YES

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


Applying the dependency pairs transformation.

After renaming modulo { (0,true)->0, (0,false)->1, (1,false)->2 },
it remains to prove termination of the 7-rule system
{ 0 1 2 1 1 2 1 -> 0 2 1 1 2 1 1 1 2 ,
  0 1 2 1 1 2 1 -> 0 1 2 1 1 1 2 ,
  0 1 2 1 1 2 1 -> 0 2 1 1 1 2 ,
  0 1 2 1 1 2 1 -> 0 1 1 2 ,
  0 1 2 1 1 2 1 -> 0 1 2 ,
  0 1 2 1 1 2 1 -> 0 2 ,
  1 1 2 1 1 2 1 ->= 1 2 1 1 2 1 1 1 2 }


The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 8:

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

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

The system is trivially terminating.