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:
 DP Processor:
  DPs:
   P#(x1) -> p#(x1)
   P#(x1) -> Q#(p(x1))
   P#(x1) -> Q#(Q(p(x1)))
   p#(p(x1)) -> q#(x1)
   p#(p(x1)) -> q#(q(x1))
   p#(Q(Q(x1))) -> p#(x1)
   p#(Q(Q(x1))) -> Q#(p(x1))
   p#(Q(Q(x1))) -> Q#(Q(p(x1)))
   Q#(p(q(x1))) -> Q#(x1)
   Q#(p(q(x1))) -> p#(Q(x1))
   Q#(p(q(x1))) -> q#(p(Q(x1)))
   q#(q(p(x1))) -> q#(x1)
   q#(q(p(x1))) -> q#(q(x1))
   q#(q(p(x1))) -> p#(q(q(x1)))
  TRS:
   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
  Matrix Interpretation Processor: dim=2
   
   interpretation:
    [q#](x0) = [2 1]x0 + [1],
    
    [Q#](x0) = [1 1]x0 + [1],
    
    [p#](x0) = [1 3]x0 + [1],
    
    [P#](x0) = [6 4]x0 + [5],
    
              [0 1]     [0]
    [q](x0) = [2 0]x0 + [1],
    
              [0 1]     [1]
    [Q](x0) = [1 0]x0 + [0],
    
              [0 2]     [1]
    [p](x0) = [1 1]x0 + [1],
    
              [0 3]     [2]
    [P](x0) = [2 1]x0 + [2]
   orientation:
    P#(x1) = [6 4]x1 + [5] >= [1 3]x1 + [1] = p#(x1)
    
    P#(x1) = [6 4]x1 + [5] >= [1 3]x1 + [3] = Q#(p(x1))
    
    P#(x1) = [6 4]x1 + [5] >= [1 3]x1 + [4] = Q#(Q(p(x1)))
    
    p#(p(x1)) = [3 5]x1 + [5] >= [2 1]x1 + [1] = q#(x1)
    
    p#(p(x1)) = [3 5]x1 + [5] >= [2 2]x1 + [2] = q#(q(x1))
    
    p#(Q(Q(x1))) = [1 3]x1 + [5] >= [1 3]x1 + [1] = p#(x1)
    
    p#(Q(Q(x1))) = [1 3]x1 + [5] >= [1 3]x1 + [3] = Q#(p(x1))
    
    p#(Q(Q(x1))) = [1 3]x1 + [5] >= [1 3]x1 + [4] = Q#(Q(p(x1)))
    
    Q#(p(q(x1))) = [6 1]x1 + [6] >= [1 1]x1 + [1] = Q#(x1)
    
    Q#(p(q(x1))) = [6 1]x1 + [6] >= [3 1]x1 + [2] = p#(Q(x1))
    
    Q#(p(q(x1))) = [6 1]x1 + [6] >= [5 1]x1 + [5] = q#(p(Q(x1)))
    
    q#(q(p(x1))) = [2 6]x1 + [6] >= [2 1]x1 + [1] = q#(x1)
    
    q#(q(p(x1))) = [2 6]x1 + [6] >= [2 2]x1 + [2] = q#(q(x1))
    
    q#(q(p(x1))) = [2 6]x1 + [6] >= [2 6]x1 + [5] = p#(q(q(x1)))
    
            [0 3]     [2]    [0 2]     [2]              
    P(x1) = [2 1]x1 + [2] >= [1 1]x1 + [2] = Q(Q(p(x1)))
    
               [2 2]     [3]    [2 0]     [1]           
    p(p(x1)) = [1 3]x1 + [3] >= [0 2]x1 + [1] = q(q(x1))
    
                  [0 2]     [3]    [0 2]     [2]              
    p(Q(Q(x1))) = [1 1]x1 + [3] >= [1 1]x1 + [2] = Q(Q(p(x1)))
    
                  [2 1]     [3]    [1 1]     [2]              
    Q(p(q(x1))) = [4 0]x1 + [3] >= [4 0]x1 + [3] = q(p(Q(x1)))
    
                  [0 4]     [3]    [0 4]     [3]              
    q(q(p(x1))) = [2 2]x1 + [3] >= [2 2]x1 + [3] = p(q(q(x1)))
    
               [1 0]     [0]           
    q(Q(x1)) = [0 2]x1 + [3] >= x1 = x1
    
               [2 0]     [2]           
    Q(q(x1)) = [0 1]x1 + [0] >= x1 = x1
    
               [4 2]     [5]           
    p(P(x1)) = [2 4]x1 + [5] >= x1 = x1
    
               [3 3]     [5]           
    P(p(x1)) = [1 5]x1 + [5] >= x1 = x1
   problem:
    DPs:
     
    TRS:
     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
   Qed