/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 3-rule system
{ 0 ⟶ ,
  0 1 ⟶ 2 1 2 0 ,
  2 2 ⟶ 2 1 0 }

The system was reversed.

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

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

After renaming modulo the bijection { (1,↑) ↦ 0, (0,↓) ↦ 1, (0,↑) ↦ 2, (2,↓) ↦ 3, (1,↓) ↦ 4, (2,↑) ↦ 5 },
it remains to prove termination of the 9-rule system
{ 0 1 ⟶ 2 3 4 3 ,
  0 1 ⟶ 5 4 3 ,
  0 1 ⟶ 0 3 ,
  0 1 ⟶ 5 ,
  5 3 ⟶ 2 4 3 ,
  5 3 ⟶ 0 3 ,
  1 →= ,
  4 1 →= 1 3 4 3 ,
  3 3 →= 1 4 3 }

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 ⎟
    ⎝     ⎠

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

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

After renaming modulo the bijection { (5,0) ↦ 0, (0,1) ↦ 1, (1,1) ↦ 2, (5,2) ↦ 3, (2,3) ↦ 4, (3,4) ↦ 5, (4,1) ↦ 6, (1,3) ↦ 7, (4,3) ↦ 8, (1,4) ↦ 9, (4,4) ↦ 10, (0,4) ↦ 11, (2,1) ↦ 12, (2,4) ↦ 13, (4,6) ↦ 14, (0,3) ↦ 15, (3,1) ↦ 16, (3,3) ↦ 17, (5,1) ↦ 18, (5,3) ↦ 19, (5,4) ↦ 20 },
it remains to prove termination of the 73-rule system
{ 0 1 2 ⟶ 3 4 5 6 ,
  0 1 7 ⟶ 3 4 5 8 ,
  0 1 9 ⟶ 3 4 5 10 ,
  0 1 2 ⟶ 0 11 6 ,
  0 1 7 ⟶ 0 11 8 ,
  0 1 9 ⟶ 0 11 10 ,
  0 1 2 ⟶ 3 12 ,
  0 1 7 ⟶ 3 4 ,
  0 1 9 ⟶ 3 13 ,
  3 13 6 ⟶ 0 11 6 ,
  3 13 8 ⟶ 0 11 8 ,
  3 13 10 ⟶ 0 11 10 ,
  3 13 14 ⟶ 0 11 14 ,
  1 2 →= 1 ,
  1 7 →= 15 ,
  1 9 →= 11 ,
  2 2 →= 2 ,
  2 7 →= 7 ,
  2 9 →= 9 ,
  12 2 →= 12 ,
  12 7 →= 4 ,
  12 9 →= 13 ,
  16 2 →= 16 ,
  16 7 →= 17 ,
  16 9 →= 5 ,
  6 2 →= 6 ,
  6 7 →= 8 ,
  6 9 →= 10 ,
  18 2 →= 18 ,
  18 7 →= 19 ,
  18 9 →= 20 ,
  15 16 2 →= 1 9 8 5 6 ,
  15 16 7 →= 1 9 8 5 8 ,
  15 16 9 →= 1 9 8 5 10 ,
  7 16 2 →= 2 9 8 5 6 ,
  7 16 7 →= 2 9 8 5 8 ,
  7 16 9 →= 2 9 8 5 10 ,
  4 16 2 →= 12 9 8 5 6 ,
  4 16 7 →= 12 9 8 5 8 ,
  4 16 9 →= 12 9 8 5 10 ,
  17 16 2 →= 16 9 8 5 6 ,
  17 16 7 →= 16 9 8 5 8 ,
  17 16 9 →= 16 9 8 5 10 ,
  8 16 2 →= 6 9 8 5 6 ,
  8 16 7 →= 6 9 8 5 8 ,
  8 16 9 →= 6 9 8 5 10 ,
  19 16 2 →= 18 9 8 5 6 ,
  19 16 7 →= 18 9 8 5 8 ,
  19 16 9 →= 18 9 8 5 10 ,
  11 10 6 →= 1 7 5 6 ,
  11 10 8 →= 1 7 5 8 ,
  11 10 10 →= 1 7 5 10 ,
  11 10 14 →= 1 7 5 14 ,
  9 10 6 →= 2 7 5 6 ,
  9 10 8 →= 2 7 5 8 ,
  9 10 10 →= 2 7 5 10 ,
  9 10 14 →= 2 7 5 14 ,
  13 10 6 →= 12 7 5 6 ,
  13 10 8 →= 12 7 5 8 ,
  13 10 10 →= 12 7 5 10 ,
  13 10 14 →= 12 7 5 14 ,
  5 10 6 →= 16 7 5 6 ,
  5 10 8 →= 16 7 5 8 ,
  5 10 10 →= 16 7 5 10 ,
  5 10 14 →= 16 7 5 14 ,
  10 10 6 →= 6 7 5 6 ,
  10 10 8 →= 6 7 5 8 ,
  10 10 10 →= 6 7 5 10 ,
  10 10 14 →= 6 7 5 14 ,
  20 10 6 →= 18 7 5 6 ,
  20 10 8 →= 18 7 5 8 ,
  20 10 10 →= 18 7 5 10 ,
  20 10 14 →= 18 7 5 14 }

