/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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YES

After renaming modulo the bijection { b ↦ 0, c ↦ 1, d ↦ 2, g ↦ 3, f ↦ 4, a ↦ 5 },
it remains to prove termination of the 8-rule system
{ 0 0 ⟶ 1 2 ,
  1 1 ⟶ 2 2 2 ,
  1 ⟶ 3 ,
  2 2 ⟶ 1 4 ,
  2 2 2 ⟶ 3 1 ,
  4 ⟶ 5 3 ,
  3 ⟶ 2 5 0 ,
  3 3 ⟶ 0 1 }

The system was reversed.

After renaming modulo the bijection { 0 ↦ 0, 2 ↦ 1, 1 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5 },
it remains to prove termination of the 8-rule system
{ 0 0 ⟶ 1 2 ,
  2 2 ⟶ 1 1 1 ,
  2 ⟶ 3 ,
  1 1 ⟶ 4 2 ,
  1 1 1 ⟶ 2 3 ,
  4 ⟶ 3 5 ,
  3 ⟶ 0 5 1 ,
  3 3 ⟶ 2 0 }

Applying sparse tiling TRFC(2) [Geser/Hofbauer/Waldmann, FSCD 2019].

After renaming modulo the bijection { (0,0) ↦ 0, (0,1) ↦ 1, (1,2) ↦ 2, (2,0) ↦ 3, (2,1) ↦ 4, (0,2) ↦ 5, (2,2) ↦ 6, (0,3) ↦ 7, (2,3) ↦ 8, (0,4) ↦ 9, (2,4) ↦ 10, (0,5) ↦ 11, (2,5) ↦ 12, (0,7) ↦ 13, (2,7) ↦ 14, (1,0) ↦ 15, (1,1) ↦ 16, (3,0) ↦ 17, (3,1) ↦ 18, (4,0) ↦ 19, (4,1) ↦ 20, (5,0) ↦ 21, (5,1) ↦ 22, (6,0) ↦ 23, (6,1) ↦ 24, (1,3) ↦ 25, (1,4) ↦ 26, (1,5) ↦ 27, (1,7) ↦ 28, (3,2) ↦ 29, (4,2) ↦ 30, (5,2) ↦ 31, (6,2) ↦ 32, (3,3) ↦ 33, (3,4) ↦ 34, (3,5) ↦ 35, (3,7) ↦ 36, (4,3) ↦ 37, (5,3) ↦ 38, (6,3) ↦ 39, (4,4) ↦ 40, (5,4) ↦ 41, (6,4) ↦ 42 },
it remains to prove termination of the 378-rule system
{ 0 0 0 ⟶ 1 2 3 ,
  0 0 1 ⟶ 1 2 4 ,
  0 0 5 ⟶ 1 2 6 ,
  0 0 7 ⟶ 1 2 8 ,
  0 0 9 ⟶ 1 2 10 ,
  0 0 11 ⟶ 1 2 12 ,
  0 0 13 ⟶ 1 2 14 ,
  15 0 0 ⟶ 16 2 3 ,
  15 0 1 ⟶ 16 2 4 ,
  15 0 5 ⟶ 16 2 6 ,
  15 0 7 ⟶ 16 2 8 ,
  15 0 9 ⟶ 16 2 10 ,
  15 0 11 ⟶ 16 2 12 ,
  15 0 13 ⟶ 16 2 14 ,
