/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 5-rule system
{ 0 ⟶ ,
  0 0 ⟶ 1 2 ,
  1 ⟶ ,
  2 ⟶ ,
  2 1 ⟶ 0 1 2 }

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

After renaming modulo the bijection { (0,↑) ↦ 0, (0,↓) ↦ 1, (1,↑) ↦ 2, (2,↓) ↦ 3, (2,↑) ↦ 4, (1,↓) ↦ 5 },
it remains to prove termination of the 10-rule system
{ 0 1 ⟶ 2 3 ,
  0 1 ⟶ 4 ,
  4 5 ⟶ 0 5 3 ,
  4 5 ⟶ 2 3 ,
  4 5 ⟶ 4 ,
  1 →= ,
  1 1 →= 5 3 ,
  5 →= ,
  3 →= ,
  3 5 →= 1 5 3 }

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 1 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

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

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

After renaming modulo the bijection { (5,0) ↦ 0, (0,1) ↦ 1, (1,1) ↦ 2, (5,2) ↦ 3, (2,1) ↦ 4, (1,3) ↦ 5, (2,3) ↦ 6, (1,4) ↦ 7, (2,4) ↦ 8, (1,6) ↦ 9, (2,6) ↦ 10, (3,1) ↦ 11, (0,3) ↦ 12, (3,4) ↦ 13, (4,1) ↦ 14, (3,3) ↦ 15, (4,3) ↦ 16, (4,4) ↦ 17, (3,6) ↦ 18, (4,6) ↦ 19, (0,4) ↦ 20, (0,6) ↦ 21, (5,1) ↦ 22, (5,3) ↦ 23, (5,4) ↦ 24, (5,6) ↦ 25 },
it remains to prove termination of the 132-rule system
{ 0 1 2 ⟶ 3 4 ,
  0 1 5 ⟶ 3 6 ,
  0 1 7 ⟶ 3 8 ,
  0 1 9 ⟶ 3 10 ,
  3 6 11 ⟶ 0 12 13 14 ,
  3 6 15 ⟶ 0 12 13 16 ,
  3 6 13 ⟶ 0 12 13 17 ,
  3 6 18 ⟶ 0 12 13 19 ,
  3 6 11 ⟶ 3 4 ,
  3 6 15 ⟶ 3 6 ,
  3 6 13 ⟶ 3 8 ,
  3 6 18 ⟶ 3 10 ,
  1 2 →= 1 ,
  1 5 →= 12 ,
  1 7 →= 20 ,
  1 9 →= 21 ,
  2 2 →= 2 ,
  2 5 →= 5 ,
  2 7 →= 7 ,
  2 9 →= 9 ,
  4 2 →= 4 ,
  4 5 →= 6 ,
  4 7 →= 8 ,
  4 9 →= 10 ,
  11 2 →= 11 ,
  11 5 →= 15 ,
  11 7 →= 13 ,
  11 9 →= 18 ,
  14 2 →= 14 ,
  14 5 →= 16 ,
  14 7 →= 17 ,
  14 9 →= 19 ,
  22 2 →= 22 ,
  22 5 →= 23 ,
  22 7 →= 24 ,
  22 9 →= 25 ,
  1 2 2 →= 12 13 14 ,
  1 2 5 →= 12 13 16 ,
  1 2 7 →= 12 13 17 ,
  1 2 9 →= 12 13 19 ,
  2 2 2 →= 5 13 14 ,
  2 2 5 →= 5 13 16 ,
  2 2 7 →= 5 13 17 ,
  2 2 9 →= 5 13 19 ,
  4 2 2 →= 6 13 14 ,
  4 2 5 →= 6 13 16 ,
  4 2 7 →= 6 13 17 ,
  4 2 9 →= 6 13 19 ,
  11 2 2 →= 15 13 14 ,
  11 2 5 →= 15 13 16 ,
  11 2 7 →= 15 13 17 ,
  11 2 9 →= 15 13 19 ,
  14 2 2 →= 16 13 14 ,
  14 2 5 →= 16 13 16 ,
  14 2 7 →= 16 13 17 ,
  14 2 9 →= 16 13 19 ,
  22 2 2 →= 23 13 14 ,
  22 2 5 →= 23 13 16 ,
  22 2 7 →= 23 13 17 ,
  22 2 9 →= 23 13 19 ,
  12 11 →= 1 ,
  12 15 →= 12 ,
  12 13 →= 20 ,
  12 18 →= 21 ,
  5 11 →= 2 ,
  5 15 →= 5 ,
  5 13 →= 7 ,
  5 18 →= 9 ,
  6 11 →= 4 ,
  6 15 →= 6 ,
  6 13 →= 8 ,
  6 18 →= 10 ,
  15 11 →= 11 ,
  15 15 →= 15 ,
  15 13 →= 13 ,
  15 18 →= 18 ,
  16 11 →= 14 ,
  16 15 →= 16 ,
  16 13 →= 17 ,
  16 18 →= 19 ,
  23 11 →= 22 ,
  23 15 →= 23 ,
  23 13 →= 24 ,
  23 18 →= 25 ,
  20 14 →= 1 ,
  20 16 →= 12 ,
  20 17 →= 20 ,
  20 19 →= 21 ,
  7 14 →= 2 ,
  7 16 →= 5 ,
  7 17 →= 7 ,
  7 19 →= 9 ,
  8 14 →= 4 ,
  8 16 →= 6 ,
  8 17 →= 8 ,
  8 19 →= 10 ,
  13 14 →= 11 ,
  13 16 →= 15 ,
  13 17 →= 13 ,
  13 19 →= 18 ,
  17 14 →= 14 ,
  17 16 →= 16 ,
  17 17 →= 17 ,
  17 19 →= 19 ,
  24 14 →= 22 ,
  24 16 →= 23 ,
  24 17 →= 24 ,
  24 19 →= 25 ,
  20 16 11 →= 1 5 13 14 ,
  20 16 15 →= 1 5 13 16 ,
  20 16 13 →= 1 5 13 17 ,
  20 16 18 →= 1 5 13 19 ,
  7 16 11 →= 2 5 13 14 ,
  7 16 15 →= 2 5 13 16 ,
  7 16 13 →= 2 5 13 17 ,
  7 16 18 →= 2 5 13 19 ,
  8 16 11 →= 4 5 13 14 ,
  8 16 15 →= 4 5 13 16 ,
  8 16 13 →= 4 5 13 17 ,
  8 16 18 →= 4 5 13 19 ,
  13 16 11 →= 11 5 13 14 ,
  13 16 15 →= 11 5 13 16 ,
  13 16 13 →= 11 5 13 17 ,
  13 16 18 →= 11 5 13 19 ,
  17 16 11 →= 14 5 13 14 ,
  17 16 15 →= 14 5 13 16 ,
  17 16 13 →= 14 5 13 17 ,
  17 16 18 →= 14 5 13 19 ,
  24 16 11 →= 22 5 13 14 ,
  24 16 15 →= 22 5 13 16 ,
  24 16 13 →= 22 5 13 17 ,
  24 16 18 →= 22 5 13 19 }

