/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 { a ↦ 0, b ↦ 1, c ↦ 2 },
it remains to prove termination of the 4-rule system
{ 0 ⟶ ,
  0 1 ⟶ 2 1 ,
  1 ⟶ 0 0 2 ,
  2 2 ⟶ 1 }

Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols.

After renaming modulo the bijection { (0,↑) ↦ 0, (1,↓) ↦ 1, (2,↑) ↦ 2, (1,↑) ↦ 3, (0,↓) ↦ 4, (2,↓) ↦ 5 },
it remains to prove termination of the 9-rule system
{ 0 1 ⟶ 2 1 ,
  3 ⟶ 0 4 5 ,
  3 ⟶ 0 5 ,
  3 ⟶ 2 ,
  2 5 ⟶ 3 ,
  4 →= ,
  4 1 →= 5 1 ,
  1 →= 4 4 5 ,
  5 5 →= 1 }

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

After renaming modulo the bijection { (6,0) ↦ 0, (0,1) ↦ 1, (1,1) ↦ 2, (6,2) ↦ 3, (2,1) ↦ 4, (1,4) ↦ 5, (1,5) ↦ 6, (1,7) ↦ 7, (6,3) ↦ 8, (3,1) ↦ 9, (0,4) ↦ 10, (4,5) ↦ 11, (5,1) ↦ 12, (3,4) ↦ 13, (5,4) ↦ 14, (3,5) ↦ 15, (5,5) ↦ 16, (3,7) ↦ 17, (5,7) ↦ 18, (0,5) ↦ 19, (2,4) ↦ 20, (2,5) ↦ 21, (2,7) ↦ 22, (4,1) ↦ 23, (4,4) ↦ 24, (6,4) ↦ 25, (6,1) ↦ 26, (6,5) ↦ 27 },
it remains to prove termination of the 125-rule system
{ 0 1 2 ⟶ 3 4 2 ,
  0 1 5 ⟶ 3 4 5 ,
  0 1 6 ⟶ 3 4 6 ,
  0 1 7 ⟶ 3 4 7 ,
  8 9 ⟶ 0 10 11 12 ,
  8 13 ⟶ 0 10 11 14 ,
  8 15 ⟶ 0 10 11 16 ,
  8 17 ⟶ 0 10 11 18 ,
  8 9 ⟶ 0 19 12 ,
  8 13 ⟶ 0 19 14 ,
  8 15 ⟶ 0 19 16 ,
  8 17 ⟶ 0 19 18 ,
  8 9 ⟶ 3 4 ,
  8 13 ⟶ 3 20 ,
  8 15 ⟶ 3 21 ,
  8 17 ⟶ 3 22 ,
  3 21 12 ⟶ 8 9 ,
  3 21 14 ⟶ 8 13 ,
  3 21 16 ⟶ 8 15 ,
  3 21 18 ⟶ 8 17 ,
  10 23 →= 1 ,
  10 24 →= 10 ,
  10 11 →= 19 ,
  5 23 →= 2 ,
  5 24 →= 5 ,
  5 11 →= 6 ,
  20 23 →= 4 ,
  20 24 →= 20 ,
  20 11 →= 21 ,
  13 23 →= 9 ,
  13 24 →= 13 ,
  13 11 →= 15 ,
  24 23 →= 23 ,
  24 24 →= 24 ,
  24 11 →= 11 ,
  14 23 →= 12 ,
  14 24 →= 14 ,
  14 11 →= 16 ,
  25 23 →= 26 ,
  25 24 →= 25 ,
  25 11 →= 27 ,
  10 23 2 →= 19 12 2 ,
  10 23 5 →= 19 12 5 ,
  10 23 6 →= 19 12 6 ,
  10 23 7 →= 19 12 7 ,
  5 23 2 →= 6 12 2 ,
  5 23 5 →= 6 12 5 ,
  5 23 6 →= 6 12 6 ,
  5 23 7 →= 6 12 7 ,
  20 23 2 →= 21 12 2 ,
  20 23 5 →= 21 12 5 ,
  20 23 6 →= 21 12 6 ,
  20 23 7 →= 21 12 7 ,
  13 23 2 →= 15 12 2 ,
  13 23 5 →= 15 12 5 ,
  13 23 6 →= 15 12 6 ,
  13 23 7 →= 15 12 7 ,
  24 23 2 →= 11 12 2 ,
  24 23 5 →= 11 12 5 ,
  24 23 6 →= 11 12 6 ,
  24 23 7 →= 11 12 7 ,
  14 23 2 →= 16 12 2 ,
  14 23 5 →= 16 12 5 ,
  14 23 6 →= 16 12 6 ,
  14 23 7 →= 16 12 7 ,
  25 23 2 →= 27 12 2 ,
  25 23 5 →= 27 12 5 ,
  25 23 6 →= 27 12 6 ,
  25 23 7 →= 27 12 7 ,
  1 2 →= 10 24 11 12 ,
  1 5 →= 10 24 11 14 ,
  1 6 →= 10 24 11 16 ,
  1 7 →= 10 24 11 18 ,
  2 2 →= 5 24 11 12 ,
  2 5 →= 5 24 11 14 ,
  2 6 →= 5 24 11 16 ,
  2 7 →= 5 24 11 18 ,
  4 2 →= 20 24 11 12 ,
  4 5 →= 20 24 11 14 ,
  4 6 →= 20 24 11 16 ,
  4 7 →= 20 24 11 18 ,
  9 2 →= 13 24 11 12 ,
  9 5 →= 13 24 11 14 ,
  9 6 →= 13 24 11 16 ,
  9 7 →= 13 24 11 18 ,
  23 2 →= 24 24 11 12 ,
  23 5 →= 24 24 11 14 ,
  23 6 →= 24 24 11 16 ,
  23 7 →= 24 24 11 18 ,
  12 2 →= 14 24 11 12 ,
  12 5 →= 14 24 11 14 ,
  12 6 →= 14 24 11 16 ,
  12 7 →= 14 24 11 18 ,
  26 2 →= 25 24 11 12 ,
  26 5 →= 25 24 11 14 ,
  26 6 →= 25 24 11 16 ,
  26 7 →= 25 24 11 18 ,
  19 16 12 →= 1 2 ,
  19 16 14 →= 1 5 ,
  19 16 16 →= 1 6 ,
  19 16 18 →= 1 7 ,
  6 16 12 →= 2 2 ,
  6 16 14 →= 2 5 ,
  6 16 16 →= 2 6 ,
  6 16 18 →= 2 7 ,
  21 16 12 →= 4 2 ,
  21 16 14 →= 4 5 ,
  21 16 16 →= 4 6 ,
  21 16 18 →= 4 7 ,
  15 16 12 →= 9 2 ,
  15 16 14 →= 9 5 ,
  15 16 16 →= 9 6 ,
  15 16 18 →= 9 7 ,
  11 16 12 →= 23 2 ,
  11 16 14 →= 23 5 ,
  11 16 16 →= 23 6 ,
  11 16 18 →= 23 7 ,
  16 16 12 →= 12 2 ,
  16 16 14 →= 12 5 ,
  16 16 16 →= 12 6 ,
  16 16 18 →= 12 7 ,
  27 16 12 →= 26 2 ,
  27 16 14 →= 26 5 ,
  27 16 16 →= 26 6 ,
  27 16 18 →= 26 7 }