  3 0 0 ⟶ 4 2 3 ,
  3 0 1 ⟶ 4 2 4 ,
  3 0 5 ⟶ 4 2 6 ,
  3 0 7 ⟶ 4 2 8 ,
  3 0 9 ⟶ 4 2 10 ,
  3 0 11 ⟶ 4 2 12 ,
  3 0 13 ⟶ 4 2 14 ,
  17 0 0 ⟶ 18 2 3 ,
  17 0 1 ⟶ 18 2 4 ,
  17 0 5 ⟶ 18 2 6 ,
  17 0 7 ⟶ 18 2 8 ,
  17 0 9 ⟶ 18 2 10 ,
  17 0 11 ⟶ 18 2 12 ,
  17 0 13 ⟶ 18 2 14 ,
  19 0 0 ⟶ 20 2 3 ,
  19 0 1 ⟶ 20 2 4 ,
  19 0 5 ⟶ 20 2 6 ,
  19 0 7 ⟶ 20 2 8 ,
  19 0 9 ⟶ 20 2 10 ,
  19 0 11 ⟶ 20 2 12 ,
  19 0 13 ⟶ 20 2 14 ,
  21 0 0 ⟶ 22 2 3 ,
  21 0 1 ⟶ 22 2 4 ,
  21 0 5 ⟶ 22 2 6 ,
  21 0 7 ⟶ 22 2 8 ,
  21 0 9 ⟶ 22 2 10 ,
  21 0 11 ⟶ 22 2 12 ,
  21 0 13 ⟶ 22 2 14 ,
  23 0 0 ⟶ 24 2 3 ,
  23 0 1 ⟶ 24 2 4 ,
  23 0 5 ⟶ 24 2 6 ,
  23 0 7 ⟶ 24 2 8 ,
  23 0 9 ⟶ 24 2 10 ,
  23 0 11 ⟶ 24 2 12 ,
  23 0 13 ⟶ 24 2 14 ,
  5 6 3 ⟶ 1 16 16 15 ,
  5 6 4 ⟶ 1 16 16 16 ,
  5 6 6 ⟶ 1 16 16 2 ,
  5 6 8 ⟶ 1 16 16 25 ,
  5 6 10 ⟶ 1 16 16 26 ,
  5 6 12 ⟶ 1 16 16 27 ,
  5 6 14 ⟶ 1 16 16 28 ,
  2 6 3 ⟶ 16 16 16 15 ,
  2 6 4 ⟶ 16 16 16 16 ,
  2 6 6 ⟶ 16 16 16 2 ,
  2 6 8 ⟶ 16 16 16 25 ,
  2 6 10 ⟶ 16 16 16 26 ,
  2 6 12 ⟶ 16 16 16 27 ,
  2 6 14 ⟶ 16 16 16 28 ,
  6 6 3 ⟶ 4 16 16 15 ,
  6 6 4 ⟶ 4 16 16 16 ,
  6 6 6 ⟶ 4 16 16 2 ,
  6 6 8 ⟶ 4 16 16 25 ,
  6 6 10 ⟶ 4 16 16 26 ,
  6 6 12 ⟶ 4 16 16 27 ,
  6 6 14 ⟶ 4 16 16 28 ,
  29 6 3 ⟶ 18 16 16 15 ,
  29 6 4 ⟶ 18 16 16 16 ,
  29 6 6 ⟶ 18 16 16 2 ,
  29 6 8 ⟶ 18 16 16 25 ,
  29 6 10 ⟶ 18 16 16 26 ,
  29 6 12 ⟶ 18 16 16 27 ,
  29 6 14 ⟶ 18 16 16 28 ,
  30 6 3 ⟶ 20 16 16 15 ,
  30 6 4 ⟶ 20 16 16 16 ,
  30 6 6 ⟶ 20 16 16 2 ,
  30 6 8 ⟶ 20 16 16 25 ,
  30 6 10 ⟶ 20 16 16 26 ,
  30 6 12 ⟶ 20 16 16 27 ,
  30 6 14 ⟶ 20 16 16 28 ,
  31 6 3 ⟶ 22 16 16 15 ,
