/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


YES

After renaming modulo { c->0, b->1, a->2 }, 
it remains to prove termination of the 6-rule system
{ 0 1 0 -> 0 2 1 ,
  0 1 1 -> 2 2 0 ,
  2 2 1 -> 1 1 0 ,
  1 1 0 -> 2 1 0 ,
  1 1 1 -> 1 2 1 ,
  0 1 2 ->= 1 2 0 }


The system was reversed.

After renaming modulo { 0->0, 1->1, 2->2 }, 
it remains to prove termination of the 6-rule system
{ 0 1 0 -> 1 2 0 ,
  1 1 0 -> 0 2 2 ,
  1 2 2 -> 0 1 1 ,
  0 1 1 -> 0 1 2 ,
  1 1 1 -> 1 2 1 ,
  2 1 0 ->= 0 2 1 }


Applying context closure of depth 1 in the following form: System R over Sigma
maps to { fold(xly) -> fold(xry) | l -> r in R, x,y in Sigma } over Sigma^2,
where fold(a_1...a_n) = (a_1,a_2)...(a_{n-1},a_{n})

After renaming modulo { [0, 0]->0, [0, 1]->1, [1, 0]->2, [1, 2]->3, [2, 0]->4, [1, 1]->5, [0, 2]->6, [2, 2]->7, [2, 1]->8 },
it remains to prove termination of the 54-rule system
{ 0 1 2 0 -> 1 3 4 0 ,
  1 5 2 0 -> 0 6 7 4 ,
  1 3 7 4 -> 0 1 5 2 ,
  0 1 5 2 -> 0 1 3 4 ,
  1 5 5 2 -> 1 3 8 2 ,
  6 8 2 0 ->= 0 6 8 2 ,
  0 1 2 1 -> 1 3 4 1 ,
  1 5 2 1 -> 0 6 7 8 ,
  1 3 7 8 -> 0 1 5 5 ,
  0 1 5 5 -> 0 1 3 8 ,
  1 5 5 5 -> 1 3 8 5 ,
  6 8 2 1 ->= 0 6 8 5 ,
  0 1 2 6 -> 1 3 4 6 ,
  1 5 2 6 -> 0 6 7 7 ,
  1 3 7 7 -> 0 1 5 3 ,
  0 1 5 3 -> 0 1 3 7 ,
  1 5 5 3 -> 1 3 8 3 ,
  6 8 2 6 ->= 0 6 8 3 ,
  2 1 2 0 -> 5 3 4 0 ,
  5 5 2 0 -> 2 6 7 4 ,
  5 3 7 4 -> 2 1 5 2 ,
  2 1 5 2 -> 2 1 3 4 ,
  5 5 5 2 -> 5 3 8 2 ,
  3 8 2 0 ->= 2 6 8 2 ,
  2 1 2 1 -> 5 3 4 1 ,
  5 5 2 1 -> 2 6 7 8 ,
  5 3 7 8 -> 2 1 5 5 ,
  2 1 5 5 -> 2 1 3 8 ,
  5 5 5 5 -> 5 3 8 5 ,
  3 8 2 1 ->= 2 6 8 5 ,
  2 1 2 6 -> 5 3 4 6 ,
  5 5 2 6 -> 2 6 7 7 ,
  5 3 7 7 -> 2 1 5 3 ,
  2 1 5 3 -> 2 1 3 7 ,
  5 5 5 3 -> 5 3 8 3 ,
  3 8 2 6 ->= 2 6 8 3 ,
  4 1 2 0 -> 8 3 4 0 ,
  8 5 2 0 -> 4 6 7 4 ,
  8 3 7 4 -> 4 1 5 2 ,
  4 1 5 2 -> 4 1 3 4 ,
  8 5 5 2 -> 8 3 8 2 ,
  7 8 2 0 ->= 4 6 8 2 ,
  4 1 2 1 -> 8 3 4 1 ,
  8 5 2 1 -> 4 6 7 8 ,
  8 3 7 8 -> 4 1 5 5 ,
  4 1 5 5 -> 4 1 3 8 ,
  8 5 5 5 -> 8 3 8 5 ,
  7 8 2 1 ->= 4 6 8 5 ,
  4 1 2 6 -> 8 3 4 6 ,
  8 5 2 6 -> 4 6 7 7 ,
  8 3 7 7 -> 4 1 5 3 ,
  4 1 5 3 -> 4 1 3 7 ,
  8 5 5 3 -> 8 3 8 3 ,
  7 8 2 6 ->= 4 6 8 3 }


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 0 |
| 0 1 |
 \   /
2 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
3 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
4 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
5 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
6 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
7 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
8 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

