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:
 Arctic Interpretation Processor:
  dimension: 1
  interpretation:
   [P](x0) = x0,
   
   [p](x0) = x0,
   
   [r](x0) = 1x0,
   
   [R](x0) = 1x0
  orientation:
   R(x1) = 1x1 >= 1x1 = r(x1)
   
   r(p(x1)) = 1x1 >= 1x1 = p(p(r(P(x1))))
   
   r(r(x1)) = 2x1 >= x1 = x1
   
   r(P(P(x1))) = 1x1 >= 1x1 = P(P(r(x1)))
   
   p(P(x1)) = x1 >= x1 = x1
   
   P(p(x1)) = x1 >= x1 = x1
   
   r(R(x1)) = 2x1 >= x1 = x1
   
   R(r(x1)) = 2x1 >= x1 = x1
  problem:
   R(x1) -> r(x1)
   r(p(x1)) -> p(p(r(P(x1))))
   r(P(P(x1))) -> P(P(r(x1)))
   p(P(x1)) -> x1
   P(p(x1)) -> x1
  Arctic Interpretation Processor:
   dimension: 2
   interpretation:
              [0 0]  
    [P](x0) = [0 0]x0,
    
              [0 0]  
    [p](x0) = [0 0]x0,
    
              [0 0]  
    [r](x0) = [0 0]x0,
    
              [2 2]  
    [R](x0) = [1 2]x0
   orientation:
            [2 2]      [0 0]          
    R(x1) = [1 2]x1 >= [0 0]x1 = r(x1)
    
               [0 0]      [0 0]                   
    r(p(x1)) = [0 0]x1 >= [0 0]x1 = p(p(r(P(x1))))
    
                  [0 0]      [0 0]                
    r(P(P(x1))) = [0 0]x1 >= [0 0]x1 = P(P(r(x1)))
    
               [0 0]             
    p(P(x1)) = [0 0]x1 >= x1 = x1
    
               [0 0]             
    P(p(x1)) = [0 0]x1 >= x1 = x1
   problem:
    r(p(x1)) -> p(p(r(P(x1))))
    r(P(P(x1))) -> P(P(r(x1)))
    p(P(x1)) -> x1
    P(p(x1)) -> x1
   String Reversal Processor:
    p(r(x1)) -> P(r(p(p(x1))))
    P(P(r(x1))) -> r(P(P(x1)))
    P(p(x1)) -> x1
    p(P(x1)) -> x1
    Matrix Interpretation Processor: dim=3
     
     interpretation:
                [1 1 0]  
      [P](x0) = [0 0 1]x0
                [0 1 0]  ,
      
                [1 0 0]  
      [p](x0) = [0 0 1]x0
                [0 1 0]  ,
      
                [1 0 0]     [0]
      [r](x0) = [0 0 0]x0 + [0]
                [0 1 1]     [1]
     orientation:
                 [1 0 0]     [0]    [1 0 0]     [0]                 
      p(r(x1)) = [0 1 1]x1 + [1] >= [0 1 1]x1 + [1] = P(r(p(p(x1))))
                 [0 0 0]     [0]    [0 0 0]     [0]                 
      
                    [1 1 1]     [1]    [1 1 1]     [0]              
      P(P(r(x1))) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = r(P(P(x1)))
                    [0 1 1]     [1]    [0 1 1]     [1]              
      
                 [1 0 1]             
      P(p(x1)) = [0 1 0]x1 >= x1 = x1
                 [0 0 1]             
      
                 [1 1 0]             
      p(P(x1)) = [0 1 0]x1 >= x1 = x1
                 [0 0 1]             
     problem:
      p(r(x1)) -> P(r(p(p(x1))))
      P(p(x1)) -> x1
      p(P(x1)) -> x1
     Arctic Interpretation Processor:
      dimension: 2
      interpretation:
                 [1 1]  
       [P](x0) = [0 0]x0,
       
                 [0  3 ]  
       [p](x0) = [-& 0 ]x0,
       
                 [0 0]  
       [r](x0) = [0 3]x0
      orientation:
                  [3 6]      [1 4]                   
       p(r(x1)) = [0 3]x1 >= [0 3]x1 = P(r(p(p(x1))))
       
                  [1 4]             
       P(p(x1)) = [0 3]x1 >= x1 = x1
       
                  [3 3]             
       p(P(x1)) = [0 0]x1 >= x1 = x1
      problem:
       p(r(x1)) -> P(r(p(p(x1))))
       p(P(x1)) -> x1
      String Reversal Processor:
       r(p(x1)) -> p(p(r(P(x1))))
       P(p(x1)) -> x1
       Matrix Interpretation Processor: dim=3
        
        interpretation:
                        [1]
         [P](x0) = x0 + [0]
                        [0],
         
                        [0]
         [p](x0) = x0 + [1]
                        [1],
         
                   [1 1 0]  
         [r](x0) = [0 1 1]x0
                   [0 1 1]  
        orientation:
                    [1 1 0]     [1]    [1 1 0]     [1]                 
         r(p(x1)) = [0 1 1]x1 + [2] >= [0 1 1]x1 + [2] = p(p(r(P(x1))))
                    [0 1 1]     [2]    [0 1 1]     [2]                 
         
                         [1]           
         P(p(x1)) = x1 + [1] >= x1 = x1
                         [1]           
        problem:
         r(p(x1)) -> p(p(r(P(x1))))
        Arctic Interpretation Processor:
         dimension: 2
         interpretation:
                    [0  -&]  
          [P](x0) = [-& -&]x0,
          
                    [1 0]  
          [p](x0) = [3 1]x0,
          
                    [1 3]  
          [r](x0) = [2 3]x0
         orientation:
                     [6 4]      [4  -&]                   
          r(p(x1)) = [6 4]x1 >= [5  -&]x1 = p(p(r(P(x1))))
         problem:
          
         Qed