MAYBE

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:
 Complexity Transformation Processor:
  strict:
   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
  weak:
   
  Matrix Interpretation Processor:
   dimension: 1
   max_matrix:
    1
    interpretation:
     [P](x0) = x0,
     
     [p](x0) = x0,
     
     [r](x0) = x0 + 1,
     
     [R](x0) = x0 + 1
    orientation:
     R(x1) = x1 + 1 >= x1 + 1 = r(x1)
     
     r(p(x1)) = x1 + 1 >= x1 + 1 = p(p(r(P(x1))))
     
     r(r(x1)) = x1 + 2 >= x1 = x1
     
     r(P(P(x1))) = x1 + 1 >= x1 + 1 = P(P(r(x1)))
     
     p(P(x1)) = x1 >= x1 = x1
     
     P(p(x1)) = x1 >= x1 = x1
     
     r(R(x1)) = x1 + 2 >= x1 = x1
     
     R(r(x1)) = x1 + 2 >= x1 = x1
    problem:
     strict:
      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
     weak:
      r(r(x1)) -> x1
      r(R(x1)) -> x1
      R(r(x1)) -> x1
    Matrix Interpretation Processor:
     dimension: 1
     max_matrix:
      1
      interpretation:
       [P](x0) = x0 + 1,
       
       [p](x0) = x0 + 1,
       
       [r](x0) = x0,
       
       [R](x0) = x0
      orientation:
       R(x1) = x1 >= x1 = r(x1)
       
       r(p(x1)) = x1 + 1 >= x1 + 3 = p(p(r(P(x1))))
       
       r(P(P(x1))) = x1 + 2 >= x1 + 2 = P(P(r(x1)))
       
       p(P(x1)) = x1 + 2 >= x1 = x1
       
       P(p(x1)) = x1 + 2 >= x1 = x1
       
       r(r(x1)) = x1 >= x1 = x1
       
       r(R(x1)) = x1 >= x1 = x1
       
       R(r(x1)) = x1 >= x1 = x1
      problem:
       strict:
        R(x1) -> r(x1)
        r(p(x1)) -> p(p(r(P(x1))))
        r(P(P(x1))) -> P(P(r(x1)))
       weak:
        p(P(x1)) -> x1
        P(p(x1)) -> x1
        r(r(x1)) -> x1
        r(R(x1)) -> x1
        R(r(x1)) -> x1
      Matrix Interpretation Processor:
       dimension: 1
       max_matrix:
        1
        interpretation:
         [P](x0) = x0,
         
         [p](x0) = x0,
         
         [r](x0) = x0,
         
         [R](x0) = x0 + 1
        orientation:
         R(x1) = x1 + 1 >= x1 = r(x1)
         
         r(p(x1)) = x1 >= x1 = p(p(r(P(x1))))
         
         r(P(P(x1))) = x1 >= x1 = P(P(r(x1)))
         
         p(P(x1)) = x1 >= x1 = x1
         
         P(p(x1)) = x1 >= x1 = x1
         
         r(r(x1)) = x1 >= x1 = x1
         
         r(R(x1)) = x1 + 1 >= x1 = x1
         
         R(r(x1)) = x1 + 1 >= x1 = x1
        problem:
         strict:
          r(p(x1)) -> p(p(r(P(x1))))
          r(P(P(x1))) -> P(P(r(x1)))
         weak:
          R(x1) -> r(x1)
          p(P(x1)) -> x1
          P(p(x1)) -> x1
          r(r(x1)) -> x1
          r(R(x1)) -> x1
          R(r(x1)) -> x1
        Open