YES

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


Applying the dependency pairs transformation.

After renaming modulo { (0,true)->0, (0,false)->1, (1,false)->2 },
it remains to prove termination of the 14-rule system
{ 0 1 -> 0 2 1 ,
  0 1 -> 0 ,
  0 1 2 1 -> 0 2 1 2 2 1 1 ,
  0 1 2 1 -> 0 2 2 1 1 ,
  0 1 2 1 -> 0 1 ,
  0 1 2 1 -> 0 ,
  0 2 1 -> 0 2 2 1 ,
  0 2 1 -> 0 ,
  0 2 2 1 2 1 -> 0 1 ,
  0 2 2 1 2 1 -> 0 ,
  1 1 ->= 1 2 1 ,
  1 1 2 1 ->= 1 2 1 2 2 1 1 ,
  1 2 1 ->= 1 2 2 1 ,
  1 2 2 1 2 1 ->= 1 1 }


The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 4:

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

After renaming modulo { 0->0, 1->1, 2->2 }, 
it remains to prove termination of the 9-rule system
{ 0 1 -> 0 2 1 ,
  0 1 -> 0 ,
  0 1 2 1 -> 0 2 1 2 2 1 1 ,
  0 1 2 1 -> 0 2 2 1 1 ,
  0 2 1 -> 0 2 2 1 ,
  0 2 1 -> 0 ,
  1 1 ->= 1 2 1 ,
  1 1 2 1 ->= 1 2 1 2 2 1 1 ,
  1 2 2 1 2 1 ->= 1 1 }


The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 4:

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

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

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

The system is trivially terminating.