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

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

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

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

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

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

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

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

15 ↦ ⎛     ⎞
     ⎜ 1 1 ⎟
     ⎜ 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 { 15 ↦ 0, 5 ↦ 1, 2 ↦ 2, 3 ↦ 3, 6 ↦ 4, 14 ↦ 5, 9 ↦ 6, 7 ↦ 7, 4 ↦ 8, 8 ↦ 9, 11 ↦ 10, 16 ↦ 11, 1 ↦ 12, 18 ↦ 13, 19 ↦ 14, 12 ↦ 15, 10 ↦ 16, 17 ↦ 17 },
it remains to prove termination of the 30-rule system
{ 0 1 2 3 →= 2 4 3 5 0 6 1 ,
  0 1 2 4 →= 2 4 3 5 0 6 7 ,
  6 1 2 3 →= 7 4 3 5 0 6 1 ,
  6 1 2 4 →= 7 4 3 5 0 6 7 ,
  8 6 →= 8 ,
  0 7 →= 2 ,
  0 6 →= 0 ,
  9 6 →= 9 ,
  6 1 →= 1 ,
  6 7 →= 7 ,
  6 6 →= 6 ,
  6 10 →= 10 ,
  11 6 →= 11 ,
  12 5 →= 12 ,
  5 5 →= 5 ,
  5 2 →= 2 ,
  5 0 →= 0 ,
  5 13 →= 13 ,
  3 5 →= 3 ,
  1 5 →= 1 ,
  14 5 →= 14 ,
  15 4 →= 15 ,
  2 4 →= 2 ,
  4 3 →= 3 ,
  4 4 →= 4 ,
  4 9 →= 9 ,
  4 16 →= 16 ,
  7 3 →= 1 ,
  7 4 →= 7 ,
  17 4 →= 17 }

The system is trivially terminating.