/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 },
it remains to prove termination of the 3-rule system
{ 0 ⟶ ,
  0 0 ⟶ 0 1 0 1 ,
  1 1 1 1 ⟶ 0 }

The system was reversed.

After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1 },
it remains to prove termination of the 3-rule system
{ 0 ⟶ ,
  0 0 ⟶ 1 0 1 0 ,
  1 1 1 1 ⟶ 0 }

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

After renaming modulo the bijection { (0,0) ↦ 0, (0,1) ↦ 1, (0,3) ↦ 2, (1,0) ↦ 3, (1,1) ↦ 4, (1,3) ↦ 5, (2,0) ↦ 6, (2,1) ↦ 7, (2,3) ↦ 8 },
it remains to prove termination of the 27-rule system
{ 0 0 ⟶ 0 ,
  0 1 ⟶ 1 ,
  0 2 ⟶ 2 ,
  3 0 ⟶ 3 ,
  3 1 ⟶ 4 ,
  3 2 ⟶ 5 ,
  6 0 ⟶ 6 ,
  6 1 ⟶ 7 ,
  6 2 ⟶ 8 ,
  0 0 0 ⟶ 1 3 1 3 0 ,
  0 0 1 ⟶ 1 3 1 3 1 ,
  0 0 2 ⟶ 1 3 1 3 2 ,
  3 0 0 ⟶ 4 3 1 3 0 ,
  3 0 1 ⟶ 4 3 1 3 1 ,
  3 0 2 ⟶ 4 3 1 3 2 ,
  6 0 0 ⟶ 7 3 1 3 0 ,
  6 0 1 ⟶ 7 3 1 3 1 ,
  6 0 2 ⟶ 7 3 1 3 2 ,
  1 4 4 4 3 ⟶ 0 0 ,
  1 4 4 4 4 ⟶ 0 1 ,
  1 4 4 4 5 ⟶ 0 2 ,
  4 4 4 4 3 ⟶ 3 0 ,
  4 4 4 4 4 ⟶ 3 1 ,
  4 4 4 4 5 ⟶ 3 2 ,
  7 4 4 4 3 ⟶ 6 0 ,
  7 4 4 4 4 ⟶ 6 1 ,
  7 4 4 4 5 ⟶ 6 2 }

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 1 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

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

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

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

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

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

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

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

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

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

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 0 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

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

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

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 1 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

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

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

After renaming modulo the bijection { 3 ↦ 0, 2 ↦ 1, 5 ↦ 2, 1 ↦ 3, 4 ↦ 4 },
it remains to prove termination of the 5-rule system
{ 0 1 →= 2 ,
  0 3 3 →= 2 0 1 0 3 ,
  0 3 1 →= 2 0 1 0 1 ,
  0 3 4 →= 2 0 1 0 4 ,
  1 2 2 2 0 →= 3 3 }

The system is trivially terminating.