/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 ,
  0 1 ⟶ 0 2 0 2 ,
  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 ⟶ ,
  0 ⟶ 1 1 ,
  1 0 ⟶ 2 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, (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 ,
  0 0 ⟶ 1 5 4 ,
  0 1 ⟶ 1 5 5 ,
  0 2 ⟶ 1 5 6 ,
  0 3 ⟶ 1 5 7 ,
  4 0 ⟶ 5 5 4 ,
  4 1 ⟶ 5 5 5 ,
  4 2 ⟶ 5 5 6 ,
  4 3 ⟶ 5 5 7 ,
  8 0 ⟶ 9 5 4 ,
  8 1 ⟶ 9 5 5 ,
  8 2 ⟶ 9 5 6 ,
  8 3 ⟶ 9 5 7 ,
  12 0 ⟶ 13 5 4 ,
  12 1 ⟶ 13 5 5 ,
  12 2 ⟶ 13 5 6 ,
  12 3 ⟶ 13 5 7 ,
  1 4 0 ⟶ 2 8 2 8 0 ,
  1 4 1 ⟶ 2 8 2 8 1 ,
  1 4 2 ⟶ 2 8 2 8 2 ,
  1 4 3 ⟶ 2 8 2 8 3 ,
  5 4 0 ⟶ 6 8 2 8 0 ,
  5 4 1 ⟶ 6 8 2 8 1 ,
  5 4 2 ⟶ 6 8 2 8 2 ,
  5 4 3 ⟶ 6 8 2 8 3 ,
  9 4 0 ⟶ 10 8 2 8 0 ,
  9 4 1 ⟶ 10 8 2 8 1 ,
  9 4 2 ⟶ 10 8 2 8 2 ,
  9 4 3 ⟶ 10 8 2 8 3 ,
  13 4 0 ⟶ 14 8 2 8 0 ,
  13 4 1 ⟶ 14 8 2 8 1 ,
  13 4 2 ⟶ 14 8 2 8 2 ,
  13 4 3 ⟶ 14 8 2 8 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 3 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

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

4 ↦ ⎛     ⎞
    ⎜ 1 2 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

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

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

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

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

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

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

12 ↦ ⎛     ⎞
     ⎜ 1 0 ⎟
     ⎜ 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, 0 ↦ 3, 9 ↦ 4, 5 ↦ 5, 4 ↦ 6, 6 ↦ 7, 1 ↦ 8, 3 ↦ 9 },
it remains to prove termination of the 13-rule system
{ 0 1 ⟶ 2 ,
  0 3 ⟶ 4 5 6 ,
  0 1 ⟶ 4 5 7 ,
  5 6 3 ⟶ 7 0 1 0 3 ,
  5 6 8 ⟶ 7 0 1 0 8 ,
  5 6 1 ⟶ 7 0 1 0 1 ,
  5 6 9 ⟶ 7 0 1 0 9 ,
  4 6 3 ⟶ 2 0 1 0 3 ,
  4 6 8 ⟶ 2 0 1 0 8 ,
  4 6 1 ⟶ 2 0 1 0 1 ,
  4 6 9 ⟶ 2 0 1 0 9 ,
  1 2 0 ⟶ 3 ,
  7 2 0 ⟶ 6 }

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

After renaming modulo the bijection { (0,↑) ↦ 0, (3,↓) ↦ 1, (4,↑) ↦ 2, (5,↓) ↦ 3, (6,↓) ↦ 4, (5,↑) ↦ 5, (1,↓) ↦ 6, (7,↓) ↦ 7, (7,↑) ↦ 8, (0,↓) ↦ 9, (1,↑) ↦ 10, (8,↓) ↦ 11, (9,↓) ↦ 12, (2,↓) ↦ 13, (4,↓) ↦ 14 },
it remains to prove termination of the 46-rule system
{ 0 1 ⟶ 2 3 4 ,
  0 1 ⟶ 5 4 ,
  0 6 ⟶ 2 3 7 ,
  0 6 ⟶ 5 7 ,
  0 6 ⟶ 8 ,
  5 4 1 ⟶ 8 9 6 9 1 ,
  5 4 1 ⟶ 0 6 9 1 ,
  5 4 1 ⟶ 10 9 1 ,
  5 4 1 ⟶ 0 1 ,
  5 4 11 ⟶ 8 9 6 9 11 ,
  5 4 11 ⟶ 0 6 9 11 ,
  5 4 11 ⟶ 10 9 11 ,
  5 4 11 ⟶ 0 11 ,
  5 4 6 ⟶ 8 9 6 9 6 ,
  5 4 6 ⟶ 0 6 9 6 ,
  5 4 6 ⟶ 10 9 6 ,
  5 4 6 ⟶ 0 6 ,
  5 4 12 ⟶ 8 9 6 9 12 ,
  5 4 12 ⟶ 0 6 9 12 ,
  5 4 12 ⟶ 10 9 12 ,
  5 4 12 ⟶ 0 12 ,
  2 4 1 ⟶ 0 6 9 1 ,
  2 4 1 ⟶ 10 9 1 ,
  2 4 1 ⟶ 0 1 ,
  2 4 11 ⟶ 0 6 9 11 ,
  2 4 11 ⟶ 10 9 11 ,
  2 4 11 ⟶ 0 11 ,
  2 4 6 ⟶ 0 6 9 6 ,
  2 4 6 ⟶ 10 9 6 ,
  2 4 6 ⟶ 0 6 ,
  2 4 12 ⟶ 0 6 9 12 ,
  2 4 12 ⟶ 10 9 12 ,
  2 4 12 ⟶ 0 12 ,
  9 6 →= 13 ,
  9 1 →= 14 3 4 ,
  9 6 →= 14 3 7 ,
  3 4 1 →= 7 9 6 9 1 ,
  3 4 11 →= 7 9 6 9 11 ,
  3 4 6 →= 7 9 6 9 6 ,
  3 4 12 →= 7 9 6 9 12 ,
  14 4 1 →= 13 9 6 9 1 ,
  14 4 11 →= 13 9 6 9 11 ,
  14 4 6 →= 13 9 6 9 6 ,
  14 4 12 →= 13 9 6 9 12 ,
  6 13 9 →= 1 ,
  7 13 9 →= 4 }

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 1 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

4 ↦ ⎛     ⎞
    ⎜ 1 2 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

6 ↦ ⎛     ⎞
    ⎜ 1 2 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

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

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

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

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

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

13 ↦ ⎛     ⎞
     ⎜ 1 2 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

14 ↦ ⎛     ⎞
     ⎜ 1 2 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

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

The system is trivially terminating.