/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 1 2 1 ,
  1 ⟶ 0 0 ,
  2 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, (1,↑) ↦ 4, (0,↓) ↦ 5 },
it remains to prove termination of the 9-rule system
{ 0 1 ⟶ 2 1 3 1 ,
  0 1 ⟶ 4 3 1 ,
  0 1 ⟶ 2 1 ,
  4 ⟶ 0 5 ,
  4 ⟶ 0 ,
  5 →= ,
  5 1 →= 3 1 3 1 ,
  1 →= 5 5 ,
  3 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, 3 ↦ 3, 5 ↦ 4 },
it remains to prove termination of the 7-rule system
{ 0 1 ⟶ 2 3 1 ,
  2 ⟶ 0 4 ,
  2 ⟶ 0 ,
  4 →= ,
  4 1 →= 3 1 3 1 ,
  1 →= 4 4 ,
  3 3 →= }

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, (2,3) ↦ 4, (3,1) ↦ 5, (1,3) ↦ 6, (1,4) ↦ 7, (1,6) ↦ 8, (2,1) ↦ 9, (0,4) ↦ 10, (4,1) ↦ 11, (4,3) ↦ 12, (2,4) ↦ 13, (4,4) ↦ 14, (2,6) ↦ 15, (4,6) ↦ 16, (0,3) ↦ 17, (0,6) ↦ 18, (3,4) ↦ 19, (3,3) ↦ 20, (3,6) ↦ 21, (5,4) ↦ 22, (5,1) ↦ 23, (5,3) ↦ 24, (5,6) ↦ 25 },
it remains to prove termination of the 108-rule system
{ 0 1 2 ⟶ 3 4 5 2 ,
  0 1 6 ⟶ 3 4 5 6 ,
  0 1 7 ⟶ 3 4 5 7 ,
  0 1 8 ⟶ 3 4 5 8 ,
  3 9 ⟶ 0 10 11 ,
  3 4 ⟶ 0 10 12 ,
  3 13 ⟶ 0 10 14 ,
  3 15 ⟶ 0 10 16 ,
  3 9 ⟶ 0 1 ,
  3 4 ⟶ 0 17 ,
  3 13 ⟶ 0 10 ,
  3 15 ⟶ 0 18 ,
  10 11 →= 1 ,
  10 12 →= 17 ,
  10 14 →= 10 ,
  10 16 →= 18 ,
  7 11 →= 2 ,
  7 12 →= 6 ,
  7 14 →= 7 ,
  7 16 →= 8 ,
  13 11 →= 9 ,
  13 12 →= 4 ,
  13 14 →= 13 ,
  13 16 →= 15 ,
  19 11 →= 5 ,
  19 12 →= 20 ,
  19 14 →= 19 ,
  19 16 →= 21 ,
  14 11 →= 11 ,
  14 12 →= 12 ,
  14 14 →= 14 ,
  14 16 →= 16 ,
  22 11 →= 23 ,
  22 12 →= 24 ,
  22 14 →= 22 ,
  22 16 →= 25 ,
  10 11 2 →= 17 5 6 5 2 ,
  10 11 6 →= 17 5 6 5 6 ,
  10 11 7 →= 17 5 6 5 7 ,
  10 11 8 →= 17 5 6 5 8 ,
  7 11 2 →= 6 5 6 5 2 ,
  7 11 6 →= 6 5 6 5 6 ,
  7 11 7 →= 6 5 6 5 7 ,
  7 11 8 →= 6 5 6 5 8 ,
  13 11 2 →= 4 5 6 5 2 ,
  13 11 6 →= 4 5 6 5 6 ,
  13 11 7 →= 4 5 6 5 7 ,
  13 11 8 →= 4 5 6 5 8 ,
  19 11 2 →= 20 5 6 5 2 ,
  19 11 6 →= 20 5 6 5 6 ,
  19 11 7 →= 20 5 6 5 7 ,
  19 11 8 →= 20 5 6 5 8 ,
  14 11 2 →= 12 5 6 5 2 ,
  14 11 6 →= 12 5 6 5 6 ,
  14 11 7 →= 12 5 6 5 7 ,
  14 11 8 →= 12 5 6 5 8 ,
  22 11 2 →= 24 5 6 5 2 ,
  22 11 6 →= 24 5 6 5 6 ,
  22 11 7 →= 24 5 6 5 7 ,
  22 11 8 →= 24 5 6 5 8 ,
  1 2 →= 10 14 11 ,
  1 6 →= 10 14 12 ,
  1 7 →= 10 14 14 ,
  1 8 →= 10 14 16 ,
  2 2 →= 7 14 11 ,
  2 6 →= 7 14 12 ,
  2 7 →= 7 14 14 ,
  2 8 →= 7 14 16 ,
  9 2 →= 13 14 11 ,
  9 6 →= 13 14 12 ,
  9 7 →= 13 14 14 ,
  9 8 →= 13 14 16 ,
  5 2 →= 19 14 11 ,
  5 6 →= 19 14 12 ,
  5 7 →= 19 14 14 ,
  5 8 →= 19 14 16 ,
  11 2 →= 14 14 11 ,
  11 6 →= 14 14 12 ,
  11 7 →= 14 14 14 ,
  11 8 →= 14 14 16 ,
  23 2 →= 22 14 11 ,
  23 6 →= 22 14 12 ,
  23 7 →= 22 14 14 ,
  23 8 →= 22 14 16 ,
  17 20 5 →= 1 ,
  17 20 20 →= 17 ,
  17 20 19 →= 10 ,
  17 20 21 →= 18 ,
  6 20 5 →= 2 ,
  6 20 20 →= 6 ,
  6 20 19 →= 7 ,
  6 20 21 →= 8 ,
  4 20 5 →= 9 ,
  4 20 20 →= 4 ,
  4 20 19 →= 13 ,
  4 20 21 →= 15 ,
  20 20 5 →= 5 ,
  20 20 20 →= 20 ,
  20 20 19 →= 19 ,
  20 20 21 →= 21 ,
  12 20 5 →= 11 ,
  12 20 20 →= 12 ,
  12 20 19 →= 14 ,
  12 20 21 →= 16 ,
  24 20 5 →= 23 ,
  24 20 20 →= 24 ,
  24 20 19 →= 22 ,
  24 20 21 →= 25 }

