/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, c->1, b->2 }, 
it remains to prove termination of the 7-rule system
{ 0 1 2 -> 2 0 2 ,
  0 2 1 -> 2 2 1 ,
  0 2 1 ->= 1 1 2 ,
  2 0 1 ->= 1 0 1 ,
  0 0 1 ->= 0 0 2 ,
  2 1 0 ->= 0 1 2 ,
  0 0 0 ->= 1 0 2 }


The system was reversed.

After renaming modulo { 2->0, 1->1, 0->2 }, 
it remains to prove termination of the 7-rule system
{ 0 1 2 -> 0 2 0 ,
  1 0 2 -> 1 0 0 ,
  1 0 2 ->= 0 1 1 ,
  1 2 0 ->= 1 2 1 ,
  1 2 2 ->= 0 2 2 ,
  2 1 0 ->= 0 1 2 ,
  2 2 2 ->= 0 2 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 0 |
| 0 1 |
 \   /
1 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
2 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /

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


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, 2]->2, [2, 0]->3, [0, 2]->4, [2, 1]->5, [1, 0]->6, [2, 2]->7, [1, 1]->8 },
it remains to prove termination of the 36-rule system
{ 0 1 2 3 -> 0 4 3 0 ,
  1 2 3 0 ->= 1 2 5 6 ,
  1 2 7 3 ->= 0 4 7 3 ,
  4 5 6 0 ->= 0 1 2 3 ,
  0 1 2 5 -> 0 4 3 1 ,
  1 2 3 1 ->= 1 2 5 8 ,
  1 2 7 5 ->= 0 4 7 5 ,
  4 5 6 1 ->= 0 1 2 5 ,
  0 1 2 7 -> 0 4 3 4 ,
  1 2 3 4 ->= 1 2 5 2 ,
  1 2 7 7 ->= 0 4 7 7 ,
  4 5 6 4 ->= 0 1 2 7 ,
  6 1 2 3 -> 6 4 3 0 ,
  8 2 3 0 ->= 8 2 5 6 ,
  8 2 7 3 ->= 6 4 7 3 ,
  2 5 6 0 ->= 6 1 2 3 ,
  6 1 2 5 -> 6 4 3 1 ,
  8 2 3 1 ->= 8 2 5 8 ,
  8 2 7 5 ->= 6 4 7 5 ,
  2 5 6 1 ->= 6 1 2 5 ,
  6 1 2 7 -> 6 4 3 4 ,
  8 2 3 4 ->= 8 2 5 2 ,
  8 2 7 7 ->= 6 4 7 7 ,
  2 5 6 4 ->= 6 1 2 7 ,
  3 1 2 3 -> 3 4 3 0 ,
  5 2 3 0 ->= 5 2 5 6 ,
  5 2 7 3 ->= 3 4 7 3 ,
  7 5 6 0 ->= 3 1 2 3 ,
  3 1 2 5 -> 3 4 3 1 ,
  5 2 3 1 ->= 5 2 5 8 ,
  5 2 7 5 ->= 3 4 7 5 ,
  7 5 6 1 ->= 3 1 2 5 ,
  3 1 2 7 -> 3 4 3 4 ,
  5 2 3 4 ->= 5 2 5 2 ,
  5 2 7 7 ->= 3 4 7 7 ,
  7 5 6 4 ->= 3 1 2 7 }