  31 6 4 ⟶ 22 16 16 16 ,
  31 6 6 ⟶ 22 16 16 2 ,
  31 6 8 ⟶ 22 16 16 25 ,
  31 6 10 ⟶ 22 16 16 26 ,
  31 6 12 ⟶ 22 16 16 27 ,
  31 6 14 ⟶ 22 16 16 28 ,
  32 6 3 ⟶ 24 16 16 15 ,
  32 6 4 ⟶ 24 16 16 16 ,
  32 6 6 ⟶ 24 16 16 2 ,
  32 6 8 ⟶ 24 16 16 25 ,
  32 6 10 ⟶ 24 16 16 26 ,
  32 6 12 ⟶ 24 16 16 27 ,
  32 6 14 ⟶ 24 16 16 28 ,
  5 3 ⟶ 7 17 ,
  5 4 ⟶ 7 18 ,
  5 6 ⟶ 7 29 ,
  5 8 ⟶ 7 33 ,
  5 10 ⟶ 7 34 ,
  5 12 ⟶ 7 35 ,
  5 14 ⟶ 7 36 ,
  2 3 ⟶ 25 17 ,
  2 4 ⟶ 25 18 ,
  2 6 ⟶ 25 29 ,
  2 8 ⟶ 25 33 ,
  2 10 ⟶ 25 34 ,
  2 12 ⟶ 25 35 ,
  2 14 ⟶ 25 36 ,
  6 3 ⟶ 8 17 ,
  6 4 ⟶ 8 18 ,
  6 6 ⟶ 8 29 ,
  6 8 ⟶ 8 33 ,
  6 10 ⟶ 8 34 ,
  6 12 ⟶ 8 35 ,
  6 14 ⟶ 8 36 ,
  29 3 ⟶ 33 17 ,
  29 4 ⟶ 33 18 ,
  29 6 ⟶ 33 29 ,
  29 8 ⟶ 33 33 ,
  29 10 ⟶ 33 34 ,
  29 12 ⟶ 33 35 ,
  29 14 ⟶ 33 36 ,
  30 3 ⟶ 37 17 ,
  30 4 ⟶ 37 18 ,
  30 6 ⟶ 37 29 ,
  30 8 ⟶ 37 33 ,
  30 10 ⟶ 37 34 ,
  30 12 ⟶ 37 35 ,
  30 14 ⟶ 37 36 ,
  31 3 ⟶ 38 17 ,
  31 4 ⟶ 38 18 ,
  31 6 ⟶ 38 29 ,
  31 8 ⟶ 38 33 ,
  31 10 ⟶ 38 34 ,
  31 12 ⟶ 38 35 ,
  31 14 ⟶ 38 36 ,
  32 3 ⟶ 39 17 ,
  32 4 ⟶ 39 18 ,
  32 6 ⟶ 39 29 ,
  32 8 ⟶ 39 33 ,
  32 10 ⟶ 39 34 ,
  32 12 ⟶ 39 35 ,
  32 14 ⟶ 39 36 ,
  1 16 15 ⟶ 9 30 3 ,
  1 16 16 ⟶ 9 30 4 ,
  1 16 2 ⟶ 9 30 6 ,
  1 16 25 ⟶ 9 30 8 ,
  1 16 26 ⟶ 9 30 10 ,
  1 16 27 ⟶ 9 30 12 ,
  1 16 28 ⟶ 9 30 14 ,
  16 16 15 ⟶ 26 30 3 ,
  16 16 16 ⟶ 26 30 4 ,
  16 16 2 ⟶ 26 30 6 ,
  16 16 25 ⟶ 26 30 8 ,
  16 16 26 ⟶ 26 30 10 ,
  16 16 27 ⟶ 26 30 12 ,
  16 16 28 ⟶ 26 30 14 ,
  4 16 15 ⟶ 10 30 3 ,
  4 16 16 ⟶ 10 30 4 ,
  4 16 2 ⟶ 10 30 6 ,
  4 16 25 ⟶ 10 30 8 ,
  4 16 26 ⟶ 10 30 10 ,
  4 16 27 ⟶ 10 30 12 ,
  4 16 28 ⟶ 10 30 14 ,
