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

The system was reversed.

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

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 17-rule system
{ 0 1 2 ⟶ 3 2 2 1 1 1 4 4 4 ,
  0 1 2 ⟶ 3 2 1 1 1 4 4 4 ,
  0 1 2 ⟶ 3 1 1 1 4 4 4 ,
  0 1 2 ⟶ 5 1 1 4 4 4 ,
  0 1 2 ⟶ 5 1 4 4 4 ,
  0 1 2 ⟶ 5 4 4 4 ,
  0 1 2 ⟶ 0 4 4 ,
  0 1 2 ⟶ 0 4 ,
  0 1 2 ⟶ 0 ,
  5 4 ⟶ 3 2 2 ,
  5 4 ⟶ 3 2 ,
  5 4 ⟶ 3 ,
  4 1 2 →= 2 2 2 1 1 1 4 4 4 ,
  1 4 →= 2 2 2 ,
  2 →= ,
  1 →= ,
  4 →= }

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, 5 ↦ 4, 3 ↦ 5 },
it remains to prove termination of the 11-rule system
{ 0 1 2 ⟶ 0 3 3 ,
  0 1 2 ⟶ 0 3 ,
  0 1 2 ⟶ 0 ,
  4 3 ⟶ 5 2 2 ,
  4 3 ⟶ 5 2 ,
  4 3 ⟶ 5 ,
  3 1 2 →= 2 2 2 1 1 1 3 3 3 ,
  1 3 →= 2 2 2 ,
  2 →= ,
  1 →= ,
  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, 2 ↦ 2, 3 ↦ 3 },
it remains to prove termination of the 8-rule system
{ 0 1 2 ⟶ 0 3 3 ,
  0 1 2 ⟶ 0 3 ,
  0 1 2 ⟶ 0 ,
  3 1 2 →= 2 2 2 1 1 1 3 3 3 ,
  1 3 →= 2 2 2 ,
  2 →= ,
  1 →= ,
  3 →= }