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 1 0 0 0 |
 \         /
2 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 1 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 0 0 |
| 0 0 0 0 0 |
| 0 0 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 1 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 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 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 34-rule system
{ 0 1 2 3 -> 0 4 3 0 ,
  1 2 3 0 ->= 1 2 5 6 ,
  1 2 7 3 ->= 0 4 7 3 ,
  4 5 6 0 ->= 0 1 2 3 ,
  0 1 2 5 -> 0 4 3 1 ,
  1 2 3 1 ->= 1 2 5 8 ,
  1 2 7 5 ->= 0 4 7 5 ,
  0 1 2 7 -> 0 4 3 4 ,
  1 2 3 4 ->= 1 2 5 2 ,
  1 2 7 7 ->= 0 4 7 7 ,
  4 5 6 4 ->= 0 1 2 7 ,
  6 1 2 3 -> 6 4 3 0 ,
  8 2 3 0 ->= 8 2 5 6 ,
  8 2 7 3 ->= 6 4 7 3 ,
  2 5 6 0 ->= 6 1 2 3 ,
  6 1 2 5 -> 6 4 3 1 ,
  8 2 3 1 ->= 8 2 5 8 ,
  8 2 7 5 ->= 6 4 7 5 ,
  2 5 6 1 ->= 6 1 2 5 ,
  6 1 2 7 -> 6 4 3 4 ,
  8 2 3 4 ->= 8 2 5 2 ,
  8 2 7 7 ->= 6 4 7 7 ,
  2 5 6 4 ->= 6 1 2 7 ,
  3 1 2 3 -> 3 4 3 0 ,
  5 2 3 0 ->= 5 2 5 6 ,
  5 2 7 3 ->= 3 4 7 3 ,
  7 5 6 0 ->= 3 1 2 3 ,
  3 1 2 5 -> 3 4 3 1 ,
  5 2 3 1 ->= 5 2 5 8 ,
  5 2 7 5 ->= 3 4 7 5 ,
  3 1 2 7 -> 3 4 3 4 ,
  5 2 3 4 ->= 5 2 5 2 ,
  5 2 7 7 ->= 3 4 7 7 ,
  7 5 6 4 ->= 3 1 2 7 }


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 1 0 0 0 |
 \         /
2 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 1 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 0 0 |
| 0 0 0 0 0 |
| 0 0 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 1 0 0 0 |
 \         /
5 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 |
 \         /
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 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 1 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 32-rule system
{ 0 1 2 3 -> 0 4 3 0 ,
  1 2 3 0 ->= 1 2 5 6 ,
  1 2 7 3 ->= 0 4 7 3 ,
  4 5 6 0 ->= 0 1 2 3 ,
  0 1 2 5 -> 0 4 3 1 ,
  1 2 3 1 ->= 1 2 5 8 ,
  1 2 7 5 ->= 0 4 7 5 ,
  0 1 2 7 -> 0 4 3 4 ,
  1 2 3 4 ->= 1 2 5 2 ,
  1 2 7 7 ->= 0 4 7 7 ,
  6 1 2 3 -> 6 4 3 0 ,
  8 2 3 0 ->= 8 2 5 6 ,
  8 2 7 3 ->= 6 4 7 3 ,
  2 5 6 0 ->= 6 1 2 3 ,
  6 1 2 5 -> 6 4 3 1 ,
  8 2 3 1 ->= 8 2 5 8 ,
  8 2 7 5 ->= 6 4 7 5 ,
  2 5 6 1 ->= 6 1 2 5 ,
  6 1 2 7 -> 6 4 3 4 ,
  8 2 3 4 ->= 8 2 5 2 ,
  8 2 7 7 ->= 6 4 7 7 ,
  2 5 6 4 ->= 6 1 2 7 ,
  3 1 2 3 -> 3 4 3 0 ,
  5 2 3 0 ->= 5 2 5 6 ,
  5 2 7 3 ->= 3 4 7 3 ,
  7 5 6 0 ->= 3 1 2 3 ,
  3 1 2 5 -> 3 4 3 1 ,
  5 2 3 1 ->= 5 2 5 8 ,
  5 2 7 5 ->= 3 4 7 5 ,
  3 1 2 7 -> 3 4 3 4 ,
  5 2 3 4 ->= 5 2 5 2 ,
  5 2 7 7 ->= 3 4 7 7 }