  18 16 15 ⟶ 34 30 3 ,
  18 16 16 ⟶ 34 30 4 ,
  18 16 2 ⟶ 34 30 6 ,
  18 16 25 ⟶ 34 30 8 ,
  18 16 26 ⟶ 34 30 10 ,
  18 16 27 ⟶ 34 30 12 ,
  18 16 28 ⟶ 34 30 14 ,
  20 16 15 ⟶ 40 30 3 ,
  20 16 16 ⟶ 40 30 4 ,
  20 16 2 ⟶ 40 30 6 ,
  20 16 25 ⟶ 40 30 8 ,
  20 16 26 ⟶ 40 30 10 ,
  20 16 27 ⟶ 40 30 12 ,
  20 16 28 ⟶ 40 30 14 ,
  22 16 15 ⟶ 41 30 3 ,
  22 16 16 ⟶ 41 30 4 ,
  22 16 2 ⟶ 41 30 6 ,
  22 16 25 ⟶ 41 30 8 ,
  22 16 26 ⟶ 41 30 10 ,
  22 16 27 ⟶ 41 30 12 ,
  22 16 28 ⟶ 41 30 14 ,
  24 16 15 ⟶ 42 30 3 ,
  24 16 16 ⟶ 42 30 4 ,
  24 16 2 ⟶ 42 30 6 ,
  24 16 25 ⟶ 42 30 8 ,
  24 16 26 ⟶ 42 30 10 ,
  24 16 27 ⟶ 42 30 12 ,
  24 16 28 ⟶ 42 30 14 ,
  1 16 16 15 ⟶ 5 8 17 ,
  1 16 16 16 ⟶ 5 8 18 ,
  1 16 16 2 ⟶ 5 8 29 ,
  1 16 16 25 ⟶ 5 8 33 ,
  1 16 16 26 ⟶ 5 8 34 ,
  1 16 16 27 ⟶ 5 8 35 ,
  1 16 16 28 ⟶ 5 8 36 ,
  16 16 16 15 ⟶ 2 8 17 ,
  16 16 16 16 ⟶ 2 8 18 ,
  16 16 16 2 ⟶ 2 8 29 ,
  16 16 16 25 ⟶ 2 8 33 ,
  16 16 16 26 ⟶ 2 8 34 ,
  16 16 16 27 ⟶ 2 8 35 ,
  16 16 16 28 ⟶ 2 8 36 ,
  4 16 16 15 ⟶ 6 8 17 ,
  4 16 16 16 ⟶ 6 8 18 ,
  4 16 16 2 ⟶ 6 8 29 ,
  4 16 16 25 ⟶ 6 8 33 ,
  4 16 16 26 ⟶ 6 8 34 ,
  4 16 16 27 ⟶ 6 8 35 ,
  4 16 16 28 ⟶ 6 8 36 ,
  18 16 16 15 ⟶ 29 8 17 ,
  18 16 16 16 ⟶ 29 8 18 ,
  18 16 16 2 ⟶ 29 8 29 ,
  18 16 16 25 ⟶ 29 8 33 ,
  18 16 16 26 ⟶ 29 8 34 ,
  18 16 16 27 ⟶ 29 8 35 ,
  18 16 16 28 ⟶ 29 8 36 ,
  20 16 16 15 ⟶ 30 8 17 ,
  20 16 16 16 ⟶ 30 8 18 ,
  20 16 16 2 ⟶ 30 8 29 ,
  20 16 16 25 ⟶ 30 8 33 ,
  20 16 16 26 ⟶ 30 8 34 ,
  20 16 16 27 ⟶ 30 8 35 ,
  20 16 16 28 ⟶ 30 8 36 ,
  22 16 16 15 ⟶ 31 8 17 ,
  22 16 16 16 ⟶ 31 8 18 ,
  22 16 16 2 ⟶ 31 8 29 ,
