YES

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


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 25 |
|  0  1 |
 \     /
1 is interpreted by
 /     \
|  1 17 |
|  0  1 |
 \     /
2 is interpreted by
 /     \
|  1 11 |
|  0  1 |
 \     /
3 is interpreted by
 /   \
| 1 7 |
| 0 1 |
 \   /

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


Applying the dependency pairs transformation.

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

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

The system is trivially terminating.