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, (3,3) ↦ 5, (3,1) ↦ 6, (2,2) ↦ 7, (3,2) ↦ 8, (2,3) ↦ 9, (2,5) ↦ 10, (3,5) ↦ 11, (0,2) ↦ 12, (0,5) ↦ 13, (1,1) ↦ 14, (1,3) ↦ 15, (4,3) ↦ 16, (4,2) ↦ 17, (4,1) ↦ 18, (1,5) ↦ 19, (4,5) ↦ 20 },
it remains to prove termination of the 112-rule system
{ 0 1 2 3 ⟶ 0 4 5 6 ,
  0 1 2 7 ⟶ 0 4 5 8 ,
  0 1 2 9 ⟶ 0 4 5 5 ,
  0 1 2 10 ⟶ 0 4 5 11 ,
  0 1 2 3 ⟶ 0 4 6 ,
  0 1 2 7 ⟶ 0 4 8 ,
  0 1 2 9 ⟶ 0 4 5 ,
  0 1 2 10 ⟶ 0 4 11 ,
  0 1 2 3 ⟶ 0 1 ,
  0 1 2 7 ⟶ 0 12 ,
  0 1 2 9 ⟶ 0 4 ,
  0 1 2 10 ⟶ 0 13 ,
  4 6 2 3 →= 12 7 7 3 14 14 15 5 5 6 ,
  4 6 2 7 →= 12 7 7 3 14 14 15 5 5 8 ,
  4 6 2 9 →= 12 7 7 3 14 14 15 5 5 5 ,
  4 6 2 10 →= 12 7 7 3 14 14 15 5 5 11 ,
  15 6 2 3 →= 2 7 7 3 14 14 15 5 5 6 ,
  15 6 2 7 →= 2 7 7 3 14 14 15 5 5 8 ,
  15 6 2 9 →= 2 7 7 3 14 14 15 5 5 5 ,
  15 6 2 10 →= 2 7 7 3 14 14 15 5 5 11 ,
  9 6 2 3 →= 7 7 7 3 14 14 15 5 5 6 ,
  9 6 2 7 →= 7 7 7 3 14 14 15 5 5 8 ,
  9 6 2 9 →= 7 7 7 3 14 14 15 5 5 5 ,
  9 6 2 10 →= 7 7 7 3 14 14 15 5 5 11 ,
  5 6 2 3 →= 8 7 7 3 14 14 15 5 5 6 ,
  5 6 2 7 →= 8 7 7 3 14 14 15 5 5 8 ,
  5 6 2 9 →= 8 7 7 3 14 14 15 5 5 5 ,
  5 6 2 10 →= 8 7 7 3 14 14 15 5 5 11 ,
  16 6 2 3 →= 17 7 7 3 14 14 15 5 5 6 ,
  16 6 2 7 →= 17 7 7 3 14 14 15 5 5 8 ,
  16 6 2 9 →= 17 7 7 3 14 14 15 5 5 5 ,
  16 6 2 10 →= 17 7 7 3 14 14 15 5 5 11 ,
  1 15 6 →= 12 7 7 3 ,
  1 15 8 →= 12 7 7 7 ,
  1 15 5 →= 12 7 7 9 ,
  1 15 11 →= 12 7 7 10 ,
  14 15 6 →= 2 7 7 3 ,
  14 15 8 →= 2 7 7 7 ,
  14 15 5 →= 2 7 7 9 ,
  14 15 11 →= 2 7 7 10 ,
  3 15 6 →= 7 7 7 3 ,
  3 15 8 →= 7 7 7 7 ,
  3 15 5 →= 7 7 7 9 ,
  3 15 11 →= 7 7 7 10 ,
  6 15 6 →= 8 7 7 3 ,
  6 15 8 →= 8 7 7 7 ,
  6 15 5 →= 8 7 7 9 ,
  6 15 11 →= 8 7 7 10 ,
  18 15 6 →= 17 7 7 3 ,
  18 15 8 →= 17 7 7 7 ,
  18 15 5 →= 17 7 7 9 ,
  18 15 11 →= 17 7 7 10 ,
  12 3 →= 1 ,
  12 7 →= 12 ,
  12 9 →= 4 ,
  12 10 →= 13 ,
  2 3 →= 14 ,
  2 7 →= 2 ,
  2 9 →= 15 ,
  2 10 →= 19 ,
  7 3 →= 3 ,
  7 7 →= 7 ,
  7 9 →= 9 ,
  7 10 →= 10 ,
  8 3 →= 6 ,
  8 7 →= 8 ,
  8 9 →= 5 ,
  8 10 →= 11 ,
  17 3 →= 18 ,
  17 7 →= 17 ,
  17 9 →= 16 ,
  17 10 →= 20 ,
  1 14 →= 1 ,
  1 2 →= 12 ,
  1 15 →= 4 ,
  1 19 →= 13 ,
  14 14 →= 14 ,
  14 2 →= 2 ,
  14 15 →= 15 ,
  14 19 →= 19 ,
  3 14 →= 3 ,
  3 2 →= 7 ,
  3 15 →= 9 ,
  3 19 →= 10 ,
  6 14 →= 6 ,
  6 2 →= 8 ,
  6 15 →= 5 ,
  6 19 →= 11 ,
  18 14 →= 18 ,
  18 2 →= 17 ,
  18 15 →= 16 ,
  18 19 →= 20 ,
  4 6 →= 1 ,
  4 8 →= 12 ,
  4 5 →= 4 ,
  4 11 →= 13 ,
  15 6 →= 14 ,
  15 8 →= 2 ,
  15 5 →= 15 ,
  15 11 →= 19 ,
  9 6 →= 3 ,
  9 8 →= 7 ,
  9 5 →= 9 ,
  9 11 →= 10 ,
  5 6 →= 6 ,
  5 8 →= 8 ,
  5 5 →= 5 ,
  5 11 →= 11 ,
  16 6 →= 18 ,
  16 8 →= 17 ,
  16 5 →= 16 ,
  16 11 →= 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 1 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