After renaming modulo { 0->0, 1->1, 2->2, 3->3, 4->4, 7->5, 5->6, 6->7, 8->8 }, 
it remains to prove termination of the 27-rule system
{ 0 1 2 0 -> 1 3 4 0 ,
  1 3 5 4 -> 0 1 6 2 ,
  7 8 2 0 ->= 0 7 8 2 ,
  0 1 2 1 -> 1 3 4 1 ,
  1 3 5 8 -> 0 1 6 6 ,
  7 8 2 1 ->= 0 7 8 6 ,
  0 1 2 7 -> 1 3 4 7 ,
  1 6 2 7 -> 0 7 5 5 ,
  0 1 6 3 -> 0 1 3 5 ,
  7 8 2 7 ->= 0 7 8 3 ,
  2 1 2 0 -> 6 3 4 0 ,
  6 3 5 4 -> 2 1 6 2 ,
  3 8 2 0 ->= 2 7 8 2 ,
  2 1 2 1 -> 6 3 4 1 ,
  6 3 5 8 -> 2 1 6 6 ,
  3 8 2 1 ->= 2 7 8 6 ,
  2 1 2 7 -> 6 3 4 7 ,
  6 6 2 7 -> 2 7 5 5 ,
  2 1 6 3 -> 2 1 3 5 ,
  3 8 2 7 ->= 2 7 8 3 ,
  4 1 2 0 -> 8 3 4 0 ,
  8 3 5 4 -> 4 1 6 2 ,
  4 1 2 1 -> 8 3 4 1 ,
  8 3 5 8 -> 4 1 6 6 ,
  4 1 2 7 -> 8 3 4 7 ,
  8 6 2 7 -> 4 7 5 5 ,
  4 1 6 3 -> 4 1 3 5 }


The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 5:

0 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /
1 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /
2 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /
3 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 1 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /
4 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 1 0 0 0 |
 \         /
5 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 1 |
| 0 0 0 0 0 |
 \         /
6 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /
7 is interpreted by
 /         \
| 1 0 0 1 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /
8 is interpreted by
 /         \
| 1 0 1 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 1 0 0 0 |
 \         /

After renaming modulo { 0->0, 1->1, 2->2, 3->3, 4->4, 5->5, 6->6, 7->7, 8->8 }, 
it remains to prove termination of the 25-rule system
{ 0 1 2 0 -> 1 3 4 0 ,
  1 3 5 4 -> 0 1 6 2 ,
  7 8 2 0 ->= 0 7 8 2 ,
  0 1 2 1 -> 1 3 4 1 ,
  1 3 5 8 -> 0 1 6 6 ,
  7 8 2 1 ->= 0 7 8 6 ,
  0 1 2 7 -> 1 3 4 7 ,
  1 6 2 7 -> 0 7 5 5 ,
  0 1 6 3 -> 0 1 3 5 ,
  7 8 2 7 ->= 0 7 8 3 ,
  2 1 2 0 -> 6 3 4 0 ,
  6 3 5 4 -> 2 1 6 2 ,
  3 8 2 0 ->= 2 7 8 2 ,
  2 1 2 1 -> 6 3 4 1 ,
  6 3 5 8 -> 2 1 6 6 ,
  3 8 2 1 ->= 2 7 8 6 ,
  2 1 2 7 -> 6 3 4 7 ,
  6 6 2 7 -> 2 7 5 5 ,
  2 1 6 3 -> 2 1 3 5 ,
  3 8 2 7 ->= 2 7 8 3 ,
  4 1 2 0 -> 8 3 4 0 ,
  4 1 2 1 -> 8 3 4 1 ,
  4 1 2 7 -> 8 3 4 7 ,
  8 6 2 7 -> 4 7 5 5 ,
  4 1 6 3 -> 4 1 3 5 }


The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 8:

