YES

Problem:
 P(x1) -> Q(Q(p(x1)))
 p(p(x1)) -> q(q(x1))
 p(Q(Q(x1))) -> Q(Q(p(x1)))
 Q(p(q(x1))) -> q(p(Q(x1)))
 q(q(p(x1))) -> p(q(q(x1)))
 q(Q(x1)) -> x1
 Q(q(x1)) -> x1
 p(P(x1)) -> x1
 P(p(x1)) -> x1

Proof:
 String Reversal Processor:
  P(x1) -> p(Q(Q(x1)))
  p(p(x1)) -> q(q(x1))
  Q(Q(p(x1))) -> p(Q(Q(x1)))
  q(p(Q(x1))) -> Q(p(q(x1)))
  p(q(q(x1))) -> q(q(p(x1)))
  Q(q(x1)) -> x1
  q(Q(x1)) -> x1
  P(p(x1)) -> x1
  p(P(x1)) -> x1
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [p](x0) = 4x0 + 2,
    
    [q](x0) = x0,
    
    [P](x0) = 4x0 + 3,
    
    [Q](x0) = x0
   orientation:
    P(x1) = 4x1 + 3 >= 4x1 + 2 = p(Q(Q(x1)))
    
    p(p(x1)) = 16x1 + 10 >= x1 = q(q(x1))
    
    Q(Q(p(x1))) = 4x1 + 2 >= 4x1 + 2 = p(Q(Q(x1)))
    
    q(p(Q(x1))) = 4x1 + 2 >= 4x1 + 2 = Q(p(q(x1)))
    
    p(q(q(x1))) = 4x1 + 2 >= 4x1 + 2 = q(q(p(x1)))
    
    Q(q(x1)) = x1 >= x1 = x1
    
    q(Q(x1)) = x1 >= x1 = x1
    
    P(p(x1)) = 16x1 + 11 >= x1 = x1
    
    p(P(x1)) = 16x1 + 14 >= x1 = x1
   problem:
    Q(Q(p(x1))) -> p(Q(Q(x1)))
    q(p(Q(x1))) -> Q(p(q(x1)))
    p(q(q(x1))) -> q(q(p(x1)))
    Q(q(x1)) -> x1
    q(Q(x1)) -> x1
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [p](x0) = x0,
     
     [q](x0) = x0,
     
     [Q](x0) = 4x0 + 1
    orientation:
     Q(Q(p(x1))) = 16x1 + 5 >= 16x1 + 5 = p(Q(Q(x1)))
     
     q(p(Q(x1))) = 4x1 + 1 >= 4x1 + 1 = Q(p(q(x1)))
     
     p(q(q(x1))) = x1 >= x1 = q(q(p(x1)))
     
     Q(q(x1)) = 4x1 + 1 >= x1 = x1
     
     q(Q(x1)) = 4x1 + 1 >= x1 = x1
    problem:
     Q(Q(p(x1))) -> p(Q(Q(x1)))
     q(p(Q(x1))) -> Q(p(q(x1)))
     p(q(q(x1))) -> q(q(p(x1)))
    String Reversal Processor:
     p(Q(Q(x1))) -> Q(Q(p(x1)))
     Q(p(q(x1))) -> q(p(Q(x1)))
     q(q(p(x1))) -> p(q(q(x1)))
     Bounds Processor:
      bound: 0
      enrichment: match
      automaton:
       final states: {8,5,1}
       transitions:
        Q0(4) -> 1*
        Q0(3) -> 4*
        Q0(2) -> 6*
        q0(7) -> 5*
        q0(9) -> 10*
        q0(2) -> 9*
        f40() -> 2*
        p0(6) -> 7*
        p0(2) -> 3*
        p0(10) -> 8*
        8 -> 9,10
        1 -> 3,7
        5 -> 6,4
      problem:
       
      Qed