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:
 Arctic Interpretation Processor:
  dimension: 1
  interpretation:
   [q](x0) = 1x0,
   
   [Q](x0) = x0,
   
   [p](x0) = 1x0,
   
   [P](x0) = 1x0
  orientation:
   P(x1) = 1x1 >= 1x1 = Q(Q(p(x1)))
   
   p(p(x1)) = 2x1 >= 2x1 = q(q(x1))
   
   p(Q(Q(x1))) = 1x1 >= 1x1 = Q(Q(p(x1)))
   
   Q(p(q(x1))) = 2x1 >= 2x1 = q(p(Q(x1)))
   
   q(q(p(x1))) = 3x1 >= 3x1 = p(q(q(x1)))
   
   q(Q(x1)) = 1x1 >= x1 = x1
   
   Q(q(x1)) = 1x1 >= x1 = x1
   
   p(P(x1)) = 2x1 >= x1 = x1
   
   P(p(x1)) = 2x1 >= x1 = x1
  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)))
  Arctic Interpretation Processor:
   dimension: 1
   interpretation:
    [q](x0) = x0,
    
    [Q](x0) = x0,
    
    [p](x0) = x0,
    
    [P](x0) = 8x0
   orientation:
    P(x1) = 8x1 >= x1 = Q(Q(p(x1)))
    
    p(p(x1)) = x1 >= x1 = q(q(x1))
    
    p(Q(Q(x1))) = x1 >= x1 = Q(Q(p(x1)))
    
    Q(p(q(x1))) = x1 >= x1 = q(p(Q(x1)))
    
    q(q(p(x1))) = x1 >= x1 = p(q(q(x1)))
   problem:
    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)))
   Arctic Interpretation Processor:
    dimension: 1
    interpretation:
     [q](x0) = x0,
     
     [Q](x0) = x0,
     
     [p](x0) = 8x0
    orientation:
     p(p(x1)) = 16x1 >= x1 = q(q(x1))
     
     p(Q(Q(x1))) = 8x1 >= 8x1 = Q(Q(p(x1)))
     
     Q(p(q(x1))) = 8x1 >= 8x1 = q(p(Q(x1)))
     
     q(q(p(x1))) = 8x1 >= 8x1 = p(q(q(x1)))
    problem:
     p(Q(Q(x1))) -> Q(Q(p(x1)))
     Q(p(q(x1))) -> q(p(Q(x1)))
     q(q(p(x1))) -> p(q(q(x1)))
    String Reversal Processor:
     Q(Q(p(x1))) -> p(Q(Q(x1)))
     q(p(Q(x1))) -> Q(p(q(x1)))
     p(q(q(x1))) -> q(q(p(x1)))
     Matrix Interpretation Processor: dim=3
      
      interpretation:
                 [1 1 0]  
       [q](x0) = [0 1 0]x0
                 [0 0 0]  ,
       
                 [1 0 0]     [0]
       [Q](x0) = [0 1 0]x0 + [1]
                 [0 0 0]     [0],
       
                 [1 0 0]  
       [p](x0) = [0 1 0]x0
                 [0 0 0]  
      orientation:
                     [1 0 0]     [0]    [1 0 0]     [0]              
       Q(Q(p(x1))) = [0 1 0]x1 + [2] >= [0 1 0]x1 + [2] = p(Q(Q(x1)))
                     [0 0 0]     [0]    [0 0 0]     [0]              
       
                     [1 1 0]     [1]    [1 1 0]     [0]              
       q(p(Q(x1))) = [0 1 0]x1 + [1] >= [0 1 0]x1 + [1] = Q(p(q(x1)))
                     [0 0 0]     [0]    [0 0 0]     [0]              
       
                     [1 2 0]      [1 2 0]                
       p(q(q(x1))) = [0 1 0]x1 >= [0 1 0]x1 = q(q(p(x1)))
                     [0 0 0]      [0 0 0]                
      problem:
       Q(Q(p(x1))) -> p(Q(Q(x1)))
       p(q(q(x1))) -> q(q(p(x1)))
      KBO Processor:
       weight function:
        w0 = 1
        w(q) = w(Q) = w(p) = 1
       precedence:
        Q > p > q
       problem:
        
       Qed