/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/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 2 ⟶ 1 1 0 0 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 ,
  2 3 ⟶ 4 1 5 5 3 ,
  2 3 ⟶ 4 5 5 3 ,
  2 3 ⟶ 0 5 3 ,
  2 3 ⟶ 0 3 ,
  5 →= ,
  5 1 →= 3 ,
  1 →= ,
  3 3 →= 1 1 5 5 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 1 ⎟
    ⎜ 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, 3 ↦ 3, 5 ↦ 4 },
it remains to prove termination of the 7-rule system
{ 0 1 ⟶ 2 ,
  2 3 ⟶ 0 4 3 ,
  2 3 ⟶ 0 3 ,
  4 →= ,
  4 1 →= 3 ,
  1 →= ,
  3 3 →= 1 1 4 4 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, (2,1) ↦ 4, (1,3) ↦ 5, (2,3) ↦ 6, (1,4) ↦ 7, (2,4) ↦ 8, (3,1) ↦ 9, (0,4) ↦ 10, (4,3) ↦ 11, (3,3) ↦ 12, (3,4) ↦ 13, (3,6) ↦ 14, (0,3) ↦ 15, (4,1) ↦ 16, (4,4) ↦ 17, (5,4) ↦ 18, (5,1) ↦ 19, (5,3) ↦ 20 },
it remains to prove termination of the 89-rule system
{ 0 1 2 ⟶ 3 4 ,
  0 1 5 ⟶ 3 6 ,
  0 1 7 ⟶ 3 8 ,
  3 6 9 ⟶ 0 10 11 9 ,
  3 6 12 ⟶ 0 10 11 12 ,
  3 6 13 ⟶ 0 10 11 13 ,
  3 6 14 ⟶ 0 10 11 14 ,
  3 6 9 ⟶ 0 15 9 ,
  3 6 12 ⟶ 0 15 12 ,
  3 6 13 ⟶ 0 15 13 ,
  3 6 14 ⟶ 0 15 14 ,
  10 16 →= 1 ,
  10 11 →= 15 ,
  10 17 →= 10 ,
  7 16 →= 2 ,
  7 11 →= 5 ,
  7 17 →= 7 ,
  8 16 →= 4 ,
  8 11 →= 6 ,
  8 17 →= 8 ,
  13 16 →= 9 ,
  13 11 →= 12 ,
  13 17 →= 13 ,
  17 16 →= 16 ,
  17 11 →= 11 ,
  17 17 →= 17 ,
  18 16 →= 19 ,
  18 11 →= 20 ,
  18 17 →= 18 ,
  10 16 2 →= 15 9 ,
  10 16 5 →= 15 12 ,
  10 16 7 →= 15 13 ,
  7 16 2 →= 5 9 ,
  7 16 5 →= 5 12 ,
  7 16 7 →= 5 13 ,
  8 16 2 →= 6 9 ,
  8 16 5 →= 6 12 ,
  8 16 7 →= 6 13 ,
  13 16 2 →= 12 9 ,
  13 16 5 →= 12 12 ,
  13 16 7 →= 12 13 ,
  17 16 2 →= 11 9 ,
  17 16 5 →= 11 12 ,
  17 16 7 →= 11 13 ,
  18 16 2 →= 20 9 ,
  18 16 5 →= 20 12 ,
  18 16 7 →= 20 13 ,
  1 2 →= 1 ,
  1 5 →= 15 ,
  1 7 →= 10 ,
  2 2 →= 2 ,
  2 5 →= 5 ,
  2 7 →= 7 ,
  4 2 →= 4 ,
  4 5 →= 6 ,
  4 7 →= 8 ,
  9 2 →= 9 ,
  9 5 →= 12 ,
  9 7 →= 13 ,
  16 2 →= 16 ,
  16 5 →= 11 ,
  16 7 →= 17 ,
  19 2 →= 19 ,
  19 5 →= 20 ,
  19 7 →= 18 ,
  15 12 9 →= 1 2 7 17 11 9 ,
  15 12 12 →= 1 2 7 17 11 12 ,
  15 12 13 →= 1 2 7 17 11 13 ,
  15 12 14 →= 1 2 7 17 11 14 ,
  5 12 9 →= 2 2 7 17 11 9 ,
  5 12 12 →= 2 2 7 17 11 12 ,
  5 12 13 →= 2 2 7 17 11 13 ,
  5 12 14 →= 2 2 7 17 11 14 ,
  6 12 9 →= 4 2 7 17 11 9 ,
  6 12 12 →= 4 2 7 17 11 12 ,
  6 12 13 →= 4 2 7 17 11 13 ,
  6 12 14 →= 4 2 7 17 11 14 ,
  12 12 9 →= 9 2 7 17 11 9 ,
  12 12 12 →= 9 2 7 17 11 12 ,
  12 12 13 →= 9 2 7 17 11 13 ,
  12 12 14 →= 9 2 7 17 11 14 ,
  11 12 9 →= 16 2 7 17 11 9 ,
  11 12 12 →= 16 2 7 17 11 12 ,
  11 12 13 →= 16 2 7 17 11 13 ,
  11 12 14 →= 16 2 7 17 11 14 ,
  20 12 9 →= 19 2 7 17 11 9 ,
  20 12 12 →= 19 2 7 17 11 12 ,
  20 12 13 →= 19 2 7 17 11 13 ,
  20 12 14 →= 19 2 7 17 11 14 }

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 3 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

3 ↦ ⎛     ⎞
    ⎜ 1 2 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

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

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

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

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

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

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

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

12 ↦ ⎛     ⎞
     ⎜ 1 6 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

13 ↦ ⎛     ⎞
     ⎜ 1 5 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

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

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

16 ↦ ⎛     ⎞
     ⎜ 1 5 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

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

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

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

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

After renaming modulo the bijection { 10 ↦ 0, 17 ↦ 1, 7 ↦ 2, 11 ↦ 3, 5 ↦ 4, 8 ↦ 5, 13 ↦ 6, 12 ↦ 7, 16 ↦ 8, 18 ↦ 9, 2 ↦ 10, 9 ↦ 11, 1 ↦ 12, 4 ↦ 13, 19 ↦ 14, 14 ↦ 15 },
it remains to prove termination of the 33-rule system
{ 0 1 →= 0 ,
  2 3 →= 4 ,
  2 1 →= 2 ,
  5 1 →= 5 ,
  6 3 →= 7 ,
  6 1 →= 6 ,
  1 8 →= 8 ,
  1 3 →= 3 ,
  1 1 →= 1 ,
  9 1 →= 9 ,
  2 8 10 →= 4 11 ,
  2 8 4 →= 4 7 ,
  2 8 2 →= 4 6 ,
  6 8 10 →= 7 11 ,
  6 8 4 →= 7 7 ,
  6 8 2 →= 7 6 ,
  1 8 10 →= 3 11 ,
  1 8 4 →= 3 7 ,
  1 8 2 →= 3 6 ,
  12 10 →= 12 ,
  10 10 →= 10 ,
  10 4 →= 4 ,
  10 2 →= 2 ,
  13 10 →= 13 ,
  11 10 →= 11 ,
  11 4 →= 7 ,
  11 2 →= 6 ,
  8 10 →= 8 ,
  14 10 →= 14 ,
  3 7 11 →= 8 10 2 1 3 11 ,
  3 7 7 →= 8 10 2 1 3 7 ,
  3 7 6 →= 8 10 2 1 3 6 ,
  3 7 15 →= 8 10 2 1 3 15 }

The system is trivially terminating.