YES

Problem:
 a(p(x1)) -> p(a(A(x1)))
 a(A(x1)) -> A(a(x1))
 p(A(A(x1))) -> a(p(x1))

Proof:
 DP Processor:
  DPs:
   a#(p(x1)) -> a#(A(x1))
   a#(p(x1)) -> p#(a(A(x1)))
   a#(A(x1)) -> a#(x1)
   p#(A(A(x1))) -> p#(x1)
   p#(A(A(x1))) -> a#(p(x1))
  TRS:
   a(p(x1)) -> p(a(A(x1)))
   a(A(x1)) -> A(a(x1))
   p(A(A(x1))) -> a(p(x1))
  TDG Processor:
   DPs:
    a#(p(x1)) -> a#(A(x1))
    a#(p(x1)) -> p#(a(A(x1)))
    a#(A(x1)) -> a#(x1)
    p#(A(A(x1))) -> p#(x1)
    p#(A(A(x1))) -> a#(p(x1))
   TRS:
    a(p(x1)) -> p(a(A(x1)))
    a(A(x1)) -> A(a(x1))
    p(A(A(x1))) -> a(p(x1))
   graph:
    p#(A(A(x1))) -> p#(x1) -> p#(A(A(x1))) -> a#(p(x1))
    p#(A(A(x1))) -> p#(x1) -> p#(A(A(x1))) -> p#(x1)
    p#(A(A(x1))) -> a#(p(x1)) -> a#(A(x1)) -> a#(x1)
    p#(A(A(x1))) -> a#(p(x1)) -> a#(p(x1)) -> p#(a(A(x1)))
    p#(A(A(x1))) -> a#(p(x1)) -> a#(p(x1)) -> a#(A(x1))
    a#(A(x1)) -> a#(x1) -> a#(A(x1)) -> a#(x1)
    a#(A(x1)) -> a#(x1) -> a#(p(x1)) -> p#(a(A(x1)))
    a#(A(x1)) -> a#(x1) -> a#(p(x1)) -> a#(A(x1))
    a#(p(x1)) -> p#(a(A(x1))) -> p#(A(A(x1))) -> a#(p(x1))
    a#(p(x1)) -> p#(a(A(x1))) -> p#(A(A(x1))) -> p#(x1)
    a#(p(x1)) -> a#(A(x1)) -> a#(A(x1)) -> a#(x1)
    a#(p(x1)) -> a#(A(x1)) -> a#(p(x1)) -> p#(a(A(x1)))
    a#(p(x1)) -> a#(A(x1)) -> a#(p(x1)) -> a#(A(x1))
   EDG Processor:
    DPs:
     a#(p(x1)) -> a#(A(x1))
     a#(p(x1)) -> p#(a(A(x1)))
     a#(A(x1)) -> a#(x1)
     p#(A(A(x1))) -> p#(x1)
     p#(A(A(x1))) -> a#(p(x1))
    TRS:
     a(p(x1)) -> p(a(A(x1)))
     a(A(x1)) -> A(a(x1))
     p(A(A(x1))) -> a(p(x1))
    graph:
     p#(A(A(x1))) -> p#(x1) -> p#(A(A(x1))) -> p#(x1)
     p#(A(A(x1))) -> p#(x1) -> p#(A(A(x1))) -> a#(p(x1))
     p#(A(A(x1))) -> a#(p(x1)) -> a#(p(x1)) -> a#(A(x1))
     p#(A(A(x1))) -> a#(p(x1)) -> a#(p(x1)) -> p#(a(A(x1)))
     p#(A(A(x1))) -> a#(p(x1)) -> a#(A(x1)) -> a#(x1)
     a#(A(x1)) -> a#(x1) -> a#(p(x1)) -> a#(A(x1))
     a#(A(x1)) -> a#(x1) -> a#(p(x1)) -> p#(a(A(x1)))
     a#(A(x1)) -> a#(x1) -> a#(A(x1)) -> a#(x1)
     a#(p(x1)) -> p#(a(A(x1))) -> p#(A(A(x1))) -> p#(x1)
     a#(p(x1)) -> p#(a(A(x1))) -> p#(A(A(x1))) -> a#(p(x1))
     a#(p(x1)) -> a#(A(x1)) -> a#(A(x1)) -> a#(x1)
    Matrix Interpretation Processor: dim=2
     
     interpretation:
      [p#](x0) = [1 0]x0 + [1],
      
      [a#](x0) = [1 0]x0 + [1],
      
                [1 2]     [2]
      [A](x0) = [0 1]x0 + [0],
      
                [1 1]  
      [a](x0) = [0 1]x0,
      
                [1 3]     [3]
      [p](x0) = [0 4]x0 + [2]
     orientation:
      a#(p(x1)) = [1 3]x1 + [4] >= [1 2]x1 + [3] = a#(A(x1))
      
      a#(p(x1)) = [1 3]x1 + [4] >= [1 3]x1 + [3] = p#(a(A(x1)))
      
      a#(A(x1)) = [1 2]x1 + [3] >= [1 0]x1 + [1] = a#(x1)
      
      p#(A(A(x1))) = [1 4]x1 + [5] >= [1 0]x1 + [1] = p#(x1)
      
      p#(A(A(x1))) = [1 4]x1 + [5] >= [1 3]x1 + [4] = a#(p(x1))
      
                 [1 7]     [5]    [1 6]     [5]              
      a(p(x1)) = [0 4]x1 + [2] >= [0 4]x1 + [2] = p(a(A(x1)))
      
                 [1 3]     [2]    [1 3]     [2]           
      a(A(x1)) = [0 1]x1 + [0] >= [0 1]x1 + [0] = A(a(x1))
      
                    [1 7]     [7]    [1 7]     [5]           
      p(A(A(x1))) = [0 4]x1 + [2] >= [0 4]x1 + [2] = a(p(x1))
     problem:
      DPs:
       
      TRS:
       a(p(x1)) -> p(a(A(x1)))
       a(A(x1)) -> A(a(x1))
       p(A(A(x1))) -> a(p(x1))
     Qed