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

The system was reversed.

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

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

After renaming modulo the bijection { (0,0) ↦ 0, (0,1) ↦ 1, (1,0) ↦ 2, (1,1) ↦ 3, (0,2) ↦ 4, (1,2) ↦ 5, (0,4) ↦ 6, (1,4) ↦ 7, (2,0) ↦ 8, (2,1) ↦ 9, (3,0) ↦ 10, (3,1) ↦ 11, (2,2) ↦ 12, (2,4) ↦ 13, (3,2) ↦ 14, (3,4) ↦ 15 },
it remains to prove termination of the 64-rule system
{ 0 0 ⟶ 1 2 ,
  0 1 ⟶ 1 3 ,
  0 4 ⟶ 1 5 ,
  0 6 ⟶ 1 7 ,
  2 0 ⟶ 3 2 ,
  2 1 ⟶ 3 3 ,
  2 4 ⟶ 3 5 ,
  2 6 ⟶ 3 7 ,
  8 0 ⟶ 9 2 ,
  8 1 ⟶ 9 3 ,
  8 4 ⟶ 9 5 ,
  8 6 ⟶ 9 7 ,
  10 0 ⟶ 11 2 ,
  10 1 ⟶ 11 3 ,
  10 4 ⟶ 11 5 ,
  10 6 ⟶ 11 7 ,
  1 2 0 ⟶ 0 ,
  1 2 1 ⟶ 1 ,
  1 2 4 ⟶ 4 ,
  1 2 6 ⟶ 6 ,
  3 2 0 ⟶ 2 ,
  3 2 1 ⟶ 3 ,
  3 2 4 ⟶ 5 ,
  3 2 6 ⟶ 7 ,
  9 2 0 ⟶ 8 ,
  9 2 1 ⟶ 9 ,
  9 2 4 ⟶ 12 ,
  9 2 6 ⟶ 13 ,
  11 2 0 ⟶ 10 ,
  11 2 1 ⟶ 11 ,
  11 2 4 ⟶ 14 ,
  11 2 6 ⟶ 15 ,
  0 1 2 ⟶ 1 2 4 8 0 ,
  0 1 3 ⟶ 1 2 4 8 1 ,
  0 1 5 ⟶ 1 2 4 8 4 ,
  0 1 7 ⟶ 1 2 4 8 6 ,
  2 1 2 ⟶ 3 2 4 8 0 ,
  2 1 3 ⟶ 3 2 4 8 1 ,
  2 1 5 ⟶ 3 2 4 8 4 ,
  2 1 7 ⟶ 3 2 4 8 6 ,
  8 1 2 ⟶ 9 2 4 8 0 ,
  8 1 3 ⟶ 9 2 4 8 1 ,
  8 1 5 ⟶ 9 2 4 8 4 ,
  8 1 7 ⟶ 9 2 4 8 6 ,
  10 1 2 ⟶ 11 2 4 8 0 ,
  10 1 3 ⟶ 11 2 4 8 1 ,
  10 1 5 ⟶ 11 2 4 8 4 ,
  10 1 7 ⟶ 11 2 4 8 6 ,
  4 12 8 ⟶ 0 ,
  4 12 9 ⟶ 1 ,
  4 12 12 ⟶ 4 ,
  4 12 13 ⟶ 6 ,
  5 12 8 ⟶ 2 ,
  5 12 9 ⟶ 3 ,
  5 12 12 ⟶ 5 ,
  5 12 13 ⟶ 7 ,
  12 12 8 ⟶ 8 ,
  12 12 9 ⟶ 9 ,
  12 12 12 ⟶ 12 ,
  12 12 13 ⟶ 13 ,
  14 12 8 ⟶ 10 ,
  14 12 9 ⟶ 11 ,
  14 12 12 ⟶ 14 ,
  14 12 13 ⟶ 15 }

The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 2:

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

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

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

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

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

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

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

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

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

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

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

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

12 ↦ ⎛     ⎞
     ⎜ 1 2 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

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

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

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

After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 12 ↦ 10 },
it remains to prove termination of the 29-rule system
{ 0 0 ⟶ 1 2 ,
  0 1 ⟶ 1 3 ,
  0 4 ⟶ 1 5 ,
  0 6 ⟶ 1 7 ,
  2 0 ⟶ 3 2 ,
  2 1 ⟶ 3 3 ,
  2 4 ⟶ 3 5 ,
  2 6 ⟶ 3 7 ,
  8 0 ⟶ 9 2 ,
  8 1 ⟶ 9 3 ,
  8 4 ⟶ 9 5 ,
  8 6 ⟶ 9 7 ,
  9 2 4 ⟶ 10 ,
  0 1 2 ⟶ 1 2 4 8 0 ,
  0 1 3 ⟶ 1 2 4 8 1 ,
  0 1 5 ⟶ 1 2 4 8 4 ,
  0 1 7 ⟶ 1 2 4 8 6 ,
  2 1 2 ⟶ 3 2 4 8 0 ,
  2 1 3 ⟶ 3 2 4 8 1 ,
  2 1 5 ⟶ 3 2 4 8 4 ,
  2 1 7 ⟶ 3 2 4 8 6 ,
  8 1 2 ⟶ 9 2 4 8 0 ,
  8 1 3 ⟶ 9 2 4 8 1 ,
  8 1 5 ⟶ 9 2 4 8 4 ,
  8 1 7 ⟶ 9 2 4 8 6 ,
  4 10 8 ⟶ 0 ,
  4 10 9 ⟶ 1 ,
  5 10 8 ⟶ 2 ,
  5 10 9 ⟶ 3 }

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

After renaming modulo the bijection { (0,↑) ↦ 0, (0,↓) ↦ 1, (2,↑) ↦ 2, (4,↓) ↦ 3, (5,↑) ↦ 4, (8,↑) ↦ 5, (9,↑) ↦ 6, (2,↓) ↦ 7, (1,↓) ↦ 8, (3,↓) ↦ 9, (5,↓) ↦ 10, (6,↓) ↦ 11, (7,↓) ↦ 12, (8,↓) ↦ 13, (4,↑) ↦ 14, (10,↓) ↦ 15, (9,↓) ↦ 16 },
it remains to prove termination of the 87-rule system
{ 0 1 ⟶ 2 ,
  0 3 ⟶ 4 ,
  2 1 ⟶ 2 ,
  2 3 ⟶ 4 ,
  5 1 ⟶ 6 7 ,
  5 1 ⟶ 2 ,
  5 8 ⟶ 6 9 ,
  5 3 ⟶ 6 10 ,
  5 3 ⟶ 4 ,
  5 11 ⟶ 6 12 ,
  0 8 7 ⟶ 2 3 13 1 ,
  0 8 7 ⟶ 14 13 1 ,
  0 8 7 ⟶ 5 1 ,
  0 8 7 ⟶ 0 ,
  0 8 9 ⟶ 2 3 13 8 ,
  0 8 9 ⟶ 14 13 8 ,
  0 8 9 ⟶ 5 8 ,
  0 8 10 ⟶ 2 3 13 3 ,
  0 8 10 ⟶ 14 13 3 ,
  0 8 10 ⟶ 5 3 ,
  0 8 10 ⟶ 14 ,
  0 8 12 ⟶ 2 3 13 11 ,
  0 8 12 ⟶ 14 13 11 ,
  0 8 12 ⟶ 5 11 ,
  2 8 7 ⟶ 2 3 13 1 ,
  2 8 7 ⟶ 14 13 1 ,
  2 8 7 ⟶ 5 1 ,
  2 8 7 ⟶ 0 ,
  2 8 9 ⟶ 2 3 13 8 ,
  2 8 9 ⟶ 14 13 8 ,
  2 8 9 ⟶ 5 8 ,
  2 8 10 ⟶ 2 3 13 3 ,
  2 8 10 ⟶ 14 13 3 ,
  2 8 10 ⟶ 5 3 ,
  2 8 10 ⟶ 14 ,
  2 8 12 ⟶ 2 3 13 11 ,
  2 8 12 ⟶ 14 13 11 ,
  2 8 12 ⟶ 5 11 ,
  5 8 7 ⟶ 6 7 3 13 1 ,
  5 8 7 ⟶ 2 3 13 1 ,
  5 8 7 ⟶ 14 13 1 ,
  5 8 7 ⟶ 5 1 ,
  5 8 7 ⟶ 0 ,
  5 8 9 ⟶ 6 7 3 13 8 ,
  5 8 9 ⟶ 2 3 13 8 ,
  5 8 9 ⟶ 14 13 8 ,
  5 8 9 ⟶ 5 8 ,
  5 8 10 ⟶ 6 7 3 13 3 ,
  5 8 10 ⟶ 2 3 13 3 ,
  5 8 10 ⟶ 14 13 3 ,
  5 8 10 ⟶ 5 3 ,
  5 8 10 ⟶ 14 ,
  5 8 12 ⟶ 6 7 3 13 11 ,
  5 8 12 ⟶ 2 3 13 11 ,
  5 8 12 ⟶ 14 13 11 ,
  5 8 12 ⟶ 5 11 ,
  14 15 13 ⟶ 0 ,
  4 15 13 ⟶ 2 ,
  1 1 →= 8 7 ,
  1 8 →= 8 9 ,
  1 3 →= 8 10 ,
  1 11 →= 8 12 ,
  7 1 →= 9 7 ,
  7 8 →= 9 9 ,
  7 3 →= 9 10 ,
  7 11 →= 9 12 ,
  13 1 →= 16 7 ,
  13 8 →= 16 9 ,
  13 3 →= 16 10 ,
  13 11 →= 16 12 ,
  16 7 3 →= 15 ,
  1 8 7 →= 8 7 3 13 1 ,
  1 8 9 →= 8 7 3 13 8 ,
  1 8 10 →= 8 7 3 13 3 ,
  1 8 12 →= 8 7 3 13 11 ,
  7 8 7 →= 9 7 3 13 1 ,
  7 8 9 →= 9 7 3 13 8 ,
  7 8 10 →= 9 7 3 13 3 ,
  7 8 12 →= 9 7 3 13 11 ,
  13 8 7 →= 16 7 3 13 1 ,
  13 8 9 →= 16 7 3 13 8 ,
  13 8 10 →= 16 7 3 13 3 ,
  13 8 12 →= 16 7 3 13 11 ,
  3 15 13 →= 1 ,
  3 15 16 →= 8 ,
  10 15 13 →= 7 ,
  10 15 16 →= 9 }

