YES

Problem:
 R(x1) -> r(x1)
 r(p(x1)) -> p(p(r(P(x1))))
 r(r(x1)) -> x1
 r(P(P(x1))) -> P(P(r(x1)))
 p(P(x1)) -> x1
 P(p(x1)) -> x1
 r(R(x1)) -> x1
 R(r(x1)) -> x1

Proof:
 DP Processor:
  DPs:
   R#(x1) -> r#(x1)
   r#(p(x1)) -> P#(x1)
   r#(p(x1)) -> r#(P(x1))
   r#(p(x1)) -> p#(r(P(x1)))
   r#(p(x1)) -> p#(p(r(P(x1))))
   r#(P(P(x1))) -> r#(x1)
   r#(P(P(x1))) -> P#(r(x1))
   r#(P(P(x1))) -> P#(P(r(x1)))
  TRS:
   R(x1) -> r(x1)
   r(p(x1)) -> p(p(r(P(x1))))
   r(r(x1)) -> x1
   r(P(P(x1))) -> P(P(r(x1)))
   p(P(x1)) -> x1
   P(p(x1)) -> x1
   r(R(x1)) -> x1
   R(r(x1)) -> x1
  Matrix Interpretation Processor: dim=3
   
   interpretation:
    [p#](x0) = [0],
    
    [P#](x0) = [0 0 1]x0,
    
    [r#](x0) = [1 1 1]x0,
    
    [R#](x0) = [1 1 1]x0 + [1],
    
                   [0]
    [P](x0) = x0 + [0]
                   [1],
    
                   [1]
    [p](x0) = x0 + [1]
                   [0],
    
              [1 1 0]  
    [r](x0) = [1 1 0]x0
              [1 0 1]  ,
    
              [1 1 1]     [0]
    [R](x0) = [1 1 0]x0 + [0]
              [1 1 1]     [1]
   orientation:
    R#(x1) = [1 1 1]x1 + [1] >= [1 1 1]x1 = r#(x1)
    
    r#(p(x1)) = [1 1 1]x1 + [2] >= [0 0 1]x1 = P#(x1)
    
    r#(p(x1)) = [1 1 1]x1 + [2] >= [1 1 1]x1 + [1] = r#(P(x1))
    
    r#(p(x1)) = [1 1 1]x1 + [2] >= [0] = p#(r(P(x1)))
    
    r#(p(x1)) = [1 1 1]x1 + [2] >= [0] = p#(p(r(P(x1))))
    
    r#(P(P(x1))) = [1 1 1]x1 + [2] >= [1 1 1]x1 = r#(x1)
    
    r#(P(P(x1))) = [1 1 1]x1 + [2] >= [1 0 1]x1 = P#(r(x1))
    
    r#(P(P(x1))) = [1 1 1]x1 + [2] >= [1 0 1]x1 + [1] = P#(P(r(x1)))
    
            [1 1 1]     [0]    [1 1 0]          
    R(x1) = [1 1 0]x1 + [0] >= [1 1 0]x1 = r(x1)
            [1 1 1]     [1]    [1 0 1]          
    
               [1 1 0]     [2]    [1 1 0]     [2]                 
    r(p(x1)) = [1 1 0]x1 + [2] >= [1 1 0]x1 + [2] = p(p(r(P(x1))))
               [1 0 1]     [1]    [1 0 1]     [1]                 
    
               [2 2 0]             
    r(r(x1)) = [2 2 0]x1 >= x1 = x1
               [2 1 1]             
    
                  [1 1 0]     [0]    [1 1 0]     [0]              
    r(P(P(x1))) = [1 1 0]x1 + [0] >= [1 1 0]x1 + [0] = P(P(r(x1)))
                  [1 0 1]     [2]    [1 0 1]     [2]              
    
                    [1]           
    p(P(x1)) = x1 + [1] >= x1 = x1
                    [1]           
    
                    [1]           
    P(p(x1)) = x1 + [1] >= x1 = x1
                    [1]           
    
               [2 2 1]     [0]           
    r(R(x1)) = [2 2 1]x1 + [0] >= x1 = x1
               [2 2 2]     [1]           
    
               [3 2 1]     [0]           
    R(r(x1)) = [2 2 0]x1 + [0] >= x1 = x1
               [3 2 1]     [1]           
   problem:
    DPs:
     
    TRS:
     R(x1) -> r(x1)
     r(p(x1)) -> p(p(r(P(x1))))
     r(r(x1)) -> x1
     r(P(P(x1))) -> P(P(r(x1)))
     p(P(x1)) -> x1
     P(p(x1)) -> x1
     r(R(x1)) -> x1
     R(r(x1)) -> x1
   Qed