/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 { b ↦ 0, d ↦ 1, c ↦ 2, a ↦ 3 },
it remains to prove termination of the 7-rule system
{ 0 1 0 ⟶ 2 1 0 ,
  0 3 2 ⟶ 0 2 ,
  3 1 ⟶ 1 2 ,
  0 0 0 ⟶ 3 0 2 ,
  1 2 ⟶ 0 1 ,
  1 2 ⟶ 1 0 1 ,
  1 3 2 ⟶ 0 0 }

The system was reversed.

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

Applying sparse untiling TRFCU(2) [Geser/Hofbauer/Waldmann, FSCD 2019].

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

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 1 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

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

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

After renaming modulo the bijection { (0,↑) ↦ 0, (1,↓) ↦ 1, (0,↓) ↦ 2, (2,↓) ↦ 3, (2,↑) ↦ 4, (3,↓) ↦ 5 },
it remains to prove termination of the 13-rule system
{ 0 1 2 ⟶ 0 1 3 ,
  0 1 2 ⟶ 4 ,
  0 2 2 ⟶ 4 2 5 ,
  0 2 2 ⟶ 0 5 ,
  4 1 ⟶ 0 ,
  4 1 ⟶ 0 1 ,
  4 5 1 ⟶ 0 2 ,
  4 5 1 ⟶ 0 ,
  2 1 2 →= 2 1 3 ,
  2 2 2 →= 3 2 5 ,
  3 1 →= 1 2 ,
  3 1 →= 1 2 1 ,
  3 5 1 →= 2 2 }

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 0 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

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

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

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

After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5 },
it remains to prove termination of the 10-rule system
{ 0 1 2 ⟶ 0 1 3 ,
  0 2 2 ⟶ 4 2 5 ,
  4 1 ⟶ 0 ,
  4 1 ⟶ 0 1 ,
  4 5 1 ⟶ 0 2 ,
  2 1 2 →= 2 1 3 ,
  2 2 2 →= 3 2 5 ,
  3 1 →= 1 2 ,
  3 1 →= 1 2 1 ,
  3 5 1 →= 2 2 }

The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 3:

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

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

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

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

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

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

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

The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 4:

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

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

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

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

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

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

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

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 0 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 7:

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

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

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

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

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

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 2 ⟶ 0 1 3 ,
  2 1 2 →= 2 1 3 ,
  2 2 2 →= 3 2 4 ,
  3 1 →= 1 2 ,
  3 1 →= 1 2 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 0 ⎟
    ⎜ 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 },
it remains to prove termination of the 4-rule system
{ 0 1 2 ⟶ 0 1 3 ,
  2 1 2 →= 2 1 3 ,
  3 1 →= 1 2 ,
  3 1 →= 1 2 1 }

The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 4:

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

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

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

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

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

The system is trivially terminating.