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 4 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

2 ↦ ⎛       ⎞
    ⎜  1 10 ⎟
    ⎜  0  1 ⎟
    ⎝       ⎠

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

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

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

6 ↦ ⎛     ⎞
    ⎜ 1 5 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

7 ↦ ⎛     ⎞
    ⎜ 1 6 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

8 ↦ ⎛     ⎞
    ⎜ 1 6 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

9 ↦ ⎛     ⎞
    ⎜ 1 4 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

11 ↦ ⎛     ⎞
     ⎜ 1 6 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

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

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

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

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

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

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

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

19 ↦ ⎛     ⎞
     ⎜ 1 4 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

20 ↦ ⎛     ⎞
     ⎜ 1 5 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

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

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

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

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

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

After renaming modulo the bijection { 10 ↦ 0, 14 ↦ 1, 7 ↦ 2, 13 ↦ 3, 19 ↦ 4, 12 ↦ 5, 20 ↦ 6, 11 ↦ 7, 16 ↦ 8, 22 ↦ 9, 2 ↦ 10, 5 ↦ 11, 6 ↦ 12, 8 ↦ 13 },
it remains to prove termination of the 22-rule system
{ 0 1 →= 0 ,
  2 1 →= 2 ,
  3 1 →= 3 ,
  4 5 →= 6 ,
  4 1 →= 4 ,
  1 7 →= 7 ,
  1 5 →= 5 ,
  1 1 →= 1 ,
  1 8 →= 8 ,
  9 1 →= 9 ,
  4 7 10 →= 6 11 12 11 10 ,
  4 7 12 →= 6 11 12 11 12 ,
  4 7 2 →= 6 11 12 11 2 ,
  4 7 13 →= 6 11 12 11 13 ,
  1 7 10 →= 5 11 12 11 10 ,
  1 7 12 →= 5 11 12 11 12 ,
  1 7 2 →= 5 11 12 11 2 ,
  1 7 13 →= 5 11 12 11 13 ,
  11 10 →= 4 1 7 ,
  11 12 →= 4 1 5 ,
  12 6 11 →= 10 ,
  5 6 11 →= 7 }

The system is trivially terminating.