/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 1 2 ⟶ 2 1 0 0 2 1 ,
  0 ⟶ ,
  1 ⟶ ,
  2 ⟶ }

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

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

After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 4 ↦ 3 },
it remains to prove termination of the 6-rule system
{ 0 1 2 ⟶ 0 3 2 1 ,
  0 1 2 ⟶ 0 2 1 ,
  3 1 2 →= 2 1 3 3 2 1 ,
  3 →= ,
  1 →= ,
  2 →= }

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

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

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 3 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

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

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

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

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

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

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

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

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

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

13 ↦ ⎛     ⎞
     ⎜ 1 2 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

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

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

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

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

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

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

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

After renaming modulo the bijection { 9 ↦ 0, 13 ↦ 1, 2 ↦ 2, 3 ↦ 3, 14 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 10 ↦ 9, 11 ↦ 10, 4 ↦ 11, 17 ↦ 12, 15 ↦ 13, 1 ↦ 14, 19 ↦ 15 },
it remains to prove termination of the 34-rule system
{ 0 1 2 3 →= 2 3 0 4 5 3 6 ,
  0 1 2 7 →= 2 3 0 4 5 3 2 ,
  0 1 2 8 →= 2 3 0 4 5 3 0 ,
  0 1 2 9 →= 2 3 0 4 5 3 10 ,
  8 1 2 3 →= 7 3 0 4 5 3 6 ,
  8 1 2 7 →= 7 3 0 4 5 3 2 ,
  8 1 2 8 →= 7 3 0 4 5 3 0 ,
  8 1 2 9 →= 7 3 0 4 5 3 10 ,
  4 1 2 3 →= 5 3 0 4 5 3 6 ,
  4 1 2 7 →= 5 3 0 4 5 3 2 ,
  4 1 2 8 →= 5 3 0 4 5 3 0 ,
  4 1 2 9 →= 5 3 0 4 5 3 10 ,
  11 4 →= 11 ,
  0 5 →= 2 ,
  0 4 →= 0 ,
  8 5 →= 7 ,
  8 4 →= 8 ,
  4 1 →= 1 ,
  4 5 →= 5 ,
  4 4 →= 4 ,
  4 12 →= 12 ,
  13 4 →= 13 ,
  14 6 →= 14 ,
  6 6 →= 6 ,
  6 2 →= 2 ,
  6 0 →= 0 ,
  6 10 →= 10 ,
  3 6 →= 3 ,
  3 2 →= 7 ,
  3 0 →= 8 ,
  3 10 →= 9 ,
  1 6 →= 1 ,
  15 6 →= 15 ,
  5 3 →= 1 }

The system is trivially terminating.