ ⎛ ⎞ ⎜ 1 17 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 46 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 47 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 48 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 49 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 5 ↦ 0, 4 ↦ 1, 2 ↦ 2, 3 ↦ 3, 10 ↦ 4, 6 ↦ 5 }, it remains to prove termination of the 8-rule system { 0 1 2 3 1 →= 2 3 1 0 1 , 3 1 2 3 1 →= 4 3 1 0 1 , 0 1 2 4 4 →= 2 3 1 2 4 , 0 1 2 4 3 →= 2 3 1 2 3 , 0 1 2 4 5 →= 2 3 1 2 5 , 3 1 2 4 4 →= 4 3 1 2 4 , 3 1 2 4 3 →= 4 3 1 2 3 , 3 1 2 4 5 →= 4 3 1 2 5 } The system is trivially terminating.