YES

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

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


Applying the dependency pairs transformation.

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

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


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 2 |
| 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 |
 \   /

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

The system is trivially terminating.