/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 { a->0, b->1, c->2 }, 
it remains to prove termination of the 4-rule system
{ 0 1 2 -> 2 1 0 0 2 1 ,
  0 -> ,
  1 -> ,
  2 -> }


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

After renaming modulo { (0,0)->0, (0,1)->1, (1,2)->2, (2,0)->3, (0,2)->4, (2,1)->5, (1,0)->6, (1,1)->7, (2,2)->8, (2,4)->9, (1,4)->10, (3,0)->11, (3,2)->12, (0,4)->13, (3,1)->14, (3,4)->15 },
it remains to prove termination of the 64-rule system
{ 0 1 2 3 -> 4 5 6 0 4 5 6 ,
  0 1 2 5 -> 4 5 6 0 4 5 7 ,
  0 1 2 8 -> 4 5 6 0 4 5 2 ,
  0 1 2 9 -> 4 5 6 0 4 5 10 ,
  6 1 2 3 -> 2 5 6 0 4 5 6 ,
  6 1 2 5 -> 2 5 6 0 4 5 7 ,
  6 1 2 8 -> 2 5 6 0 4 5 2 ,
  6 1 2 9 -> 2 5 6 0 4 5 10 ,
  3 1 2 3 -> 8 5 6 0 4 5 6 ,
  3 1 2 5 -> 8 5 6 0 4 5 7 ,
  3 1 2 8 -> 8 5 6 0 4 5 2 ,
  3 1 2 9 -> 8 5 6 0 4 5 10 ,
  11 1 2 3 -> 12 5 6 0 4 5 6 ,
  11 1 2 5 -> 12 5 6 0 4 5 7 ,
  11 1 2 8 -> 12 5 6 0 4 5 2 ,
  11 1 2 9 -> 12 5 6 0 4 5 10 ,
  0 0 -> 0 ,
  0 1 -> 1 ,
  0 4 -> 4 ,
  0 13 -> 13 ,
  6 0 -> 6 ,
  6 1 -> 7 ,
  6 4 -> 2 ,
  6 13 -> 10 ,
  3 0 -> 3 ,
  3 1 -> 5 ,
  3 4 -> 8 ,
  3 13 -> 9 ,
  11 0 -> 11 ,
  11 1 -> 14 ,
  11 4 -> 12 ,
  11 13 -> 15 ,
  1 6 -> 0 ,
  1 7 -> 1 ,
  1 2 -> 4 ,
  1 10 -> 13 ,
  7 6 -> 6 ,
  7 7 -> 7 ,
  7 2 -> 2 ,
  7 10 -> 10 ,
  5 6 -> 3 ,
  5 7 -> 5 ,
  5 2 -> 8 ,
  5 10 -> 9 ,
  14 6 -> 11 ,
  14 7 -> 14 ,
  14 2 -> 12 ,
  14 10 -> 15 ,
  4 3 -> 0 ,
  4 5 -> 1 ,
  4 8 -> 4 ,
  4 9 -> 13 ,
  2 3 -> 6 ,
  2 5 -> 7 ,
  2 8 -> 2 ,
  2 9 -> 10 ,
  8 3 -> 3 ,
  8 5 -> 5 ,
  8 8 -> 8 ,
  8 9 -> 9 ,
  12 3 -> 11 ,
  12 5 -> 14 ,
  12 8 -> 12 ,
  12 9 -> 15 }


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

0 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
1 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
2 is interpreted by
 /   \
| 1 2 |
| 0 1 |
 \   /
3 is interpreted by
 /   \
| 1 3 |
| 0 1 |
 \   /
4 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
5 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
6 is interpreted by
 /   \
| 1 2 |
| 0 1 |
 \   /
7 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
8 is interpreted by
 /   \
| 1 3 |
| 0 1 |
 \   /
9 is interpreted by
 /   \
| 1 2 |
| 0 1 |
 \   /
10 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
11 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
12 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
13 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
14 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
15 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

