YES

Problem:
 a(a(x1)) -> b(b(b(x1)))
 a(x1) -> c(d(x1))
 b(b(x1)) -> c(c(c(x1)))
 c(c(x1)) -> d(d(d(x1)))
 e(d(x1)) -> a(b(c(d(e(x1)))))
 b(x1) -> d(d(x1))
 e(c(x1)) -> b(a(a(e(x1))))
 c(d(d(x1))) -> a(x1)

Proof:
 DP Processor:
  DPs:
   a#(a(x1)) -> b#(x1)
   a#(a(x1)) -> b#(b(x1))
   a#(a(x1)) -> b#(b(b(x1)))
   a#(x1) -> c#(d(x1))
   b#(b(x1)) -> c#(x1)
   b#(b(x1)) -> c#(c(x1))
   b#(b(x1)) -> c#(c(c(x1)))
   e#(d(x1)) -> e#(x1)
   e#(d(x1)) -> c#(d(e(x1)))
   e#(d(x1)) -> b#(c(d(e(x1))))
   e#(d(x1)) -> a#(b(c(d(e(x1)))))
   e#(c(x1)) -> e#(x1)
   e#(c(x1)) -> a#(e(x1))
   e#(c(x1)) -> a#(a(e(x1)))
   e#(c(x1)) -> b#(a(a(e(x1))))
   c#(d(d(x1))) -> a#(x1)
  TRS:
   a(a(x1)) -> b(b(b(x1)))
   a(x1) -> c(d(x1))
   b(b(x1)) -> c(c(c(x1)))
   c(c(x1)) -> d(d(d(x1)))
   e(d(x1)) -> a(b(c(d(e(x1)))))
   b(x1) -> d(d(x1))
   e(c(x1)) -> b(a(a(e(x1))))
   c(d(d(x1))) -> a(x1)
  TDG Processor:
   DPs:
    a#(a(x1)) -> b#(x1)
    a#(a(x1)) -> b#(b(x1))
    a#(a(x1)) -> b#(b(b(x1)))
    a#(x1) -> c#(d(x1))
    b#(b(x1)) -> c#(x1)
    b#(b(x1)) -> c#(c(x1))
    b#(b(x1)) -> c#(c(c(x1)))
    e#(d(x1)) -> e#(x1)
    e#(d(x1)) -> c#(d(e(x1)))
    e#(d(x1)) -> b#(c(d(e(x1))))
    e#(d(x1)) -> a#(b(c(d(e(x1)))))
    e#(c(x1)) -> e#(x1)
    e#(c(x1)) -> a#(e(x1))
    e#(c(x1)) -> a#(a(e(x1)))
    e#(c(x1)) -> b#(a(a(e(x1))))
    c#(d(d(x1))) -> a#(x1)
   TRS:
    a(a(x1)) -> b(b(b(x1)))
    a(x1) -> c(d(x1))
    b(b(x1)) -> c(c(c(x1)))
    c(c(x1)) -> d(d(d(x1)))
    e(d(x1)) -> a(b(c(d(e(x1)))))
    b(x1) -> d(d(x1))
    e(c(x1)) -> b(a(a(e(x1))))
    c(d(d(x1))) -> a(x1)
   graph:
    e#(c(x1)) -> e#(x1) -> e#(c(x1)) -> b#(a(a(e(x1))))
    e#(c(x1)) -> e#(x1) -> e#(c(x1)) -> a#(a(e(x1)))
    e#(c(x1)) -> e#(x1) -> e#(c(x1)) -> a#(e(x1))
    e#(c(x1)) -> e#(x1) -> e#(c(x1)) -> e#(x1)
    e#(c(x1)) -> e#(x1) -> e#(d(x1)) -> a#(b(c(d(e(x1)))))
    e#(c(x1)) -> e#(x1) -> e#(d(x1)) -> b#(c(d(e(x1))))
    e#(c(x1)) -> e#(x1) -> e#(d(x1)) -> c#(d(e(x1)))
    e#(c(x1)) -> e#(x1) -> e#(d(x1)) -> e#(x1)
    e#(c(x1)) -> b#(a(a(e(x1)))) -> b#(b(x1)) -> c#(c(c(x1)))
    e#(c(x1)) -> b#(a(a(e(x1)))) -> b#(b(x1)) -> c#(c(x1))
    e#(c(x1)) -> b#(a(a(e(x1)))) -> b#(b(x1)) -> c#(x1)
    e#(c(x1)) -> a#(e(x1)) -> a#(x1) -> c#(d(x1))
    e#(c(x1)) -> a#(e(x1)) -> a#(a(x1)) -> b#(b(b(x1)))
    e#(c(x1)) -> a#(e(x1)) -> a#(a(x1)) -> b#(b(x1))
    e#(c(x1)) -> a#(e(x1)) -> a#(a(x1)) -> b#(x1)
    e#(c(x1)) -> a#(a(e(x1))) -> a#(x1) -> c#(d(x1))
