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

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

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

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

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

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

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

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

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

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

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

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

8 ↦ ⎛     ⎞
    ⎜ 1 2 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

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

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

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

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

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

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

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

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

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

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 2 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

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

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

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

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

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

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

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

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

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

The system is trivially terminating.