After renaming modulo { 0->0, 1->1, 2->2, 3->3, 4->4, 5->5, 6->6, 7->7, 8->8, 9->9, 10->10, 13->11, 11->12, 14->13 }, 
it remains to prove termination of the 32-rule system
{ 0 1 2 3 -> 4 5 6 0 4 5 6 ,
  0 1 2 5 -> 4 5 6 0 4 5 7 ,
  0 1 2 8 -> 4 5 6 0 4 5 2 ,
  0 1 2 9 -> 4 5 6 0 4 5 10 ,
  6 1 2 3 -> 2 5 6 0 4 5 6 ,
  6 1 2 5 -> 2 5 6 0 4 5 7 ,
  6 1 2 8 -> 2 5 6 0 4 5 2 ,
  6 1 2 9 -> 2 5 6 0 4 5 10 ,
  3 1 2 3 -> 8 5 6 0 4 5 6 ,
  3 1 2 5 -> 8 5 6 0 4 5 7 ,
  3 1 2 8 -> 8 5 6 0 4 5 2 ,
  3 1 2 9 -> 8 5 6 0 4 5 10 ,
  0 0 -> 0 ,
  0 1 -> 1 ,
  0 4 -> 4 ,
  0 11 -> 11 ,
  6 0 -> 6 ,
  6 4 -> 2 ,
  3 0 -> 3 ,
  3 4 -> 8 ,
  12 0 -> 12 ,
  1 7 -> 1 ,
  7 6 -> 6 ,
  7 7 -> 7 ,
  7 2 -> 2 ,
  7 10 -> 10 ,
  5 6 -> 3 ,
  5 7 -> 5 ,
  5 2 -> 8 ,
  5 10 -> 9 ,
  13 7 -> 13 ,
  4 5 -> 1 }


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

0 is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 1 0 |
 \     /
1 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
2 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
3 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
4 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
5 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
6 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
7 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
8 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
9 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
10 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
11 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
12 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
13 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /

After renaming modulo { 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 }, 
it remains to prove termination of the 31-rule system
{ 0 1 2 3 -> 4 5 6 0 4 5 6 ,
  0 1 2 5 -> 4 5 6 0 4 5 7 ,
  0 1 2 8 -> 4 5 6 0 4 5 2 ,
  0 1 2 9 -> 4 5 6 0 4 5 10 ,
  6 1 2 3 -> 2 5 6 0 4 5 6 ,
  6 1 2 5 -> 2 5 6 0 4 5 7 ,
  6 1 2 8 -> 2 5 6 0 4 5 2 ,
  6 1 2 9 -> 2 5 6 0 4 5 10 ,
  3 1 2 3 -> 8 5 6 0 4 5 6 ,
  3 1 2 5 -> 8 5 6 0 4 5 7 ,
  3 1 2 8 -> 8 5 6 0 4 5 2 ,
  3 1 2 9 -> 8 5 6 0 4 5 10 ,
  0 1 -> 1 ,
  0 4 -> 4 ,
  0 11 -> 11 ,
  6 0 -> 6 ,
  6 4 -> 2 ,
  3 0 -> 3 ,
  3 4 -> 8 ,
  12 0 -> 12 ,
  1 7 -> 1 ,
  7 6 -> 6 ,
  7 7 -> 7 ,
  7 2 -> 2 ,
  7 10 -> 10 ,
  5 6 -> 3 ,
  5 7 -> 5 ,
  5 2 -> 8 ,
  5 10 -> 9 ,
  13 7 -> 13 ,
  4 5 -> 1 }


Applying the dependency pairs transformation.

