YES

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


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 11 |
|  0  1 |
 \     /
1 is interpreted by
 /     \
|  1 16 |
|  0  1 |
 \     /
2 is interpreted by
 /     \
|  1 23 |
|  0  1 |
 \     /
3 is interpreted by
 /     \
|  1 24 |
|  0  1 |
 \     /

After renaming modulo {  }, 
it remains to prove termination of the 0-rule system
{ }

The system is trivially terminating.