/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 { 0 ↦ 0, * ↦ 1, 1 ↦ 2, # ↦ 3, $ ↦ 4 },
it remains to prove termination of the 7-rule system
{ 0 0 1 1 ⟶ 1 1 2 2 ,
  2 2 1 1 ⟶ 0 0 3 3 ,
  3 3 0 0 ⟶ 0 0 3 3 ,
  3 3 2 2 ⟶ 2 2 3 3 ,
  3 3 4 4 ⟶ 1 1 4 4 ,
  3 3 3 3 ⟶ 3 3 ,
  3 3 1 1 ⟶ 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 1 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

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

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

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

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

After renaming modulo the bijection { (0,↑) ↦ 0, (0,↓) ↦ 1, (1,↓) ↦ 2, (2,↑) ↦ 3, (2,↓) ↦ 4, (3,↓) ↦ 5, (3,↑) ↦ 6, (4,↓) ↦ 7 },
it remains to prove termination of the 19-rule system
{ 0 1 2 2 ⟶ 3 4 ,
  0 1 2 2 ⟶ 3 ,
  3 4 2 2 ⟶ 0 1 5 5 ,
  3 4 2 2 ⟶ 0 5 5 ,
  3 4 2 2 ⟶ 6 5 ,
  3 4 2 2 ⟶ 6 ,
  6 5 1 1 ⟶ 0 1 5 5 ,
  6 5 1 1 ⟶ 0 5 5 ,
  6 5 1 1 ⟶ 6 5 ,
  6 5 1 1 ⟶ 6 ,
  6 5 4 4 ⟶ 3 4 5 5 ,
  6 5 4 4 ⟶ 3 5 5 ,
  6 5 4 4 ⟶ 6 5 ,
  6 5 4 4 ⟶ 6 ,
  1 1 2 2 →= 2 2 4 4 ,
  4 4 2 2 →= 1 1 5 5 ,
  5 5 1 1 →= 1 1 5 5 ,
  5 5 4 4 →= 4 4 5 5 ,
  5 5 7 7 →= 2 2 7 7 }

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 1 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

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

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

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

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

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

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

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 1 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

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

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

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

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

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

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

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 1 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

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

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

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

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

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

The system is trivially terminating.