YES

After renaming modulo { R->0, r->1, p->2, P->3 }, 
it remains to prove termination of the 8-rule system
{ 0 -> 1 ,
  1 2 -> 2 2 1 3 ,
  1 1 -> ,
  1 3 3 -> 3 3 1 ,
  2 3 -> ,
  3 2 -> ,
  1 0 -> ,
  0 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 { 1->0, 2->1, 3->2 }, 
it remains to prove termination of the 5-rule system
{ 0 1 -> 1 1 0 2 ,
  0 0 -> ,
  0 2 2 -> 2 2 0 ,
  1 2 -> ,
  2 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 |
 \   /

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


The system was reversed.

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


Applying the dependency pairs transformation.

After renaming modulo { (0,true)->0, (1,false)->1, (2,true)->2, (0,false)->3, (2,false)->4 },
it remains to prove termination of the 9-rule system
{ 0 1 -> 2 1 3 3 ,
  0 1 -> 0 3 ,
  0 1 -> 0 ,
  2 4 1 -> 2 4 ,
  2 4 1 -> 2 ,
  3 1 ->= 4 1 3 3 ,
  4 4 1 ->= 1 4 4 ,
  4 3 ->= ,
  3 4 ->= }


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

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

The system is trivially terminating.