    e#(c(x1)) -> a#(a(e(x1))) -> a#(a(x1)) -> b#(b(b(x1)))
    e#(c(x1)) -> a#(a(e(x1))) -> a#(a(x1)) -> b#(b(x1))
    e#(c(x1)) -> a#(a(e(x1))) -> a#(a(x1)) -> b#(x1)
    e#(d(x1)) -> e#(x1) -> e#(c(x1)) -> b#(a(a(e(x1))))
    e#(d(x1)) -> e#(x1) -> e#(c(x1)) -> a#(a(e(x1)))
    e#(d(x1)) -> e#(x1) -> e#(c(x1)) -> a#(e(x1))
    e#(d(x1)) -> e#(x1) -> e#(c(x1)) -> e#(x1)
    e#(d(x1)) -> e#(x1) -> e#(d(x1)) -> a#(b(c(d(e(x1)))))
    e#(d(x1)) -> e#(x1) -> e#(d(x1)) -> b#(c(d(e(x1))))
    e#(d(x1)) -> e#(x1) -> e#(d(x1)) -> c#(d(e(x1)))
    e#(d(x1)) -> e#(x1) -> e#(d(x1)) -> e#(x1)
    e#(d(x1)) -> c#(d(e(x1))) -> c#(d(d(x1))) -> a#(x1)
    e#(d(x1)) -> b#(c(d(e(x1)))) -> b#(b(x1)) -> c#(c(c(x1)))
    e#(d(x1)) -> b#(c(d(e(x1)))) -> b#(b(x1)) -> c#(c(x1))
    e#(d(x1)) -> b#(c(d(e(x1)))) -> b#(b(x1)) -> c#(x1)
    e#(d(x1)) -> a#(b(c(d(e(x1))))) -> a#(x1) -> c#(d(x1))
    e#(d(x1)) -> a#(b(c(d(e(x1))))) -> a#(a(x1)) -> b#(b(b(x1)))
    e#(d(x1)) -> a#(b(c(d(e(x1))))) -> a#(a(x1)) -> b#(b(x1))
    e#(d(x1)) -> a#(b(c(d(e(x1))))) -> a#(a(x1)) -> b#(x1)
    c#(d(d(x1))) -> a#(x1) -> a#(x1) -> c#(d(x1))
    c#(d(d(x1))) -> a#(x1) -> a#(a(x1)) -> b#(b(b(x1)))
    c#(d(d(x1))) -> a#(x1) -> a#(a(x1)) -> b#(b(x1))
    c#(d(d(x1))) -> a#(x1) -> a#(a(x1)) -> b#(x1)
    b#(b(x1)) -> c#(c(c(x1))) -> c#(d(d(x1))) -> a#(x1)
    b#(b(x1)) -> c#(c(x1)) -> c#(d(d(x1))) -> a#(x1)
    b#(b(x1)) -> c#(x1) -> c#(d(d(x1))) -> a#(x1)
    a#(a(x1)) -> b#(b(b(x1))) -> b#(b(x1)) -> c#(c(c(x1)))
    a#(a(x1)) -> b#(b(b(x1))) -> b#(b(x1)) -> c#(c(x1))
    a#(a(x1)) -> b#(b(b(x1))) -> b#(b(x1)) -> c#(x1)
    a#(a(x1)) -> b#(b(x1)) -> b#(b(x1)) -> c#(c(c(x1)))
    a#(a(x1)) -> b#(b(x1)) -> b#(b(x1)) -> c#(c(x1))
    a#(a(x1)) -> b#(b(x1)) -> b#(b(x1)) -> c#(x1)
    a#(a(x1)) -> b#(x1) -> b#(b(x1)) -> c#(c(c(x1)))
    a#(a(x1)) -> b#(x1) -> b#(b(x1)) -> c#(c(x1))
    a#(a(x1)) -> b#(x1) -> b#(b(x1)) -> c#(x1)
    a#(x1) -> c#(d(x1)) -> c#(d(d(x1))) -> a#(x1)
   SCC Processor:
    #sccs: 2
    #rules: 10
    #arcs: 52/256
    DPs:
     e#(c(x1)) -> e#(x1)
     e#(d(x1)) -> e#(x1)
    TRS:
     a(a(x1)) -> b(b(b(x1)))
     a(x1) -> c(d(x1))
     b(b(x1)) -> c(c(c(x1)))
     c(c(x1)) -> d(d(d(x1)))
     e(d(x1)) -> a(b(c(d(e(x1)))))
     b(x1) -> d(d(x1))
     e(c(x1)) -> b(a(a(e(x1))))
     c(d(d(x1))) -> a(x1)
    Usable Rule Processor:
     DPs:
      e#(c(x1)) -> e#(x1)
      e#(d(x1)) -> e#(x1)
     TRS:
      