After renaming modulo the bijection { 4 ↦ 0, 6 ↦ 1, 2 ↦ 2, 3 ↦ 3, 12 ↦ 4, 7 ↦ 5, 14 ↦ 6, 15 ↦ 7, 5 ↦ 8, 8 ↦ 9, 9 ↦ 10, 10 ↦ 11, 11 ↦ 12, 16 ↦ 13, 17 ↦ 14, 1 ↦ 15, 19 ↦ 16, 18 ↦ 17 },
it remains to prove termination of the 60-rule system
{ 0 1 2 3 →= 4 5 5 3 6 6 7 8 8 1 ,
  0 1 2 5 →= 4 5 5 3 6 6 7 8 8 9 ,
  0 1 2 10 →= 4 5 5 3 6 6 7 8 8 8 ,
  0 1 2 11 →= 4 5 5 3 6 6 7 8 8 12 ,
  7 1 2 3 →= 2 5 5 3 6 6 7 8 8 1 ,
  7 1 2 5 →= 2 5 5 3 6 6 7 8 8 9 ,
  7 1 2 10 →= 2 5 5 3 6 6 7 8 8 8 ,
  7 1 2 11 →= 2 5 5 3 6 6 7 8 8 12 ,
  10 1 2 3 →= 5 5 5 3 6 6 7 8 8 1 ,
  10 1 2 5 →= 5 5 5 3 6 6 7 8 8 9 ,
  10 1 2 10 →= 5 5 5 3 6 6 7 8 8 8 ,
  10 1 2 11 →= 5 5 5 3 6 6 7 8 8 12 ,
  8 1 2 3 →= 9 5 5 3 6 6 7 8 8 1 ,
  8 1 2 5 →= 9 5 5 3 6 6 7 8 8 9 ,
  8 1 2 10 →= 9 5 5 3 6 6 7 8 8 8 ,
  8 1 2 11 →= 9 5 5 3 6 6 7 8 8 12 ,
  13 1 2 3 →= 14 5 5 3 6 6 7 8 8 1 ,
  13 1 2 5 →= 14 5 5 3 6 6 7 8 8 9 ,
  13 1 2 10 →= 14 5 5 3 6 6 7 8 8 8 ,
  13 1 2 11 →= 14 5 5 3 6 6 7 8 8 12 ,
  6 7 1 →= 2 5 5 3 ,
  6 7 9 →= 2 5 5 5 ,
  6 7 8 →= 2 5 5 10 ,
  6 7 12 →= 2 5 5 11 ,
  4 5 →= 4 ,
  4 10 →= 0 ,
  2 5 →= 2 ,
  2 10 →= 7 ,
  5 3 →= 3 ,
  5 5 →= 5 ,
  5 10 →= 10 ,
  5 11 →= 11 ,
  9 3 →= 1 ,
  9 5 →= 9 ,
  9 10 →= 8 ,
  9 11 →= 12 ,
  14 5 →= 14 ,
  14 10 →= 13 ,
  15 6 →= 15 ,
  6 6 →= 6 ,
  6 2 →= 2 ,
  6 7 →= 7 ,
  6 16 →= 16 ,
  3 6 →= 3 ,
  1 6 →= 1 ,
  17 6 →= 17 ,
  0 9 →= 4 ,
  0 8 →= 0 ,
  7 9 →= 2 ,
  7 8 →= 7 ,
  10 1 →= 3 ,
  10 9 →= 5 ,
  10 8 →= 10 ,
  10 12 →= 11 ,
  8 1 →= 1 ,
  8 9 →= 9 ,
  8 8 →= 8 ,
  8 12 →= 12 ,
  13 9 →= 14 ,
  13 8 →= 13 }

The system is trivially terminating.