After renaming modulo { (0,true)->0, (1,false)->1, (2,false)->2, (3,false)->3, (4,true)->4, (5,false)->5, (6,false)->6, (0,false)->7, (4,false)->8, (5,true)->9, (6,true)->10, (7,false)->11, (7,true)->12, (8,false)->13, (9,false)->14, (10,false)->15, (3,true)->16, (1,true)->17, (12,true)->18, (13,true)->19, (11,false)->20, (12,false)->21, (13,false)->22 },
it remains to prove termination of the 113-rule system
{ 0 1 2 3 -> 4 5 6 7 8 5 6 ,
  0 1 2 3 -> 9 6 7 8 5 6 ,
  0 1 2 3 -> 10 7 8 5 6 ,
  0 1 2 3 -> 0 8 5 6 ,
  0 1 2 3 -> 4 5 6 ,
  0 1 2 3 -> 9 6 ,
  0 1 2 3 -> 10 ,
  0 1 2 5 -> 4 5 6 7 8 5 11 ,
  0 1 2 5 -> 9 6 7 8 5 11 ,
  0 1 2 5 -> 10 7 8 5 11 ,
  0 1 2 5 -> 0 8 5 11 ,
  0 1 2 5 -> 4 5 11 ,
  0 1 2 5 -> 9 11 ,
  0 1 2 5 -> 12 ,
  0 1 2 13 -> 4 5 6 7 8 5 2 ,
  0 1 2 13 -> 9 6 7 8 5 2 ,
  0 1 2 13 -> 10 7 8 5 2 ,
  0 1 2 13 -> 0 8 5 2 ,
  0 1 2 13 -> 4 5 2 ,
  0 1 2 13 -> 9 2 ,
  0 1 2 14 -> 4 5 6 7 8 5 15 ,
  0 1 2 14 -> 9 6 7 8 5 15 ,
  0 1 2 14 -> 10 7 8 5 15 ,
  0 1 2 14 -> 0 8 5 15 ,
  0 1 2 14 -> 4 5 15 ,
  0 1 2 14 -> 9 15 ,
  10 1 2 3 -> 9 6 7 8 5 6 ,
  10 1 2 3 -> 10 7 8 5 6 ,
  10 1 2 3 -> 0 8 5 6 ,
  10 1 2 3 -> 4 5 6 ,
  10 1 2 3 -> 9 6 ,
  10 1 2 3 -> 10 ,
  10 1 2 5 -> 9 6 7 8 5 11 ,
  10 1 2 5 -> 10 7 8 5 11 ,
  10 1 2 5 -> 0 8 5 11 ,
  10 1 2 5 -> 4 5 11 ,
  10 1 2 5 -> 9 11 ,
  10 1 2 5 -> 12 ,
  10 1 2 13 -> 9 6 7 8 5 2 ,
  10 1 2 13 -> 10 7 8 5 2 ,
  10 1 2 13 -> 0 8 5 2 ,
  10 1 2 13 -> 4 5 2 ,
  10 1 2 13 -> 9 2 ,
  10 1 2 14 -> 9 6 7 8 5 15 ,
  10 1 2 14 -> 10 7 8 5 15 ,
  10 1 2 14 -> 0 8 5 15 ,
  10 1 2 14 -> 4 5 15 ,
  10 1 2 14 -> 9 15 ,
  16 1 2 3 -> 9 6 7 8 5 6 ,
  16 1 2 3 -> 10 7 8 5 6 ,
  16 1 2 3 -> 0 8 5 6 ,
  16 1 2 3 -> 4 5 6 ,
  16 1 2 3 -> 9 6 ,
  16 1 2 3 -> 10 ,
  16 1 2 5 -> 9 6 7 8 5 11 ,
  16 1 2 5 -> 10 7 8 5 11 ,
  16 1 2 5 -> 0 8 5 11 ,
  16 1 2 5 -> 4 5 11 ,
  16 1 2 5 -> 9 11 ,
  16 1 2 5 -> 12 ,
  16 1 2 13 -> 9 6 7 8 5 2 ,
  16 1 2 13 -> 10 7 8 5 2 ,
  16 1 2 13 -> 0 8 5 2 ,
  16 1 2 13 -> 4 5 2 ,
  16 1 2 13 -> 9 2 ,
  16 1 2 14 -> 9 6 7 8 5 15 ,
  16 1 2 14 -> 10 7 8 5 15 ,
  16 1 2 14 -> 0 8 5 15 ,
  16 1 2 14 -> 4 5 15 ,
  16 1 2 14 -> 9 15 ,
  0 1 -> 17 ,
  0 8 -> 4 ,
  10 7 -> 10 ,
  16 7 -> 16 ,
  18 7 -> 18 ,
  17 11 -> 17 ,
  12 6 -> 10 ,
  12 11 -> 12 ,
  9 6 -> 16 ,
  9 11 -> 9 ,
  19 11 -> 19 ,
  4 5 -> 17 ,
  7 1 2 3 ->= 8 5 6 7 8 5 6 ,
  7 1 2 5 ->= 8 5 6 7 8 5 11 ,
  7 1 2 13 ->= 8 5 6 7 8 5 2 ,
  7 1 2 14 ->= 8 5 6 7 8 5 15 ,
  6 1 2 3 ->= 2 5 6 7 8 5 6 ,
  6 1 2 5 ->= 2 5 6 7 8 5 11 ,
  6 1 2 13 ->= 2 5 6 7 8 5 2 ,
  6 1 2 14 ->= 2 5 6 7 8 5 15 ,
  3 1 2 3 ->= 13 5 6 7 8 5 6 ,
  3 1 2 5 ->= 13 5 6 7 8 5 11 ,
  3 1 2 13 ->= 13 5 6 7 8 5 2 ,
  3 1 2 14 ->= 13 5 6 7 8 5 15 ,
  7 1 ->= 1 ,
  7 8 ->= 8 ,
  7 20 ->= 20 ,
  6 7 ->= 6 ,
  6 8 ->= 2 ,
  3 7 ->= 3 ,
  3 8 ->= 13 ,
  21 7 ->= 21 ,
  1 11 ->= 1 ,
  11 6 ->= 6 ,
  11 11 ->= 11 ,
  11 2 ->= 2 ,
  11 15 ->= 15 ,
  5 6 ->= 3 ,
  5 11 ->= 5 ,
  5 2 ->= 13 ,
  5 15 ->= 14 ,
  22 11 ->= 22 ,
  8 5 ->= 1 }


