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
  Usable Rule 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(p(x1)) -> q(q(x1))
    p(Q(Q(x1))) -> Q(Q(p(x1)))
    p(P(x1)) -> x1
    q(q(p(x1))) -> p(q(q(x1)))
    q(Q(x1)) -> x1
    Q(p(q(x1))) -> q(p(Q(x1)))
    Q(q(x1)) -> x1
   Matrix Interpretation Processor: dim=2
    
    interpretation:
     [q#](x0) = [1 1]x0,
     
     [Q#](x0) = [1 1]x0 + [2],
     
     [p#](x0) = [4 1]x0,
     
     [P#](x0) = [7 4]x0 + [6],
     
               [0 1]  
     [q](x0) = [1 0]x0,
     
               [0 1]     [0]
     [Q](x0) = [1 0]x0 + [1],
     
               [0 1]     [1]
     [p](x0) = [4 0]x0 + [0],
     
               [0 1]     [0]
     [P](x0) = [4 6]x0 + [4]
    orientation:
     P#(x1) = [7 4]x1 + [6] >= [4 1]x1 = p#(x1)
     
     P#(x1) = [7 4]x1 + [6] >= [4 1]x1 + [3] = Q#(p(x1))
     
     P#(x1) = [7 4]x1 + [6] >= [4 1]x1 + [4] = Q#(Q(p(x1)))
     
     p#(p(x1)) = [4 4]x1 + [4] >= [1 1]x1 = q#(x1)
     
     p#(p(x1)) = [4 4]x1 + [4] >= [1 1]x1 = q#(q(x1))
     
     p#(Q(Q(x1))) = [4 1]x1 + [5] >= [4 1]x1 = p#(x1)
     
     p#(Q(Q(x1))) = [4 1]x1 + [5] >= [4 1]x1 + [3] = Q#(p(x1))
     
     p#(Q(Q(x1))) = [4 1]x1 + [5] >= [4 1]x1 + [4] = Q#(Q(p(x1)))
     
     Q#(p(q(x1))) = [1 4]x1 + [3] >= [1 1]x1 + [2] = Q#(x1)
     
     Q#(p(q(x1))) = [1 4]x1 + [3] >= [1 4]x1 + [1] = p#(Q(x1))
     
     Q#(p(q(x1))) = [1 4]x1 + [3] >= [1 4]x1 + [2] = q#(p(Q(x1)))
     
     q#(q(p(x1))) = [4 1]x1 + [1] >= [1 1]x1 = q#(x1)
     
     q#(q(p(x1))) = [4 1]x1 + [1] >= [1 1]x1 = q#(q(x1))
     
     q#(q(p(x1))) = [4 1]x1 + [1] >= [4 1]x1 = p#(q(q(x1)))
     
                [4 0]     [1]                 
     p(p(x1)) = [0 4]x1 + [4] >= x1 = q(q(x1))
     
                   [0 1]     [2]    [0 1]     [2]              
     p(Q(Q(x1))) = [4 0]x1 + [4] >= [4 0]x1 + [1] = Q(Q(p(x1)))
     
                [4 6]     [5]           
     p(P(x1)) = [0 4]x1 + [0] >= x1 = x1
     
                   [0 1]     [1]    [0 1]     [1]              
     q(q(p(x1))) = [4 0]x1 + [0] >= [4 0]x1 + [0] = p(q(q(x1)))
     
                     [1]           
     q(Q(x1)) = x1 + [0] >= x1 = x1
     
                   [0 4]     [0]    [0 4]     [0]              
     Q(p(q(x1))) = [1 0]x1 + [2] >= [1 0]x1 + [2] = q(p(Q(x1)))
     
                     [0]           
     Q(q(x1)) = x1 + [1] >= x1 = x1
    problem:
     DPs:
      
     TRS:
      p(p(x1)) -> q(q(x1))
      p(Q(Q(x1))) -> Q(Q(p(x1)))
      p(P(x1)) -> x1
      q(q(p(x1))) -> p(q(q(x1)))
      q(Q(x1)) -> x1
      Q(p(q(x1))) -> q(p(Q(x1)))
      Q(q(x1)) -> x1
    Qed