  22 16 16 25 ⟶ 31 8 33 ,
  22 16 16 26 ⟶ 31 8 34 ,
  22 16 16 27 ⟶ 31 8 35 ,
  22 16 16 28 ⟶ 31 8 36 ,
  24 16 16 15 ⟶ 32 8 17 ,
  24 16 16 16 ⟶ 32 8 18 ,
  24 16 16 2 ⟶ 32 8 29 ,
  24 16 16 25 ⟶ 32 8 33 ,
  24 16 16 26 ⟶ 32 8 34 ,
  24 16 16 27 ⟶ 32 8 35 ,
  24 16 16 28 ⟶ 32 8 36 ,
  9 19 ⟶ 7 35 21 ,
  9 20 ⟶ 7 35 22 ,
  9 30 ⟶ 7 35 31 ,
  9 37 ⟶ 7 35 38 ,
  9 40 ⟶ 7 35 41 ,
  26 19 ⟶ 25 35 21 ,
  26 20 ⟶ 25 35 22 ,
  26 30 ⟶ 25 35 31 ,
  26 37 ⟶ 25 35 38 ,
  26 40 ⟶ 25 35 41 ,
  10 19 ⟶ 8 35 21 ,
  10 20 ⟶ 8 35 22 ,
  10 30 ⟶ 8 35 31 ,
  10 37 ⟶ 8 35 38 ,
  10 40 ⟶ 8 35 41 ,
  34 19 ⟶ 33 35 21 ,
  34 20 ⟶ 33 35 22 ,
  34 30 ⟶ 33 35 31 ,
  34 37 ⟶ 33 35 38 ,
  34 40 ⟶ 33 35 41 ,
  40 19 ⟶ 37 35 21 ,
  40 20 ⟶ 37 35 22 ,
  40 30 ⟶ 37 35 31 ,
  40 37 ⟶ 37 35 38 ,
  40 40 ⟶ 37 35 41 ,
  41 19 ⟶ 38 35 21 ,
  41 20 ⟶ 38 35 22 ,
  41 30 ⟶ 38 35 31 ,
  41 37 ⟶ 38 35 38 ,
  41 40 ⟶ 38 35 41 ,
  42 19 ⟶ 39 35 21 ,
  42 20 ⟶ 39 35 22 ,
  42 30 ⟶ 39 35 31 ,
  42 37 ⟶ 39 35 38 ,
  42 40 ⟶ 39 35 41 ,
  7 17 ⟶ 0 11 22 15 ,
  7 18 ⟶ 0 11 22 16 ,
  7 29 ⟶ 0 11 22 2 ,
  7 33 ⟶ 0 11 22 25 ,
  7 34 ⟶ 0 11 22 26 ,
  7 35 ⟶ 0 11 22 27 ,
  7 36 ⟶ 0 11 22 28 ,
  25 17 ⟶ 15 11 22 15 ,
  25 18 ⟶ 15 11 22 16 ,
  25 29 ⟶ 15 11 22 2 ,
  25 33 ⟶ 15 11 22 25 ,
  25 34 ⟶ 15 11 22 26 ,
  25 35 ⟶ 15 11 22 27 ,
  25 36 ⟶ 15 11 22 28 ,
  8 17 ⟶ 3 11 22 15 ,
  8 18 ⟶ 3 11 22 16 ,
  8 29 ⟶ 3 11 22 2 ,
  8 33 ⟶ 3 11 22 25 ,
  8 34 ⟶ 3 11 22 26 ,
  8 35 ⟶ 3 11 22 27 ,
  8 36 ⟶ 3 11 22 28 ,
  33 17 ⟶ 17 11 22 15 ,
  33 18 ⟶ 17 11 22 16 ,
  33 29 ⟶ 17 11 22 2 ,
  33 33 ⟶ 17 11 22 25 ,
  33 34 ⟶ 17 11 22 26 ,
