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

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

After renaming modulo the bijection { (0,↑) ↦ 0, (1,↓) ↦ 1, (2,↑) ↦ 2, (3,↓) ↦ 3, (3,↑) ↦ 4, (0,↓) ↦ 5, (2,↓) ↦ 6, (4,↓) ↦ 7 },
it remains to prove termination of the 12-rule system
{ 0 1 ⟶ 2 ,
  2 1 ⟶ 0 3 ,
  2 1 ⟶ 4 ,
  4 5 ⟶ 0 3 ,
  4 5 ⟶ 4 ,
  4 6 ⟶ 2 3 ,
  4 6 ⟶ 4 ,
  5 1 →= 1 6 ,
  6 1 →= 5 3 ,
  3 5 →= 5 3 ,
  3 6 →= 6 3 ,
  3 7 →= 1 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 { 2 ↦ 0, 1 ↦ 1, 0 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7 },
it remains to prove termination of the 8-rule system
{ 0 1 ⟶ 2 3 ,
  4 5 ⟶ 2 3 ,
  4 6 ⟶ 0 3 ,
  5 1 →= 1 6 ,
  6 1 →= 5 3 ,
  3 5 →= 5 3 ,
  3 6 →= 6 3 ,
  3 7 →= 1 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 0 ⎟
    ⎜ 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 { 4 ↦ 0, 6 ↦ 1, 0 ↦ 2, 3 ↦ 3, 5 ↦ 4, 1 ↦ 5, 7 ↦ 6 },
it remains to prove termination of the 6-rule system
{ 0 1 ⟶ 2 3 ,
  4 5 →= 5 1 ,
  1 5 →= 4 3 ,
  3 4 →= 4 3 ,
  3 1 →= 1 3 ,
  3 6 →= 5 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 0 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

3 ↦ ⎛     ⎞
    ⎜ 1 1 ⎟
    ⎜ 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, 1 ↦ 2, 3 ↦ 3, 6 ↦ 4 },
it remains to prove termination of the 5-rule system
{ 0 1 →= 1 2 ,
  2 1 →= 0 3 ,
  3 0 →= 0 3 ,
  3 2 →= 2 3 ,
  3 4 →= 1 4 }

The system is trivially terminating.