/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 0 0 2 ,
  1 ⟶ ,
  2 2 ⟶ 1 1 }

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

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

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 ⎟
    ⎝     ⎠

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

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, 2 ↦ 2, 3 ↦ 3 },
it remains to prove termination of the 6-rule system
{ 0 1 ⟶ 0 2 3 ,
  0 1 ⟶ 0 3 ,
  2 →= ,
  2 1 →= 3 2 2 3 ,
  1 →= ,
  3 3 →= 1 1 }

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

After renaming modulo the bijection { (4,0) ↦ 0, (0,1) ↦ 1, (1,1) ↦ 2, (0,2) ↦ 3, (2,3) ↦ 4, (3,1) ↦ 5, (1,2) ↦ 6, (3,2) ↦ 7, (1,3) ↦ 8, (3,3) ↦ 9, (1,5) ↦ 10, (3,5) ↦ 11, (0,3) ↦ 12, (2,1) ↦ 13, (2,2) ↦ 14, (2,5) ↦ 15, (0,5) ↦ 16, (4,2) ↦ 17, (4,1) ↦ 18, (4,3) ↦ 19, (4,5) ↦ 20 },
it remains to prove termination of the 88-rule system
{ 0 1 2 ⟶ 0 3 4 5 ,
  0 1 6 ⟶ 0 3 4 7 ,
  0 1 8 ⟶ 0 3 4 9 ,
  0 1 10 ⟶ 0 3 4 11 ,
  0 1 2 ⟶ 0 12 5 ,
  0 1 6 ⟶ 0 12 7 ,
  0 1 8 ⟶ 0 12 9 ,
  0 1 10 ⟶ 0 12 11 ,
  3 13 →= 1 ,
  3 14 →= 3 ,
  3 4 →= 12 ,
  3 15 →= 16 ,
  6 13 →= 2 ,
  6 14 →= 6 ,
  6 4 →= 8 ,
  6 15 →= 10 ,
  14 13 →= 13 ,
  14 14 →= 14 ,
  14 4 →= 4 ,
  14 15 →= 15 ,
  7 13 →= 5 ,
  7 14 →= 7 ,
  7 4 →= 9 ,
  7 15 →= 11 ,
  17 13 →= 18 ,
  17 14 →= 17 ,
  17 4 →= 19 ,
  17 15 →= 20 ,
  3 13 2 →= 12 7 14 4 5 ,
  3 13 6 →= 12 7 14 4 7 ,
  3 13 8 →= 12 7 14 4 9 ,
  3 13 10 →= 12 7 14 4 11 ,
  6 13 2 →= 8 7 14 4 5 ,
  6 13 6 →= 8 7 14 4 7 ,
  6 13 8 →= 8 7 14 4 9 ,
  6 13 10 →= 8 7 14 4 11 ,
  14 13 2 →= 4 7 14 4 5 ,
  14 13 6 →= 4 7 14 4 7 ,
  14 13 8 →= 4 7 14 4 9 ,
  14 13 10 →= 4 7 14 4 11 ,
  7 13 2 →= 9 7 14 4 5 ,
  7 13 6 →= 9 7 14 4 7 ,
  7 13 8 →= 9 7 14 4 9 ,
  7 13 10 →= 9 7 14 4 11 ,
  17 13 2 →= 19 7 14 4 5 ,
  17 13 6 →= 19 7 14 4 7 ,
  17 13 8 →= 19 7 14 4 9 ,
  17 13 10 →= 19 7 14 4 11 ,
  1 2 →= 1 ,
  1 6 →= 3 ,
  1 8 →= 12 ,
  1 10 →= 16 ,
  2 2 →= 2 ,
  2 6 →= 6 ,
  2 8 →= 8 ,
  2 10 →= 10 ,
  13 2 →= 13 ,
  13 6 →= 14 ,
  13 8 →= 4 ,
  13 10 →= 15 ,
  5 2 →= 5 ,
  5 6 →= 7 ,
  5 8 →= 9 ,
  5 10 →= 11 ,
  18 2 →= 18 ,
  18 6 →= 17 ,
  18 8 →= 19 ,
  18 10 →= 20 ,
  12 9 5 →= 1 2 2 ,
  12 9 7 →= 1 2 6 ,
  12 9 9 →= 1 2 8 ,
  12 9 11 →= 1 2 10 ,
  8 9 5 →= 2 2 2 ,
  8 9 7 →= 2 2 6 ,
  8 9 9 →= 2 2 8 ,
  8 9 11 →= 2 2 10 ,
  4 9 5 →= 13 2 2 ,
  4 9 7 →= 13 2 6 ,
  4 9 9 →= 13 2 8 ,
  4 9 11 →= 13 2 10 ,
  9 9 5 →= 5 2 2 ,
  9 9 7 →= 5 2 6 ,
  9 9 9 →= 5 2 8 ,
  9 9 11 →= 5 2 10 ,
  19 9 5 →= 18 2 2 ,
  19 9 7 →= 18 2 6 ,
  19 9 9 →= 18 2 8 ,
  19 9 11 →= 18 2 10 }

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 2 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

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

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

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

6 ↦ ⎛     ⎞
    ⎜ 1 3 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

7 ↦ ⎛     ⎞
    ⎜ 1 3 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

8 ↦ ⎛     ⎞
    ⎜ 1 3 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

9 ↦ ⎛     ⎞
    ⎜ 1 3 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

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

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

13 ↦ ⎛     ⎞
     ⎜ 1 3 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

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

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

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

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

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

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

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

After renaming modulo the bijection { 3 ↦ 0, 14 ↦ 1, 6 ↦ 2, 4 ↦ 3, 8 ↦ 4, 13 ↦ 5, 15 ↦ 6, 7 ↦ 7, 9 ↦ 8, 17 ↦ 9, 2 ↦ 10, 5 ↦ 11, 10 ↦ 12, 11 ↦ 13, 1 ↦ 14, 18 ↦ 15 },
it remains to prove termination of the 37-rule system
{ 0 1 →= 0 ,
  2 1 →= 2 ,
  2 3 →= 4 ,
  1 5 →= 5 ,
  1 1 →= 1 ,
  1 3 →= 3 ,
  1 6 →= 6 ,
  7 1 →= 7 ,
  7 3 →= 8 ,
  9 1 →= 9 ,
  2 5 10 →= 4 7 1 3 11 ,
  2 5 2 →= 4 7 1 3 7 ,
  2 5 4 →= 4 7 1 3 8 ,
  2 5 12 →= 4 7 1 3 13 ,
  1 5 10 →= 3 7 1 3 11 ,
  1 5 2 →= 3 7 1 3 7 ,
  1 5 4 →= 3 7 1 3 8 ,
  1 5 12 →= 3 7 1 3 13 ,
  7 5 10 →= 8 7 1 3 11 ,
  7 5 2 →= 8 7 1 3 7 ,
  7 5 4 →= 8 7 1 3 8 ,
  7 5 12 →= 8 7 1 3 13 ,
  14 10 →= 14 ,
  10 10 →= 10 ,
  10 2 →= 2 ,
  10 4 →= 4 ,
  10 12 →= 12 ,
  5 10 →= 5 ,
  11 10 →= 11 ,
  11 2 →= 7 ,
  11 4 →= 8 ,
  11 12 →= 13 ,
  15 10 →= 15 ,
  3 8 11 →= 5 10 10 ,
  3 8 7 →= 5 10 2 ,
  3 8 8 →= 5 10 4 ,
  3 8 13 →= 5 10 12 }

The system is trivially terminating.