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 6 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

2 ↦ ⎛     ⎞
    ⎜ 1 9 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

3 ↦ ⎛     ⎞
    ⎜ 1 5 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

5 ↦ ⎛       ⎞
    ⎜  1 11 ⎟
    ⎜  0  1 ⎟
    ⎝       ⎠

6 ↦ ⎛       ⎞
    ⎜  1 11 ⎟
    ⎜  0  1 ⎟
    ⎝       ⎠

7 ↦ ⎛     ⎞
    ⎜ 1 4 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

8 ↦ ⎛     ⎞
    ⎜ 1 6 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

9 ↦ ⎛     ⎞
    ⎜ 1 5 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

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

12 ↦ ⎛     ⎞
     ⎜ 1 7 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

13 ↦ ⎛     ⎞
     ⎜ 1 6 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

14 ↦ ⎛     ⎞
     ⎜ 1 9 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

15 ↦ ⎛     ⎞
     ⎜ 1 5 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

16 ↦ ⎛     ⎞
     ⎜ 1 9 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

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

18 ↦ ⎛     ⎞
     ⎜ 1 2 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

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

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

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

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

23 ↦ ⎛     ⎞
     ⎜ 1 7 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

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

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

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

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

After renaming modulo the bijection { 10 ↦ 0, 24 ↦ 1, 5 ↦ 2, 11 ↦ 3, 6 ↦ 4, 20 ↦ 5, 13 ↦ 6, 23 ↦ 7, 14 ↦ 8, 16 ↦ 9, 25 ↦ 10, 2 ↦ 11, 12 ↦ 12, 7 ↦ 13, 18 ↦ 14 },
it remains to prove termination of the 35-rule system
{ 0 1 →= 0 ,
  2 1 →= 2 ,
  2 3 →= 4 ,
  5 1 →= 5 ,
  6 1 →= 6 ,
  1 7 →= 7 ,
  1 1 →= 1 ,
  1 3 →= 3 ,
  8 1 →= 8 ,
  8 3 →= 9 ,
  10 1 →= 10 ,
  2 7 11 →= 4 12 11 ,
  2 7 2 →= 4 12 2 ,
  2 7 4 →= 4 12 4 ,
  2 7 13 →= 4 12 13 ,
  1 7 11 →= 3 12 11 ,
  1 7 2 →= 3 12 2 ,
  1 7 4 →= 3 12 4 ,
  1 7 13 →= 3 12 13 ,
  8 7 11 →= 9 12 11 ,
  8 7 2 →= 9 12 2 ,
  8 7 4 →= 9 12 4 ,
  8 7 13 →= 9 12 13 ,
  11 11 →= 2 1 3 12 ,
  11 2 →= 2 1 3 8 ,
  11 4 →= 2 1 3 9 ,
  11 13 →= 2 1 3 14 ,
  12 11 →= 8 1 3 12 ,
  12 2 →= 8 1 3 8 ,
  12 4 →= 8 1 3 9 ,
  12 13 →= 8 1 3 14 ,
  3 9 12 →= 7 11 ,
  3 9 8 →= 7 2 ,
  3 9 9 →= 7 4 ,
  3 9 14 →= 7 13 }

The system is trivially terminating.