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

0 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
1 is interpreted by
 /   \
| 1 2 |
| 0 1 |
 \   /
2 is interpreted by
 /   \
| 1 2 |
| 0 1 |
 \   /
3 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
4 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
5 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
6 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
7 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
8 is interpreted by
 /   \
| 1 2 |
| 0 1 |
 \   /
9 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
10 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
11 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
12 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
13 is interpreted by
 /   \
| 1 2 |
| 0 1 |
 \   /
14 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
15 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
16 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
17 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
18 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
19 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
20 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
21 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
22 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

After renaming modulo { 10->0, 7->1, 16->2, 18->3, 17->4, 11->5, 12->6, 9->7, 19->8, 1->9, 2->10, 3->11, 8->12, 5->13, 6->14, 13->15, 14->16, 15->17, 20->18, 21->19, 22->20 }, 
it remains to prove termination of the 38-rule system
{ 0 1 -> 0 ,
  2 1 -> 2 ,
  3 1 -> 3 ,
  4 5 -> 4 ,
  6 5 -> 6 ,
  7 5 -> 7 ,
  8 5 -> 8 ,
  1 9 10 11 ->= 12 13 14 1 12 13 14 ,
  1 9 10 13 ->= 12 13 14 1 12 13 5 ,
  1 9 10 15 ->= 12 13 14 1 12 13 10 ,
  1 9 10 16 ->= 12 13 14 1 12 13 17 ,
  14 9 10 11 ->= 10 13 14 1 12 13 14 ,
  14 9 10 13 ->= 10 13 14 1 12 13 5 ,
  14 9 10 15 ->= 10 13 14 1 12 13 10 ,
  14 9 10 16 ->= 10 13 14 1 12 13 17 ,
  11 9 10 11 ->= 15 13 14 1 12 13 14 ,
  11 9 10 13 ->= 15 13 14 1 12 13 5 ,
  11 9 10 15 ->= 15 13 14 1 12 13 10 ,
  11 9 10 16 ->= 15 13 14 1 12 13 17 ,
  1 9 ->= 9 ,
  1 12 ->= 12 ,
  1 18 ->= 18 ,
  14 1 ->= 14 ,
  14 12 ->= 10 ,
  11 1 ->= 11 ,
  11 12 ->= 15 ,
  19 1 ->= 19 ,
  9 5 ->= 9 ,
  5 14 ->= 14 ,
  5 5 ->= 5 ,
  5 10 ->= 10 ,
  5 17 ->= 17 ,
  13 14 ->= 11 ,
  13 5 ->= 13 ,
  13 10 ->= 15 ,
  13 17 ->= 16 ,
  20 5 ->= 20 ,
  12 13 ->= 9 }


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

0 is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 0 0 |
 \     /
1 is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 1 0 |
 \     /
2 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
3 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
4 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
5 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
6 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
7 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
8 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
9 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
10 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
11 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
12 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
13 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
14 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
15 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
16 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
17 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
18 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
19 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
20 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /