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 2 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

9 ↦ ⎛     ⎞
    ⎜ 1 2 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

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

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

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

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

15 ↦ ⎛     ⎞
     ⎜ 1 2 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

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

After renaming modulo the bijection { 1 ↦ 0, 8 ↦ 1, 7 ↦ 2, 9 ↦ 3, 3 ↦ 4, 10 ↦ 5, 11 ↦ 6, 12 ↦ 7, 13 ↦ 8, 16 ↦ 9, 15 ↦ 10 },
it remains to prove termination of the 29-rule system
{ 0 0 →= 1 2 ,
  0 1 →= 1 3 ,
  0 4 →= 1 5 ,
  0 6 →= 1 7 ,
  2 0 →= 3 2 ,
  2 1 →= 3 3 ,
  2 4 →= 3 5 ,
  2 6 →= 3 7 ,
  8 0 →= 9 2 ,
  8 1 →= 9 3 ,
  8 4 →= 9 5 ,
  8 6 →= 9 7 ,
  9 2 4 →= 10 ,
  0 1 2 →= 1 2 4 8 0 ,
  0 1 3 →= 1 2 4 8 1 ,
  0 1 5 →= 1 2 4 8 4 ,
  0 1 7 →= 1 2 4 8 6 ,
  2 1 2 →= 3 2 4 8 0 ,
  2 1 3 →= 3 2 4 8 1 ,
  2 1 5 →= 3 2 4 8 4 ,
  2 1 7 →= 3 2 4 8 6 ,
  8 1 2 →= 9 2 4 8 0 ,
  8 1 3 →= 9 2 4 8 1 ,
  8 1 5 →= 9 2 4 8 4 ,
  8 1 7 →= 9 2 4 8 6 ,
  4 10 8 →= 0 ,
  4 10 9 →= 1 ,
  5 10 8 →= 2 ,
  5 10 9 →= 3 }

The system is trivially terminating.