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 4 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

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

5 ↦ ⎛     ⎞
    ⎜ 1 2 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

6 ↦ ⎛     ⎞
    ⎜ 1 2 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

7 ↦ ⎛     ⎞
    ⎜ 1 2 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

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

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

11 ↦ ⎛     ⎞
     ⎜ 1 2 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

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

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

14 ↦ ⎛     ⎞
     ⎜ 1 2 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

15 ↦ ⎛     ⎞
     ⎜ 1 4 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

16 ↦ ⎛     ⎞
     ⎜ 1 4 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

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

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

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

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

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

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

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

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

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

After renaming modulo the bijection { 11 ↦ 0, 5 ↦ 1, 15 ↦ 2, 14 ↦ 3, 16 ↦ 4, 2 ↦ 5, 13 ↦ 6, 7 ↦ 7, 20 ↦ 8, 17 ↦ 9, 8 ↦ 10, 19 ↦ 11, 18 ↦ 12, 24 ↦ 13 },
it remains to prove termination of the 31-rule system
{ 0 1 →= 2 ,
  3 1 →= 4 ,
  0 5 1 →= 2 6 4 ,
  3 5 1 →= 4 6 4 ,
  1 0 →= 5 ,
  1 6 →= 7 ,
  8 9 →= 8 ,
  7 3 →= 5 ,
  7 9 →= 7 ,
  10 9 →= 10 ,
  6 3 →= 0 ,
  6 4 →= 2 ,
  6 9 →= 6 ,
  6 11 →= 12 ,
  9 3 →= 3 ,
  9 4 →= 4 ,
  9 9 →= 9 ,
  9 11 →= 11 ,
  13 9 →= 13 ,
  7 4 0 →= 5 1 6 3 ,
  7 4 2 →= 5 1 6 4 ,
  7 4 6 →= 5 1 6 9 ,
  7 4 12 →= 5 1 6 11 ,
  6 4 0 →= 0 1 6 3 ,
  6 4 2 →= 0 1 6 4 ,
  6 4 6 →= 0 1 6 9 ,
  6 4 12 →= 0 1 6 11 ,
  9 4 0 →= 3 1 6 3 ,
  9 4 2 →= 3 1 6 4 ,
  9 4 6 →= 3 1 6 9 ,
  9 4 12 →= 3 1 6 11 }

The system is trivially terminating.