After renaming modulo { 2->0, 1->1, 3->2, 4->3, 5->4, 6->5, 7->6, 8->7, 9->8, 10->9, 11->10, 12->11, 13->12, 14->13, 15->14, 16->15, 17->16, 18->17, 19->18, 20->19 }, 
it remains to prove termination of the 37-rule system
{ 0 1 -> 0 ,
  2 1 -> 2 ,
  3 4 -> 3 ,
  5 4 -> 5 ,
  6 4 -> 6 ,
  7 4 -> 7 ,
  1 8 9 10 ->= 11 12 13 1 11 12 13 ,
  1 8 9 12 ->= 11 12 13 1 11 12 4 ,
  1 8 9 14 ->= 11 12 13 1 11 12 9 ,
  1 8 9 15 ->= 11 12 13 1 11 12 16 ,
  13 8 9 10 ->= 9 12 13 1 11 12 13 ,
  13 8 9 12 ->= 9 12 13 1 11 12 4 ,
  13 8 9 14 ->= 9 12 13 1 11 12 9 ,
  13 8 9 15 ->= 9 12 13 1 11 12 16 ,
  10 8 9 10 ->= 14 12 13 1 11 12 13 ,
  10 8 9 12 ->= 14 12 13 1 11 12 4 ,
  10 8 9 14 ->= 14 12 13 1 11 12 9 ,
  10 8 9 15 ->= 14 12 13 1 11 12 16 ,
  1 8 ->= 8 ,
  1 11 ->= 11 ,
  1 17 ->= 17 ,
  13 1 ->= 13 ,
  13 11 ->= 9 ,
  10 1 ->= 10 ,
  10 11 ->= 14 ,
  18 1 ->= 18 ,
  8 4 ->= 8 ,
  4 13 ->= 13 ,
  4 4 ->= 4 ,
  4 9 ->= 9 ,
  4 16 ->= 16 ,
  12 13 ->= 10 ,
  12 4 ->= 12 ,
  12 9 ->= 14 ,
  12 16 ->= 15 ,
  19 4 ->= 19 ,
  11 12 ->= 8 }


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

0 is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 0 0 |
 \     /
1 is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 1 0 |
 \     /
2 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
3 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
4 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
5 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
6 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
7 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
8 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
9 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
10 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
11 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
12 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
13 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
14 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
15 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
16 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
17 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
18 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
19 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /

After renaming modulo { 2->0, 1->1, 3->2, 4->3, 5->4, 6->5, 7->6, 8->7, 9->8, 10->9, 11->10, 12->11, 13->12, 14->13, 15->14, 16->15, 17->16, 18->17, 19->18 }, 
it remains to prove termination of the 36-rule system
{ 0 1 -> 0 ,
  2 3 -> 2 ,
  4 3 -> 4 ,
  5 3 -> 5 ,
  6 3 -> 6 ,
  1 7 8 9 ->= 10 11 12 1 10 11 12 ,
  1 7 8 11 ->= 10 11 12 1 10 11 3 ,
  1 7 8 13 ->= 10 11 12 1 10 11 8 ,
  1 7 8 14 ->= 10 11 12 1 10 11 15 ,
  12 7 8 9 ->= 8 11 12 1 10 11 12 ,
  12 7 8 11 ->= 8 11 12 1 10 11 3 ,
  12 7 8 13 ->= 8 11 12 1 10 11 8 ,
  12 7 8 14 ->= 8 11 12 1 10 11 15 ,
  9 7 8 9 ->= 13 11 12 1 10 11 12 ,
  9 7 8 11 ->= 13 11 12 1 10 11 3 ,
  9 7 8 13 ->= 13 11 12 1 10 11 8 ,
  9 7 8 14 ->= 13 11 12 1 10 11 15 ,
  1 7 ->= 7 ,
  1 10 ->= 10 ,
  1 16 ->= 16 ,
  12 1 ->= 12 ,
  12 10 ->= 8 ,
  9 1 ->= 9 ,
  9 10 ->= 13 ,
  17 1 ->= 17 ,
  7 3 ->= 7 ,
  3 12 ->= 12 ,
  3 3 ->= 3 ,
  3 8 ->= 8 ,
  3 15 ->= 15 ,
  11 12 ->= 9 ,
  11 3 ->= 11 ,
  11 8 ->= 13 ,
  11 15 ->= 14 ,
  18 3 ->= 18 ,
  10 11 ->= 7 }


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

0 is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 0 0 |
 \     /
1 is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 1 0 |
 \     /
2 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
3 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
4 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
5 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
6 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
7 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
8 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
9 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
10 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
11 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
12 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
13 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
14 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
15 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
16 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
17 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
18 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /

