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)
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [p#](x0) = 3x0,
      
      [a#](x0) = 1x0 + -8,
      
      [A](x0) = x0 + -10,
      
      [a](x0) = x0 + -10,
      
      [p](x0) = 2x0 + -8
     orientation:
      a#(p(x1)) = 3x1 + -7 >= 1x1 + -8 = a#(A(x1))
      
      a#(p(x1)) = 3x1 + -7 >= 3x1 + -7 = p#(a(A(x1)))
      
      a#(A(x1)) = 1x1 + -8 >= 1x1 + -8 = a#(x1)
      
      p#(A(A(x1))) = 3x1 + -7 >= 3x1 = p#(x1)
      
      p#(A(A(x1))) = 3x1 + -7 >= 3x1 + -7 = a#(p(x1))
      
      a(p(x1)) = 2x1 + -8 >= 2x1 + -8 = p(a(A(x1)))
      
      a(A(x1)) = x1 + -10 >= x1 + -10 = A(a(x1))
      
      p(A(A(x1))) = 2x1 + -8 >= 2x1 + -8 = a(p(x1))
     problem:
      DPs:
       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))
     EDG Processor:
      DPs:
       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)))
       a#(A(x1)) -> a#(x1) -> a#(A(x1)) -> a#(x1)
       a#(A(x1)) -> a#(x1) -> a#(p(x1)) -> p#(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)
      Arctic Interpretation Processor:
       dimension: 1
       interpretation:
        [p#](x0) = 1,
        
        [a#](x0) = 1x0 + 0,
        
        [A](x0) = 1x0 + 1,
        
        [a](x0) = x0 + 0,
        
        [p](x0) = 0
       orientation:
        a#(p(x1)) = 1 >= 1 = p#(a(A(x1)))
        
        a#(A(x1)) = 2x1 + 2 >= 1x1 + 0 = a#(x1)
        
        p#(A(A(x1))) = 1 >= 1 = p#(x1)
        
        p#(A(A(x1))) = 1 >= 1 = a#(p(x1))
        
        a(p(x1)) = 0 >= 0 = p(a(A(x1)))
        
        a(A(x1)) = 1x1 + 1 >= 1x1 + 1 = A(a(x1))
        
        p(A(A(x1))) = 0 >= 0 = a(p(x1))
       problem:
        DPs:
         a#(p(x1)) -> p#(a(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))
       EDG Processor:
        DPs:
         a#(p(x1)) -> p#(a(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)) -> p#(a(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))
        Matrix Interpretation Processor: dim=3
         
         interpretation:
          [p#](x0) = [0 0 1]x0,
          
          [a#](x0) = [0 1 0]x0 + [1],
          
                    [0 0 0]     [0]
          [A](x0) = [0 0 0]x0 + [0]
                    [0 0 1]     [1],
          
                    [0 0 0]     [1]
          [a](x0) = [0 1 0]x0 + [1]
                    [0 0 1]     [0],
          
                    [0 0 0]     [1]
          [p](x0) = [0 0 1]x0 + [0]
                    [0 0 0]     [0]
         orientation:
          a#(p(x1)) = [0 0 1]x1 + [1] >= [0 0 1]x1 + [1] = p#(a(A(x1)))
          
          p#(A(A(x1))) = [0 0 1]x1 + [2] >= [0 0 1]x1 = p#(x1)
          
          p#(A(A(x1))) = [0 0 1]x1 + [2] >= [0 0 1]x1 + [1] = a#(p(x1))
          
                     [0 0 0]     [1]    [0 0 0]     [1]              
          a(p(x1)) = [0 0 1]x1 + [1] >= [0 0 1]x1 + [1] = p(a(A(x1)))
                     [0 0 0]     [0]    [0 0 0]     [0]              
          
                     [0 0 0]     [1]    [0 0 0]     [0]           
          a(A(x1)) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [0] = A(a(x1))
                     [0 0 1]     [1]    [0 0 1]     [1]           
          
                        [0 0 0]     [1]    [0 0 0]     [1]           
          p(A(A(x1))) = [0 0 1]x1 + [2] >= [0 0 1]x1 + [1] = a(p(x1))
                        [0 0 0]     [0]    [0 0 0]     [0]           
         problem:
          DPs:
           a#(p(x1)) -> p#(a(A(x1)))
          TRS:
           a(p(x1)) -> p(a(A(x1)))
           a(A(x1)) -> A(a(x1))
           p(A(A(x1))) -> a(p(x1))
         EDG Processor:
          DPs:
           a#(p(x1)) -> p#(a(A(x1)))
          TRS:
           a(p(x1)) -> p(a(A(x1)))
           a(A(x1)) -> A(a(x1))
           p(A(A(x1))) -> a(p(x1))
          graph:
           
          Qed