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 1 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 1 0 0 0 |
 \         /
2 is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 1 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 0 0 |
| 0 0 0 0 0 |
| 0 0 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 1 0 0 |
 \         /
5 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 |
 \         /
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 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 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 30-rule system
{ 0 1 2 3 -> 0 4 3 0 ,
  1 2 3 0 ->= 1 2 5 6 ,
  1 2 7 3 ->= 0 4 7 3 ,
  0 1 2 5 -> 0 4 3 1 ,
  1 2 3 1 ->= 1 2 5 8 ,
  1 2 7 5 ->= 0 4 7 5 ,
  0 1 2 7 -> 0 4 3 4 ,
  1 2 3 4 ->= 1 2 5 2 ,
  1 2 7 7 ->= 0 4 7 7 ,
  6 1 2 3 -> 6 4 3 0 ,
  8 2 3 0 ->= 8 2 5 6 ,
  8 2 7 3 ->= 6 4 7 3 ,
  2 5 6 0 ->= 6 1 2 3 ,
  6 1 2 5 -> 6 4 3 1 ,
  8 2 3 1 ->= 8 2 5 8 ,
  8 2 7 5 ->= 6 4 7 5 ,
  2 5 6 1 ->= 6 1 2 5 ,
  6 1 2 7 -> 6 4 3 4 ,
  8 2 3 4 ->= 8 2 5 2 ,
  8 2 7 7 ->= 6 4 7 7 ,
  2 5 6 4 ->= 6 1 2 7 ,
  3 1 2 3 -> 3 4 3 0 ,
  5 2 3 0 ->= 5 2 5 6 ,
  5 2 7 3 ->= 3 4 7 3 ,
  3 1 2 5 -> 3 4 3 1 ,
  5 2 3 1 ->= 5 2 5 8 ,
  5 2 7 5 ->= 3 4 7 5 ,
  3 1 2 7 -> 3 4 3 4 ,
  5 2 3 4 ->= 5 2 5 2 ,
  5 2 7 7 ->= 3 4 7 7 }


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 1 |
| 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 1 |
| 0 1 |
 \   /

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

After renaming modulo { 0->0, 1->1, 2->2, 3->3, 4->4, 5->5, 6->6, 8->7, 7->8 }, 
it remains to prove termination of the 21-rule system
{ 0 1 2 3 -> 0 4 3 0 ,
  1 2 5 3 ->= 0 4 5 3 ,
  0 1 2 6 -> 0 4 3 1 ,
  1 2 5 6 ->= 0 4 5 6 ,
  0 1 2 5 -> 0 4 3 4 ,
  1 2 3 4 ->= 1 2 6 2 ,
  1 2 5 5 ->= 0 4 5 5 ,
  7 1 2 3 -> 7 4 3 0 ,
  2 6 7 0 ->= 7 1 2 3 ,
  7 1 2 6 -> 7 4 3 1 ,
  2 6 7 1 ->= 7 1 2 6 ,
  7 1 2 5 -> 7 4 3 4 ,
  8 2 3 4 ->= 8 2 6 2 ,
  2 6 7 4 ->= 7 1 2 5 ,
  3 1 2 3 -> 3 4 3 0 ,
  6 2 5 3 ->= 3 4 5 3 ,
  3 1 2 6 -> 3 4 3 1 ,
  6 2 5 6 ->= 3 4 5 6 ,
  3 1 2 5 -> 3 4 3 4 ,
  6 2 3 4 ->= 6 2 6 2 ,
  6 2 5 5 ->= 3 4 5 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 1 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /
2 is interpreted by
 /         \
| 1 1 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 1 |
| 0 0 0 0 0 |
 \         /
3 is interpreted by
 /         \
| 1 1 0 0 0 |
| 0 1 0 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 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 1 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 2 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, 6->3, 4->4, 3->5, 5->6, 7->7, 8->8 }, 
it remains to prove termination of the 13-rule system
{ 0 1 2 3 -> 0 4 5 1 ,
  0 1 2 6 -> 0 4 5 4 ,
  1 2 5 4 ->= 1 2 3 2 ,
  2 3 7 0 ->= 7 1 2 5 ,
  7 1 2 3 -> 7 4 5 1 ,
  8 2 5 4 ->= 8 2 3 2 ,
  2 3 7 4 ->= 7 1 2 6 ,
  3 2 6 5 ->= 5 4 6 5 ,
  5 1 2 3 -> 5 4 5 1 ,
  3 2 6 3 ->= 5 4 6 3 ,
  5 1 2 6 -> 5 4 5 4 ,
  3 2 5 4 ->= 3 2 3 2 ,
  3 2 6 6 ->= 5 4 6 6 }