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


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

0 is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 0 0 |
 \     /
1 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 1 1 |
 \     /
2 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
3 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
4 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
5 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
6 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
7 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
8 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
9 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
10 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
11 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
12 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
13 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
14 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
15 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
16 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
17 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /

After renaming modulo { 2->0, 1->1, 3->2, 4->3, 5->4, 6->5, 7->6, 8->7, 9->8, 10->9, 11->10, 12->11, 13->12, 14->13, 15->14, 16->15, 17->16 }, 
it remains to prove termination of the 34-rule system
{ 0 1 -> 0 ,
  2 1 -> 2 ,
  3 1 -> 3 ,
  4 5 6 7 ->= 8 9 10 4 8 9 10 ,
  4 5 6 9 ->= 8 9 10 4 8 9 1 ,
  4 5 6 11 ->= 8 9 10 4 8 9 6 ,
  4 5 6 12 ->= 8 9 10 4 8 9 13 ,
  10 5 6 7 ->= 6 9 10 4 8 9 10 ,
  10 5 6 9 ->= 6 9 10 4 8 9 1 ,
  10 5 6 11 ->= 6 9 10 4 8 9 6 ,
  10 5 6 12 ->= 6 9 10 4 8 9 13 ,
  7 5 6 7 ->= 11 9 10 4 8 9 10 ,
  7 5 6 9 ->= 11 9 10 4 8 9 1 ,
  7 5 6 11 ->= 11 9 10 4 8 9 6 ,
  7 5 6 12 ->= 11 9 10 4 8 9 13 ,
  4 5 ->= 5 ,
  4 8 ->= 8 ,
  4 14 ->= 14 ,
  10 4 ->= 10 ,
  10 8 ->= 6 ,
  7 4 ->= 7 ,
  7 8 ->= 11 ,
  15 4 ->= 15 ,
  5 1 ->= 5 ,
  1 10 ->= 10 ,
  1 1 ->= 1 ,
  1 6 ->= 6 ,
  1 13 ->= 13 ,
  9 10 ->= 7 ,
  9 1 ->= 9 ,
  9 6 ->= 11 ,
  9 13 ->= 12 ,
  16 1 ->= 16 ,
  8 9 ->= 5 }


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

0 is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 0 0 |
 \     /
1 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 1 1 |
 \     /
2 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
3 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
4 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
5 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
6 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
7 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
8 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
9 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
10 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
11 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
12 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
13 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
14 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
15 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
16 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /

After renaming modulo { 2->0, 1->1, 3->2, 4->3, 5->4, 6->5, 7->6, 8->7, 9->8, 10->9, 11->10, 12->11, 13->12, 14->13, 15->14, 16->15 }, 
it remains to prove termination of the 33-rule system
{ 0 1 -> 0 ,
  2 1 -> 2 ,
  3 4 5 6 ->= 7 8 9 3 7 8 9 ,
  3 4 5 8 ->= 7 8 9 3 7 8 1 ,
  3 4 5 10 ->= 7 8 9 3 7 8 5 ,
  3 4 5 11 ->= 7 8 9 3 7 8 12 ,
  9 4 5 6 ->= 5 8 9 3 7 8 9 ,
  9 4 5 8 ->= 5 8 9 3 7 8 1 ,
  9 4 5 10 ->= 5 8 9 3 7 8 5 ,
  9 4 5 11 ->= 5 8 9 3 7 8 12 ,
  6 4 5 6 ->= 10 8 9 3 7 8 9 ,
  6 4 5 8 ->= 10 8 9 3 7 8 1 ,
  6 4 5 10 ->= 10 8 9 3 7 8 5 ,
  6 4 5 11 ->= 10 8 9 3 7 8 12 ,
  3 4 ->= 4 ,
  3 7 ->= 7 ,
  3 13 ->= 13 ,
  9 3 ->= 9 ,
  9 7 ->= 5 ,
  6 3 ->= 6 ,
  6 7 ->= 10 ,
  14 3 ->= 14 ,
  4 1 ->= 4 ,
  1 9 ->= 9 ,
  1 1 ->= 1 ,
  1 5 ->= 5 ,
  1 12 ->= 12 ,
  8 9 ->= 6 ,
  8 1 ->= 8 ,
  8 5 ->= 10 ,
  8 12 ->= 11 ,
  15 1 ->= 15 ,
  7 8 ->= 4 }


