YES

After renaming modulo { a->0, b->1 }, 
it remains to prove termination of the 15-rule system
{ 0 0 -> 1 ,
  0 0 0 -> 0 1 0 ,
  0 1 0 -> 1 1 1 ,
  0 0 0 0 -> 0 0 1 0 0 ,
  0 0 1 0 -> 0 1 1 0 1 ,
  0 1 0 0 -> 1 0 1 1 0 ,
  0 1 1 0 -> 1 1 1 1 1 ,
  0 0 0 0 0 -> 0 0 0 1 0 0 0 ,
  0 0 0 1 0 -> 0 0 1 1 0 0 1 ,
  0 0 1 0 0 -> 0 1 0 1 0 1 0 ,
  0 0 1 1 0 -> 0 1 1 1 0 1 1 ,
  0 1 0 0 0 -> 1 0 0 1 1 0 0 ,
  0 1 0 1 0 -> 1 0 1 1 1 0 1 ,
  0 1 1 0 0 -> 1 1 0 1 1 1 0 ,
  0 1 1 1 0 -> 1 1 1 1 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 1 0 |
 \     /
1 is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /

After renaming modulo { 0->0, 1->1 }, 
it remains to prove termination of the 7-rule system
{ 0 1 0 -> 1 1 1 ,
  0 1 1 0 -> 1 1 1 1 1 ,
  0 0 0 0 0 -> 0 0 0 1 0 0 0 ,
  0 0 0 1 0 -> 0 0 1 1 0 0 1 ,
  0 1 0 0 0 -> 1 0 0 1 1 0 0 ,
  0 1 0 1 0 -> 1 0 1 1 1 0 1 ,
  0 1 1 1 0 -> 1 1 1 1 1 1 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 1 0 0 |
| 0 0 0 0 1 0 |
| 0 0 0 0 0 1 |
| 0 1 0 0 0 0 |
 \           /
1 is interpreted by
 /           \
| 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 |
 \           /

After renaming modulo { 0->0, 1->1 }, 
it remains to prove termination of the 6-rule system
{ 0 1 0 -> 1 1 1 ,
  0 1 1 0 -> 1 1 1 1 1 ,
  0 0 0 1 0 -> 0 0 1 1 0 0 1 ,
  0 1 0 0 0 -> 1 0 0 1 1 0 0 ,
  0 1 0 1 0 -> 1 0 1 1 1 0 1 ,
  0 1 1 1 0 -> 1 1 1 1 1 1 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 |
 \   /

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


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

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


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 1 0 |
| 0 0 0 0 0 1 |
| 0 1 0 0 0 0 |
 \           /
1 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 0 0 |
| 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
 \           /

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

The system is trivially terminating.