YES

After renaming modulo { a->0, b->1, C->2, c->3, A->4, B->5 }, 
it remains to prove termination of the 24-rule system
{ 0 0 1 1 -> 2 2 ,
  1 1 3 3 -> 4 4 ,
  3 3 0 0 -> 5 5 ,
  4 4 2 2 -> 1 1 ,
  2 2 5 5 -> 0 0 ,
  5 5 4 4 -> 3 3 ,
  0 0 0 0 0 0 0 0 0 0 -> 4 4 4 4 4 4 ,
  4 4 4 4 4 4 4 4 -> 0 0 0 0 0 0 0 0 ,
  1 1 1 1 1 1 1 1 1 1 -> 5 5 5 5 5 5 ,
  5 5 5 5 5 5 5 5 -> 1 1 1 1 1 1 1 1 ,
  3 3 3 3 3 3 3 3 3 3 -> 2 2 2 2 2 2 ,
  2 2 2 2 2 2 2 2 -> 3 3 3 3 3 3 3 3 ,
  5 5 0 0 0 0 0 0 0 0 -> 3 3 4 4 4 4 4 4 ,
  4 4 4 4 4 4 1 1 -> 0 0 0 0 0 0 0 0 2 2 ,
  2 2 1 1 1 1 1 1 1 1 -> 0 0 5 5 5 5 5 5 ,
  5 5 5 5 5 5 3 3 -> 1 1 1 1 1 1 1 1 4 4 ,
  4 4 3 3 3 3 3 3 3 3 -> 1 1 2 2 2 2 2 2 ,
  2 2 2 2 2 2 0 0 -> 3 3 3 3 3 3 3 3 5 5 ,
  0 0 4 4 -> ,
  4 4 0 0 -> ,
  1 1 5 5 -> ,
  5 5 1 1 -> ,
  3 3 2 2 -> ,
  2 2 3 3 -> }


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

After renaming modulo { 5->0, 0->1, 3->2, 4->3, 1->4, 2->5 }, 
it remains to prove termination of the 6-rule system
{ 0 0 1 1 1 1 1 1 1 1 -> 2 2 3 3 3 3 3 3 ,
  3 3 3 3 3 3 4 4 -> 1 1 1 1 1 1 1 1 5 5 ,
  5 5 4 4 4 4 4 4 4 4 -> 1 1 0 0 0 0 0 0 ,
  0 0 0 0 0 0 2 2 -> 4 4 4 4 4 4 4 4 3 3 ,
  3 3 2 2 2 2 2 2 2 2 -> 4 4 5 5 5 5 5 5 ,
  5 5 5 5 5 5 1 1 -> 2 2 2 2 2 2 2 2 0 0 }


Applying the dependency pairs transformation.

After renaming modulo { (0,true)->0, (0,false)->1, (1,false)->2, (3,true)->3, (3,false)->4, (4,false)->5, (5,true)->6, (5,false)->7, (2,false)->8 },
it remains to prove termination of the 30-rule system
{ 0 1 2 2 2 2 2 2 2 2 -> 3 4 4 4 4 4 ,
  0 1 2 2 2 2 2 2 2 2 -> 3 4 4 4 4 ,
  0 1 2 2 2 2 2 2 2 2 -> 3 4 4 4 ,
  0 1 2 2 2 2 2 2 2 2 -> 3 4 4 ,
  0 1 2 2 2 2 2 2 2 2 -> 3 4 ,
  0 1 2 2 2 2 2 2 2 2 -> 3 ,
  3 4 4 4 4 4 5 5 -> 6 7 ,
  3 4 4 4 4 4 5 5 -> 6 ,
  6 7 5 5 5 5 5 5 5 5 -> 0 1 1 1 1 1 ,
  6 7 5 5 5 5 5 5 5 5 -> 0 1 1 1 1 ,
  6 7 5 5 5 5 5 5 5 5 -> 0 1 1 1 ,
  6 7 5 5 5 5 5 5 5 5 -> 0 1 1 ,
  6 7 5 5 5 5 5 5 5 5 -> 0 1 ,
  6 7 5 5 5 5 5 5 5 5 -> 0 ,
  0 1 1 1 1 1 8 8 -> 3 4 ,
  0 1 1 1 1 1 8 8 -> 3 ,
  3 4 8 8 8 8 8 8 8 8 -> 6 7 7 7 7 7 ,
  3 4 8 8 8 8 8 8 8 8 -> 6 7 7 7 7 ,
  3 4 8 8 8 8 8 8 8 8 -> 6 7 7 7 ,
  3 4 8 8 8 8 8 8 8 8 -> 6 7 7 ,
  3 4 8 8 8 8 8 8 8 8 -> 6 7 ,
  3 4 8 8 8 8 8 8 8 8 -> 6 ,
  6 7 7 7 7 7 2 2 -> 0 1 ,
  6 7 7 7 7 7 2 2 -> 0 ,
  1 1 2 2 2 2 2 2 2 2 ->= 8 8 4 4 4 4 4 4 ,
  4 4 4 4 4 4 5 5 ->= 2 2 2 2 2 2 2 2 7 7 ,
  7 7 5 5 5 5 5 5 5 5 ->= 2 2 1 1 1 1 1 1 ,
  1 1 1 1 1 1 8 8 ->= 5 5 5 5 5 5 5 5 4 4 ,
  4 4 8 8 8 8 8 8 8 8 ->= 5 5 7 7 7 7 7 7 ,
  7 7 7 7 7 7 2 2 ->= 8 8 8 8 8 8 8 8 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 3 |
| 0 1 |
 \   /
1 is interpreted by
 /   \
| 1 3 |
| 0 1 |
 \   /
2 is interpreted by
 /   \
| 1 2 |
| 0 1 |
 \   /
3 is interpreted by
 /   \
| 1 6 |
| 0 1 |
 \   /
4 is interpreted by
 /   \
| 1 3 |
| 0 1 |
 \   /
5 is interpreted by
 /   \
| 1 2 |
| 0 1 |
 \   /
6 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
7 is interpreted by
 /   \
| 1 3 |
| 0 1 |
 \   /
8 is interpreted by
 /   \
| 1 2 |
| 0 1 |
 \   /

After renaming modulo { 1->0, 2->1, 8->2, 4->3, 5->4, 7->5 }, 
it remains to prove termination of the 6-rule system
{ 0 0 1 1 1 1 1 1 1 1 ->= 2 2 3 3 3 3 3 3 ,
  3 3 3 3 3 3 4 4 ->= 1 1 1 1 1 1 1 1 5 5 ,
  5 5 4 4 4 4 4 4 4 4 ->= 1 1 0 0 0 0 0 0 ,
  0 0 0 0 0 0 2 2 ->= 4 4 4 4 4 4 4 4 3 3 ,
  3 3 2 2 2 2 2 2 2 2 ->= 4 4 5 5 5 5 5 5 ,
  5 5 5 5 5 5 1 1 ->= 2 2 2 2 2 2 2 2 0 0 }

The system is trivially terminating.