0 is interpreted by
 /               \
| 1 0 1 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 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 |
 \               /
1 is interpreted by
 /               \
| 1 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 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 |
 \               /
2 is interpreted by
 /               \
| 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 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 |
 \               /
3 is interpreted by
 /               \
| 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 0 0 0 0 0 1 0 |
| 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 |
 \               /
4 is interpreted by
 /               \
| 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 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 |
| 0 1 0 0 0 0 0 0 |
 \               /
5 is interpreted by
 /               \
| 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 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 1 |
| 0 0 0 0 0 0 0 0 |
 \               /
6 is interpreted by
 /               \
| 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 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 |
 \               /
7 is interpreted by
 /               \
| 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 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 |
 \               /
8 is interpreted by
 /               \
| 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 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 |
| 0 1 0 0 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 }, 
it remains to prove termination of the 24-rule system
{ 0 1 2 0 -> 1 3 4 0 ,
  1 3 5 4 -> 0 1 6 2 ,
  7 8 2 0 ->= 0 7 8 2 ,
  1 3 5 8 -> 0 1 6 6 ,
  7 8 2 1 ->= 0 7 8 6 ,
  0 1 2 7 -> 1 3 4 7 ,
  1 6 2 7 -> 0 7 5 5 ,
  0 1 6 3 -> 0 1 3 5 ,
  7 8 2 7 ->= 0 7 8 3 ,
  2 1 2 0 -> 6 3 4 0 ,
  6 3 5 4 -> 2 1 6 2 ,
  3 8 2 0 ->= 2 7 8 2 ,
  2 1 2 1 -> 6 3 4 1 ,
  6 3 5 8 -> 2 1 6 6 ,
  3 8 2 1 ->= 2 7 8 6 ,
  2 1 2 7 -> 6 3 4 7 ,
  6 6 2 7 -> 2 7 5 5 ,
  2 1 6 3 -> 2 1 3 5 ,
  3 8 2 7 ->= 2 7 8 3 ,
  4 1 2 0 -> 8 3 4 0 ,
  4 1 2 1 -> 8 3 4 1 ,
  4 1 2 7 -> 8 3 4 7 ,
  8 6 2 7 -> 4 7 5 5 ,
  4 1 6 3 -> 4 1 3 5 }


The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 5:

0 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 1 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /
1 is interpreted by
 /         \
| 1 0 0 1 0 |
| 0 1 0 0 0 |
| 0 1 0 1 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /
2 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 1 0 0 |
| 0 0 0 0 0 |
 \         /
3 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 1 0 0 0 |
 \         /
4 is interpreted by
 /         \
| 1 0 1 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /
5 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /
6 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 1 |
| 0 0 0 0 0 |
 \         /
7 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /
8 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 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 }, 
it remains to prove termination of the 18-rule system
{ 0 1 2 0 -> 1 3 4 0 ,
  1 3 5 4 -> 0 1 6 2 ,
  7 8 2 0 ->= 0 7 8 2 ,
  1 3 5 8 -> 0 1 6 6 ,
  7 8 2 1 ->= 0 7 8 6 ,
  0 1 2 7 -> 1 3 4 7 ,
  1 6 2 7 -> 0 7 5 5 ,
  7 8 2 7 ->= 0 7 8 3 ,
  2 1 2 0 -> 6 3 4 0 ,
  6 3 5 4 -> 2 1 6 2 ,
  3 8 2 0 ->= 2 7 8 2 ,
  2 1 2 1 -> 6 3 4 1 ,
  6 3 5 8 -> 2 1 6 6 ,
  3 8 2 1 ->= 2 7 8 6 ,
  2 1 2 7 -> 6 3 4 7 ,
  6 6 2 7 -> 2 7 5 5 ,
  3 8 2 7 ->= 2 7 8 3 ,
  8 6 2 7 -> 4 7 5 5 }


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 0 |
| 0 1 |
 \   /
2 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
3 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
4 is interpreted by
 /   \
| 1 0 |
| 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 1 |
| 0 1 |
 \   /

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


The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 5:

0 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /
1 is interpreted by
 /         \
| 1 0 1 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /
2 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /
3 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 1 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /
4 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 1 0 0 0 |
 \         /
5 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 1 |
| 0 0 0 0 0 |
 \         /
6 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /
7 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /
8 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /

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


The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 5:

0 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /
1 is interpreted by
 /         \
| 1 0 1 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /
2 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /
3 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 1 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /
4 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 1 0 0 0 |
 \         /
5 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /
6 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /
7 is interpreted by
 /         \
| 1 0 1 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /
8 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 1 |
| 0 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 }, 
it remains to prove termination of the 13-rule system
{ 0 1 2 0 -> 1 3 4 0 ,
  5 6 2 0 ->= 0 5 6 2 ,
  5 6 2 1 ->= 0 5 6 7 ,
  0 1 2 5 -> 1 3 4 5 ,
  1 7 2 5 -> 0 5 8 8 ,
  5 6 2 5 ->= 0 5 6 3 ,
  2 1 2 0 -> 7 3 4 0 ,
  3 6 2 0 ->= 2 5 6 2 ,
  2 1 2 1 -> 7 3 4 1 ,
  3 6 2 1 ->= 2 5 6 7 ,
  2 1 2 5 -> 7 3 4 5 ,
  7 7 2 5 -> 2 5 8 8 ,
  3 6 2 5 ->= 2 5 6 3 }


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 1 |
| 0 1 |
 \   /
2 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
3 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
4 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
5 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
6 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
7 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
8 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

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

The system is trivially terminating.