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

After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13, 15 ↦ 14, 16 ↦ 15, 17 ↦ 16, 18 ↦ 17, 19 ↦ 18, 20 ↦ 19 },
it remains to prove termination of the 66-rule system
{ 0 1 2 ⟶ 3 4 5 6 ,
  0 1 7 ⟶ 3 4 5 8 ,
  0 1 9 ⟶ 3 4 5 10 ,
  0 1 2 ⟶ 0 11 6 ,
  0 1 7 ⟶ 0 11 8 ,
  0 1 9 ⟶ 0 11 10 ,
  0 1 2 ⟶ 3 12 ,
  0 1 7 ⟶ 3 4 ,
  0 1 9 ⟶ 3 13 ,
  3 13 6 ⟶ 0 11 6 ,
  3 13 8 ⟶ 0 11 8 ,
  3 13 10 ⟶ 0 11 10 ,
  1 2 →= 1 ,
  1 7 →= 14 ,
  1 9 →= 11 ,
  2 2 →= 2 ,
  2 7 →= 7 ,
  2 9 →= 9 ,
  12 2 →= 12 ,
  12 7 →= 4 ,
  12 9 →= 13 ,
  15 2 →= 15 ,
  15 7 →= 16 ,
  15 9 →= 5 ,
  6 2 →= 6 ,
  6 7 →= 8 ,
  6 9 →= 10 ,
  17 2 →= 17 ,
  17 7 →= 18 ,
  17 9 →= 19 ,
  14 15 2 →= 1 9 8 5 6 ,
  14 15 7 →= 1 9 8 5 8 ,
  14 15 9 →= 1 9 8 5 10 ,
  7 15 2 →= 2 9 8 5 6 ,
  7 15 7 →= 2 9 8 5 8 ,
  7 15 9 →= 2 9 8 5 10 ,
  4 15 2 →= 12 9 8 5 6 ,
  4 15 7 →= 12 9 8 5 8 ,
  4 15 9 →= 12 9 8 5 10 ,
  16 15 2 →= 15 9 8 5 6 ,
  16 15 7 →= 15 9 8 5 8 ,
  16 15 9 →= 15 9 8 5 10 ,
  8 15 2 →= 6 9 8 5 6 ,
  8 15 7 →= 6 9 8 5 8 ,
  8 15 9 →= 6 9 8 5 10 ,
  18 15 2 →= 17 9 8 5 6 ,
  18 15 7 →= 17 9 8 5 8 ,
  18 15 9 →= 17 9 8 5 10 ,
  11 10 6 →= 1 7 5 6 ,
  11 10 8 →= 1 7 5 8 ,
  11 10 10 →= 1 7 5 10 ,
  9 10 6 →= 2 7 5 6 ,
  9 10 8 →= 2 7 5 8 ,
  9 10 10 →= 2 7 5 10 ,
  13 10 6 →= 12 7 5 6 ,
  13 10 8 →= 12 7 5 8 ,
  13 10 10 →= 12 7 5 10 ,
  5 10 6 →= 15 7 5 6 ,
  5 10 8 →= 15 7 5 8 ,
  5 10 10 →= 15 7 5 10 ,
  10 10 6 →= 6 7 5 6 ,
  10 10 8 →= 6 7 5 8 ,
  10 10 10 →= 6 7 5 10 ,
  19 10 6 →= 17 7 5 6 ,
  19 10 8 →= 17 7 5 8 ,
  19 10 10 →= 17 7 5 10 }

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 1 ⎟
    ⎜ 0 1 ⎟
    ⎝     ⎠

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

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

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

15 ↦ ⎛     ⎞
     ⎜ 1 2 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

16 ↦ ⎛     ⎞
     ⎜ 1 2 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

17 ↦ ⎛     ⎞
     ⎜ 1 0 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

18 ↦ ⎛     ⎞
     ⎜ 1 0 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

19 ↦ ⎛     ⎞
     ⎜ 1 0 ⎟
     ⎜ 0 1 ⎟
     ⎝     ⎠

After renaming modulo the bijection { 1 ↦ 0, 2 ↦ 1, 7 ↦ 2, 14 ↦ 3, 9 ↦ 4, 12 ↦ 5, 4 ↦ 6, 15 ↦ 7, 16 ↦ 8, 6 ↦ 9, 8 ↦ 10, 10 ↦ 11, 17 ↦ 12, 18 ↦ 13, 5 ↦ 14 },
it remains to prove termination of the 35-rule system
{ 0 1 →= 0 ,
  0 2 →= 3 ,
  1 1 →= 1 ,
  1 2 →= 2 ,
  1 4 →= 4 ,
  5 1 →= 5 ,
  5 2 →= 6 ,
  7 1 →= 7 ,
  7 2 →= 8 ,
  9 1 →= 9 ,
  9 2 →= 10 ,
  9 4 →= 11 ,
  12 1 →= 12 ,
  12 2 →= 13 ,
  3 7 1 →= 0 4 10 14 9 ,
  3 7 2 →= 0 4 10 14 10 ,
  3 7 4 →= 0 4 10 14 11 ,
  2 7 1 →= 1 4 10 14 9 ,
  2 7 2 →= 1 4 10 14 10 ,
  2 7 4 →= 1 4 10 14 11 ,
  6 7 1 →= 5 4 10 14 9 ,
  6 7 2 →= 5 4 10 14 10 ,
  6 7 4 →= 5 4 10 14 11 ,
  8 7 1 →= 7 4 10 14 9 ,
  8 7 2 →= 7 4 10 14 10 ,
  8 7 4 →= 7 4 10 14 11 ,
  10 7 1 →= 9 4 10 14 9 ,
  10 7 2 →= 9 4 10 14 10 ,
  10 7 4 →= 9 4 10 14 11 ,
  13 7 1 →= 12 4 10 14 9 ,
  13 7 2 →= 12 4 10 14 10 ,
  13 7 4 →= 12 4 10 14 11 ,
  14 11 9 →= 7 2 14 9 ,
  14 11 10 →= 7 2 14 10 ,
  14 11 11 →= 7 2 14 11 }

The system is trivially terminating.