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

0 is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 0 0 |
 \     /
1 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 1 1 |
 \     /
2 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
3 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
4 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
5 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
6 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
7 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
8 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
9 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
10 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
11 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
12 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
13 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
14 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
15 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /

After renaming modulo { 2->0, 1->1, 3->2, 4->3, 5->4, 6->5, 7->6, 8->7, 9->8, 10->9, 11->10, 12->11, 13->12, 14->13, 15->14 }, 
it remains to prove termination of the 32-rule system
{ 0 1 -> 0 ,
  2 3 4 5 ->= 6 7 8 2 6 7 8 ,
  2 3 4 7 ->= 6 7 8 2 6 7 1 ,
  2 3 4 9 ->= 6 7 8 2 6 7 4 ,
  2 3 4 10 ->= 6 7 8 2 6 7 11 ,
  8 3 4 5 ->= 4 7 8 2 6 7 8 ,
  8 3 4 7 ->= 4 7 8 2 6 7 1 ,
  8 3 4 9 ->= 4 7 8 2 6 7 4 ,
  8 3 4 10 ->= 4 7 8 2 6 7 11 ,
  5 3 4 5 ->= 9 7 8 2 6 7 8 ,
  5 3 4 7 ->= 9 7 8 2 6 7 1 ,
  5 3 4 9 ->= 9 7 8 2 6 7 4 ,
  5 3 4 10 ->= 9 7 8 2 6 7 11 ,
  2 3 ->= 3 ,
  2 6 ->= 6 ,
  2 12 ->= 12 ,
  8 2 ->= 8 ,
  8 6 ->= 4 ,
  5 2 ->= 5 ,
  5 6 ->= 9 ,
  13 2 ->= 13 ,
  3 1 ->= 3 ,
  1 8 ->= 8 ,
  1 1 ->= 1 ,
  1 4 ->= 4 ,
  1 11 ->= 11 ,
  7 8 ->= 5 ,
  7 1 ->= 7 ,
  7 4 ->= 9 ,
  7 11 ->= 10 ,
  14 1 ->= 14 ,
  6 7 ->= 3 }


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

0 is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 0 0 |
 \     /
1 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 1 1 |
 \     /
2 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
3 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
4 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
5 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
6 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
7 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
8 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
9 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
10 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
11 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
12 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
13 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
14 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /

After renaming modulo { 2->0, 3->1, 4->2, 5->3, 6->4, 7->5, 8->6, 1->7, 9->8, 10->9, 11->10, 12->11, 13->12, 14->13 }, 
it remains to prove termination of the 31-rule system
{ 0 1 2 3 ->= 4 5 6 0 4 5 6 ,
  0 1 2 5 ->= 4 5 6 0 4 5 7 ,
  0 1 2 8 ->= 4 5 6 0 4 5 2 ,
  0 1 2 9 ->= 4 5 6 0 4 5 10 ,
  6 1 2 3 ->= 2 5 6 0 4 5 6 ,
  6 1 2 5 ->= 2 5 6 0 4 5 7 ,
  6 1 2 8 ->= 2 5 6 0 4 5 2 ,
  6 1 2 9 ->= 2 5 6 0 4 5 10 ,
  3 1 2 3 ->= 8 5 6 0 4 5 6 ,
  3 1 2 5 ->= 8 5 6 0 4 5 7 ,
  3 1 2 8 ->= 8 5 6 0 4 5 2 ,
  3 1 2 9 ->= 8 5 6 0 4 5 10 ,
  0 1 ->= 1 ,
  0 4 ->= 4 ,
  0 11 ->= 11 ,
  6 0 ->= 6 ,
  6 4 ->= 2 ,
  3 0 ->= 3 ,
  3 4 ->= 8 ,
  12 0 ->= 12 ,
  1 7 ->= 1 ,
  7 6 ->= 6 ,
  7 7 ->= 7 ,
  7 2 ->= 2 ,
  7 10 ->= 10 ,
  5 6 ->= 3 ,
  5 7 ->= 5 ,
  5 2 ->= 8 ,
  5 10 ->= 9 ,
  13 7 ->= 13 ,
  4 5 ->= 1 }

The system is trivially terminating.