YES

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


Applying the dependency pairs transformation.

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

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


The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 5:

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

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

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


The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 6:

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

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

The system is trivially terminating.