  33 35 ⟶ 17 11 22 27 ,
  33 36 ⟶ 17 11 22 28 ,
  37 17 ⟶ 19 11 22 15 ,
  37 18 ⟶ 19 11 22 16 ,
  37 29 ⟶ 19 11 22 2 ,
  37 33 ⟶ 19 11 22 25 ,
  37 34 ⟶ 19 11 22 26 ,
  37 35 ⟶ 19 11 22 27 ,
  37 36 ⟶ 19 11 22 28 ,
  38 17 ⟶ 21 11 22 15 ,
  38 18 ⟶ 21 11 22 16 ,
  38 29 ⟶ 21 11 22 2 ,
  38 33 ⟶ 21 11 22 25 ,
  38 34 ⟶ 21 11 22 26 ,
  38 35 ⟶ 21 11 22 27 ,
  38 36 ⟶ 21 11 22 28 ,
  39 17 ⟶ 23 11 22 15 ,
  39 18 ⟶ 23 11 22 16 ,
  39 29 ⟶ 23 11 22 2 ,
  39 33 ⟶ 23 11 22 25 ,
  39 34 ⟶ 23 11 22 26 ,
  39 35 ⟶ 23 11 22 27 ,
  39 36 ⟶ 23 11 22 28 ,
  7 33 17 ⟶ 5 3 0 ,
  7 33 18 ⟶ 5 3 1 ,
  7 33 29 ⟶ 5 3 5 ,
  7 33 33 ⟶ 5 3 7 ,
  7 33 34 ⟶ 5 3 9 ,
  7 33 35 ⟶ 5 3 11 ,
  7 33 36 ⟶ 5 3 13 ,
  25 33 17 ⟶ 2 3 0 ,
  25 33 18 ⟶ 2 3 1 ,
  25 33 29 ⟶ 2 3 5 ,
  25 33 33 ⟶ 2 3 7 ,
  25 33 34 ⟶ 2 3 9 ,
  25 33 35 ⟶ 2 3 11 ,
  25 33 36 ⟶ 2 3 13 ,
  8 33 17 ⟶ 6 3 0 ,
  8 33 18 ⟶ 6 3 1 ,
  8 33 29 ⟶ 6 3 5 ,
  8 33 33 ⟶ 6 3 7 ,
  8 33 34 ⟶ 6 3 9 ,
  8 33 35 ⟶ 6 3 11 ,
  8 33 36 ⟶ 6 3 13 ,
  33 33 17 ⟶ 29 3 0 ,
  33 33 18 ⟶ 29 3 1 ,
  33 33 29 ⟶ 29 3 5 ,
  33 33 33 ⟶ 29 3 7 ,
  33 33 34 ⟶ 29 3 9 ,
  33 33 35 ⟶ 29 3 11 ,
  33 33 36 ⟶ 29 3 13 ,
  37 33 17 ⟶ 30 3 0 ,
  37 33 18 ⟶ 30 3 1 ,
  37 33 29 ⟶ 30 3 5 ,
  37 33 33 ⟶ 30 3 7 ,
  37 33 34 ⟶ 30 3 9 ,
  37 33 35 ⟶ 30 3 11 ,
  37 33 36 ⟶ 30 3 13 ,
  38 33 17 ⟶ 31 3 0 ,
  38 33 18 ⟶ 31 3 1 ,
  38 33 29 ⟶ 31 3 5 ,
  38 33 33 ⟶ 31 3 7 ,
  38 33 34 ⟶ 31 3 9 ,
  38 33 35 ⟶ 31 3 11 ,
  38 33 36 ⟶ 31 3 13 ,
  39 33 17 ⟶ 32 3 0 ,
  39 33 18 ⟶ 32 3 1 ,
  39 33 29 ⟶ 32 3 5 ,
  39 33 33 ⟶ 32 3 7 ,
  39 33 34 ⟶ 32 3 9 ,
  39 33 35 ⟶ 32 3 11 ,
  39 33 36 ⟶ 32 3 13 }

