YES

After renaming modulo { a->0, b->1, c->2, d->3 }, 
it remains to prove termination of the 8-rule system
{ 0 0 -> 0 1 0 1 0 ,
  2 0 -> 0 1 0 0 2 ,
  1 1 1 -> 0 1 ,
  2 1 -> 0 0 2 ,
  2 1 -> 1 0 3 ,
  3 3 -> 3 1 3 1 3 ,
  2 2 -> 2 3 2 ,
  0 0 0 -> 0 1 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 |
 \   /
3 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

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

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


The system was reversed.

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


Applying the dependency pairs transformation.

After renaming modulo { (0,true)->0, (0,false)->1, (1,false)->2, (1,true)->3, (2,false)->4, (2,true)->5, (3,false)->6 },
it remains to prove termination of the 26-rule system
{ 0 1 -> 0 2 1 2 1 ,
  0 1 -> 3 1 2 1 ,
  0 1 -> 0 2 1 ,
  0 1 -> 3 1 ,
  0 1 -> 0 ,
  0 4 -> 5 1 1 2 1 ,
  0 4 -> 0 1 2 1 ,
  0 4 -> 0 2 1 ,
  0 4 -> 3 1 ,
  0 4 -> 0 ,
  3 2 2 -> 3 1 ,
  3 2 2 -> 0 ,
  3 4 -> 5 1 1 ,
  3 4 -> 0 1 ,
  3 4 -> 0 ,
  5 4 -> 5 6 4 ,
  5 4 -> 5 ,
  0 1 1 -> 3 2 1 ,
  0 1 1 -> 3 1 ,
  0 1 1 -> 0 ,
  1 1 ->= 1 2 1 2 1 ,
  1 4 ->= 4 1 1 2 1 ,
  2 2 2 ->= 2 1 ,
  2 4 ->= 4 1 1 ,
  4 4 ->= 4 6 4 ,
  1 1 1 ->= 2 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 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 |
 \   /

After renaming modulo { 0->0, 1->1, 2->2, 3->3, 5->4, 4->5, 6->6 }, 
it remains to prove termination of the 17-rule system
{ 0 1 -> 0 2 1 2 1 ,
  0 1 -> 3 1 2 1 ,
  0 1 -> 0 2 1 ,
  0 1 -> 3 1 ,
  0 1 -> 0 ,
  3 2 2 -> 3 1 ,
  3 2 2 -> 0 ,
  4 5 -> 4 6 5 ,
  0 1 1 -> 3 2 1 ,
  0 1 1 -> 3 1 ,
  0 1 1 -> 0 ,
  1 1 ->= 1 2 1 2 1 ,
  1 5 ->= 5 1 1 2 1 ,
  2 2 2 ->= 2 1 ,
  2 5 ->= 5 1 1 ,
  5 5 ->= 5 6 5 ,
  1 1 1 ->= 2 2 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 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
1 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 1 0 |
 \     /
2 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 1 0 |
 \     /
3 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 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 |
 \     /

After renaming modulo { 0->0, 1->1, 2->2, 3->3, 5->4, 6->5 }, 
it remains to prove termination of the 16-rule system
{ 0 1 -> 0 2 1 2 1 ,
  0 1 -> 3 1 2 1 ,
  0 1 -> 0 2 1 ,
  0 1 -> 3 1 ,
  0 1 -> 0 ,
  3 2 2 -> 3 1 ,
  3 2 2 -> 0 ,
  0 1 1 -> 3 2 1 ,
  0 1 1 -> 3 1 ,
  0 1 1 -> 0 ,
  1 1 ->= 1 2 1 2 1 ,
  1 4 ->= 4 1 1 2 1 ,
  2 2 2 ->= 2 1 ,
  2 4 ->= 4 1 1 ,
  4 4 ->= 4 5 4 ,
  1 1 1 ->= 2 2 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 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 1 |
 \     /
3 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
4 is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 1 0 |
 \     /
5 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 }, 
it remains to prove termination of the 15-rule system
{ 0 1 -> 0 2 1 2 1 ,
  0 1 -> 3 1 2 1 ,
  0 1 -> 0 2 1 ,
  0 1 -> 3 1 ,
  0 1 -> 0 ,
  3 2 2 -> 3 1 ,
  3 2 2 -> 0 ,
  0 1 1 -> 3 2 1 ,
  0 1 1 -> 3 1 ,
  0 1 1 -> 0 ,
  1 1 ->= 1 2 1 2 1 ,
  1 4 ->= 4 1 1 2 1 ,
  2 2 2 ->= 2 1 ,
  2 4 ->= 4 1 1 ,
  1 1 1 ->= 2 2 1 }


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

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

The system is trivially terminating.