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 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 1 |
| 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 12-rule system
{ 0 1 2 3 -> 0 4 5 1 ,
  0 1 2 6 -> 0 4 5 4 ,
  1 2 5 4 ->= 1 2 3 2 ,
  7 1 2 3 -> 7 4 5 1 ,
  8 2 5 4 ->= 8 2 3 2 ,
  2 3 7 4 ->= 7 1 2 6 ,
  3 2 6 5 ->= 5 4 6 5 ,
  5 1 2 3 -> 5 4 5 1 ,
  3 2 6 3 ->= 5 4 6 3 ,
  5 1 2 6 -> 5 4 5 4 ,
  3 2 5 4 ->= 3 2 3 2 ,
  3 2 6 6 ->= 5 4 6 6 }


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 1 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 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 0 0 1 |
| 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 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 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 0 |
| 0 1 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, 7->6, 8->7, 6->8 }, 
it remains to prove termination of the 11-rule system
{ 0 1 2 3 -> 0 4 5 1 ,
  1 2 5 4 ->= 1 2 3 2 ,
  6 1 2 3 -> 6 4 5 1 ,
  7 2 5 4 ->= 7 2 3 2 ,
  2 3 6 4 ->= 6 1 2 8 ,
  3 2 8 5 ->= 5 4 8 5 ,
  5 1 2 3 -> 5 4 5 1 ,
  3 2 8 3 ->= 5 4 8 3 ,
  5 1 2 8 -> 5 4 5 4 ,
  3 2 5 4 ->= 3 2 3 2 ,
  3 2 8 8 ->= 5 4 8 8 }


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 1 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 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 0 0 1 |
| 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 0 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 1 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 { 1->0, 2->1, 5->2, 4->3, 3->4, 6->5, 7->6, 8->7 }, 
it remains to prove termination of the 10-rule system
{ 0 1 2 3 ->= 0 1 4 1 ,
  5 0 1 4 -> 5 3 2 0 ,
  6 1 2 3 ->= 6 1 4 1 ,
  1 4 5 3 ->= 5 0 1 7 ,
  4 1 7 2 ->= 2 3 7 2 ,
  2 0 1 4 -> 2 3 2 0 ,
  4 1 7 4 ->= 2 3 7 4 ,
  2 0 1 7 -> 2 3 2 3 ,
  4 1 2 3 ->= 4 1 4 1 ,
  4 1 7 7 ->= 2 3 7 7 }


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 1 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 1 |
| 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 1 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 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 1 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 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, 6->5, 5->6, 7->7 }, 
it remains to prove termination of the 9-rule system
{ 0 1 2 3 ->= 0 1 4 1 ,
  5 1 2 3 ->= 5 1 4 1 ,
  1 4 6 3 ->= 6 0 1 7 ,
  4 1 7 2 ->= 2 3 7 2 ,
  2 0 1 4 -> 2 3 2 0 ,
  4 1 7 4 ->= 2 3 7 4 ,
  2 0 1 7 -> 2 3 2 3 ,
  4 1 2 3 ->= 4 1 4 1 ,
  4 1 7 7 ->= 2 3 7 7 }


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


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 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 1 |
| 0 1 |
 \   /
6 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /

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


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

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

The system is trivially terminating.