The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 2:

0 ↦ ⎛       ⎞
    ⎜  1 30 ⎟
    ⎜  0  1 ⎟
    ⎝       ⎠

1 ↦ ⎛       ⎞
    ⎜  1 23 ⎟
    ⎜  0  1 ⎟
    ⎝       ⎠

2 ↦ ⎛       ⎞
    ⎜  1 23 ⎟
    ⎜  0  1 ⎟
    ⎝       ⎠

3 ↦ ⎛       ⎞
    ⎜  1 41 ⎟
    ⎜  0  1 ⎟
    ⎝       ⎠

4 ↦ ⎛       ⎞
    ⎜  1 35 ⎟
    ⎜  0  1 ⎟
    ⎝       ⎠

5 ↦ ⎛       ⎞
    ⎜  1 23 ⎟
    ⎜  0  1 ⎟
    ⎝       ⎠

6 ↦ ⎛       ⎞
    ⎜  1 35 ⎟
    ⎜  0  1 ⎟
    ⎝       ⎠

7 ↦ ⎛       ⎞
    ⎜  1 32 ⎟
    ⎜  0  1 ⎟
    ⎝       ⎠

8 ↦ ⎛       ⎞
    ⎜  1 44 ⎟
    ⎜  0  1 ⎟
    ⎝       ⎠

9 ↦ ⎛       ⎞
    ⎜  1 23 ⎟
    ⎜  0  1 ⎟
    ⎝       ⎠

10 ↦ ⎛       ⎞
     ⎜  1 35 ⎟
     ⎜  0  1 ⎟
     ⎝       ⎠

11 ↦ ⎛     ⎞
     ⎜ 1 0 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

12 ↦ ⎛       ⎞
     ⎜  1 12 ⎟
     ⎜  0  1 ⎟
     ⎝       ⎠

13 ↦ ⎛     ⎞
     ⎜ 1 1 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

14 ↦ ⎛       ⎞
     ⎜  1 12 ⎟
     ⎜  0  1 ⎟
     ⎝       ⎠

15 ↦ ⎛       ⎞
     ⎜  1 29 ⎟
     ⎜  0  1 ⎟
     ⎝       ⎠

16 ↦ ⎛       ⎞
     ⎜  1 23 ⎟
     ⎜  0  1 ⎟
     ⎝       ⎠

17 ↦ ⎛       ⎞
     ⎜  1 30 ⎟
     ⎜  0  1 ⎟
     ⎝       ⎠

18 ↦ ⎛       ⎞
     ⎜  1 24 ⎟
     ⎜  0  1 ⎟
     ⎝       ⎠

19 ↦ ⎛       ⎞
     ⎜  1 16 ⎟
     ⎜  0  1 ⎟
     ⎝       ⎠

20 ↦ ⎛       ⎞
     ⎜  1 10 ⎟
     ⎜  0  1 ⎟
     ⎝       ⎠

21 ↦ ⎛     ⎞
     ⎜ 1 6 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

22 ↦ ⎛     ⎞
     ⎜ 1 0 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

23 ↦ ⎛     ⎞
     ⎜ 1 6 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

24 ↦ ⎛     ⎞
     ⎜ 1 0 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

25 ↦ ⎛       ⎞
     ⎜  1 32 ⎟
     ⎜  0  1 ⎟
     ⎝       ⎠

26 ↦ ⎛       ⎞
     ⎜  1 23 ⎟
     ⎜  0  1 ⎟
     ⎝       ⎠

27 ↦ ⎛     ⎞
     ⎜ 1 0 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

28 ↦ ⎛     ⎞
     ⎜ 1 0 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

29 ↦ ⎛       ⎞
     ⎜  1 24 ⎟
     ⎜  0  1 ⎟
     ⎝       ⎠

30 ↦ ⎛       ⎞
     ⎜  1 10 ⎟
     ⎜  0  1 ⎟
     ⎝       ⎠

31 ↦ ⎛     ⎞
     ⎜ 1 0 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

32 ↦ ⎛     ⎞
     ⎜ 1 0 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

33 ↦ ⎛       ⎞
     ⎜  1 33 ⎟
     ⎜  0  1 ⎟
     ⎝       ⎠

34 ↦ ⎛       ⎞
     ⎜  1 24 ⎟
     ⎜  0  1 ⎟
     ⎝       ⎠

35 ↦ ⎛     ⎞
     ⎜ 1 0 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

36 ↦ ⎛     ⎞
     ⎜ 1 1 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

37 ↦ ⎛       ⎞
     ⎜  1 19 ⎟
     ⎜  0  1 ⎟
     ⎝       ⎠

38 ↦ ⎛     ⎞
     ⎜ 1 9 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

39 ↦ ⎛     ⎞
     ⎜ 1 9 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

40 ↦ ⎛       ⎞
     ⎜  1 10 ⎟
     ⎜  0  1 ⎟
     ⎝       ⎠

41 ↦ ⎛     ⎞
     ⎜ 1 0 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

42 ↦ ⎛     ⎞
     ⎜ 1 0 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

After renaming modulo the bijection {  },
it remains to prove termination of the 0-rule system
{ }

The system is trivially terminating.