YES

After renaming modulo { a->0, b->1, c->2, d->3, f->4, g->5 }, 
it remains to prove termination of the 9-rule system
{ 0 1 2 0 -> 1 0 2 1 0 1 ,
  0 3 -> 2 ,
  0 4 4 -> 5 ,
  1 5 -> 5 1 ,
  2 -> 4 4 ,
  2 0 2 -> 1 2 0 1 2 ,
  2 3 -> 0 0 ,
  5 -> 2 0 ,
  5 -> 3 3 3 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 12 |
|  0  1 |
 \     /
1 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
2 is interpreted by
 /     \
|  1 18 |
|  0  1 |
 \     /
3 is interpreted by
 /   \
| 1 7 |
| 0 1 |
 \   /
4 is interpreted by
 /   \
| 1 9 |
| 0 1 |
 \   /
5 is interpreted by
 /     \
|  1 30 |
|  0  1 |
 \     /

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


Applying the dependency pairs transformation.

After renaming modulo { (0,true)->0, (1,false)->1, (2,false)->2, (0,false)->3, (1,true)->4, (2,true)->5, (3,false)->6, (4,true)->7, (4,false)->8 },
it remains to prove termination of the 22-rule system
{ 0 1 2 3 -> 4 3 2 1 3 1 ,
  0 1 2 3 -> 0 2 1 3 1 ,
  0 1 2 3 -> 5 1 3 1 ,
  0 1 2 3 -> 4 3 1 ,
  0 1 2 3 -> 0 1 ,
  0 1 2 3 -> 4 ,
  0 6 6 -> 7 ,
  4 8 -> 7 1 ,
  4 8 -> 4 ,
  5 3 2 -> 4 2 3 1 2 ,
  5 3 2 -> 5 3 1 2 ,
  5 3 2 -> 0 1 2 ,
  5 3 2 -> 4 2 ,
  5 3 2 -> 5 ,
  7 -> 5 3 ,
  7 -> 0 ,
  3 1 2 3 ->= 1 3 2 1 3 1 ,
  3 6 6 ->= 8 ,
  1 8 ->= 8 1 ,
  2 ->= 6 6 ,
  2 3 2 ->= 1 2 3 1 2 ,
  8 ->= 2 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 0 |
| 0 1 |
 \   /
2 is interpreted by
 /   \
| 1 2 |
| 0 1 |
 \   /
3 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
4 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
5 is interpreted by
 /   \
| 1 2 |
| 0 1 |
 \   /
6 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
7 is interpreted by
 /   \
| 1 2 |
| 0 1 |
 \   /
8 is interpreted by
 /   \
| 1 2 |
| 0 1 |
 \   /

After renaming modulo { 0->0, 1->1, 2->2, 3->3, 4->4, 8->5, 7->6, 5->7, 6->8 }, 
it remains to prove termination of the 11-rule system
{ 0 1 2 3 -> 0 2 1 3 1 ,
  4 5 -> 6 1 ,
  7 3 2 -> 4 2 3 1 2 ,
  7 3 2 -> 7 3 1 2 ,
  6 -> 7 3 ,
  3 1 2 3 ->= 1 3 2 1 3 1 ,
  3 8 8 ->= 5 ,
  1 5 ->= 5 1 ,
  2 ->= 8 8 ,
  2 3 2 ->= 1 2 3 1 2 ,
  5 ->= 2 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 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
1 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 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 1 0 |
 \     /
4 is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 0 0 |
 \     /
5 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 1 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 |
 \     /

After renaming modulo { 0->0, 1->1, 2->2, 3->3, 7->4, 4->5, 6->6, 8->7, 5->8 }, 
it remains to prove termination of the 10-rule system
{ 0 1 2 3 -> 0 2 1 3 1 ,
  4 3 2 -> 5 2 3 1 2 ,
  4 3 2 -> 4 3 1 2 ,
  6 -> 4 3 ,
  3 1 2 3 ->= 1 3 2 1 3 1 ,
  3 7 7 ->= 8 ,
  1 8 ->= 8 1 ,
  2 ->= 7 7 ,
  2 3 2 ->= 1 2 3 1 2 ,
  8 ->= 2 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 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 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, 7->6, 8->7 }, 
it remains to prove termination of the 9-rule system
{ 0 1 2 3 -> 0 2 1 3 1 ,
  4 3 2 -> 5 2 3 1 2 ,
  4 3 2 -> 4 3 1 2 ,
  3 1 2 3 ->= 1 3 2 1 3 1 ,
  3 6 6 ->= 7 ,
  1 7 ->= 7 1 ,
  2 ->= 6 6 ,
  2 3 2 ->= 1 2 3 1 2 ,
  7 ->= 2 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 0 |
| 0 1 |
 \   /
2 is interpreted by
 /   \
| 1 0 |
| 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 |
 \   /

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

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

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

The system is trivially terminating.