     Arctic Interpretation Processor:
      dimension: 1
      usable rules:
       
      interpretation:
       [e#](x0) = x0,
       
       [c](x0) = x0 + 4,
       
       [d](x0) = 1x0 + 13
      orientation:
       e#(c(x1)) = x1 + 4 >= x1 = e#(x1)
       
       e#(d(x1)) = 1x1 + 13 >= x1 = e#(x1)
      problem:
       DPs:
        e#(c(x1)) -> e#(x1)
       TRS:
        
      Restore Modifier:
       DPs:
        e#(c(x1)) -> e#(x1)
       TRS:
        a(a(x1)) -> b(b(b(x1)))
        a(x1) -> c(d(x1))
        b(b(x1)) -> c(c(c(x1)))
        c(c(x1)) -> d(d(d(x1)))
        e(d(x1)) -> a(b(c(d(e(x1)))))
        b(x1) -> d(d(x1))
        e(c(x1)) -> b(a(a(e(x1))))
        c(d(d(x1))) -> a(x1)
       EDG Processor:
        DPs:
         e#(c(x1)) -> e#(x1)
        TRS:
         a(a(x1)) -> b(b(b(x1)))
         a(x1) -> c(d(x1))
         b(b(x1)) -> c(c(c(x1)))
         c(c(x1)) -> d(d(d(x1)))
         e(d(x1)) -> a(b(c(d(e(x1)))))
         b(x1) -> d(d(x1))
         e(c(x1)) -> b(a(a(e(x1))))
         c(d(d(x1))) -> a(x1)
        graph:
         e#(c(x1)) -> e#(x1) -> e#(c(x1)) -> e#(x1)
        Usable Rule Processor:
         DPs:
          e#(c(x1)) -> e#(x1)
         TRS:
          
         Arctic Interpretation Processor:
          dimension: 4
          usable rules:
           
          interpretation:
           [e#](x0) = [0  -& -& -&]x0,
           
                     [1 0 0 0]     [0]
                     [0 0 0 0]     [0]
           [c](x0) = [1 0 0 0]x0 + [0]
                     [0 0 0 0]     [0]
          orientation:
           e#(c(x1)) = [1 0 0 0]x1 + [0] >= [0  -& -& -&]x1 = e#(x1)
          problem:
           DPs:
            
           TRS:
            
          Qed
    
    DPs:
     c#(d(d(x1))) -> a#(x1)
     a#(a(x1)) -> b#(x1)
     b#(b(x1)) -> c#(x1)
     b#(b(x1)) -> c#(c(x1))
     b#(b(x1)) -> c#(c(c(x1)))
     a#(a(x1)) -> b#(b(x1))
     a#(a(x1)) -> b#(b(b(x1)))
     a#(x1) -> c#(d(x1))
    TRS:
     a(a(x1)) -> b(b(b(x1)))
     a(x1) -> c(d(x1))
     b(b(x1)) -> c(c(c(x1)))
     c(c(x1)) -> d(d(d(x1)))
     e(d(x1)) -> a(b(c(d(e(x1)))))
     b(x1) -> d(d(x1))
     e(c(x1)) -> b(a(a(e(x1))))
     c(d(d(x1))) -> a(x1)
    Usable Rule Processor:
     DPs:
      c#(d(d(x1))) -> a#(x1)
      a#(a(x1)) -> b#(x1)
      b#(b(x1)) -> c#(x1)
      b#(b(x1)) -> c#(c(x1))
      b#(b(x1)) -> c#(c(c(x1)))
      a#(a(x1)) -> b#(b(x1))
      a#(a(x1)) -> b#(b(b(x1)))
      a#(x1) -> c#(d(x1))
     TRS:
      c(c(x1)) -> d(d(d(x1)))
      c(d(d(x1))) -> a(x1)
      a(a(x1)) -> b(b(b(x1)))
      a(x1) -> c(d(x1))
      b(b(x1)) -> c(c(c(x1)))
      b(x1) -> d(d(x1))
     Arctic Interpretation Processor:
      dimension: 1
      usable rules:
       c(c(x1)) -> d(d(d(x1)))
       c(d(d(x1))) -> a(x1)
       a(a(x1)) -> b(b(b(x1)))
       a(x1) -> c(d(x1))
       b(b(x1)) -> c(c(c(x1)))
       b(x1) -> d(d(x1))
      interpretation:
       [b](x0) = 9x0,
       
       [c](x0) = 6x0,
       
       [a#](x0) = 8x0,
       
       [a](x0) = 14x0,
       
       [c#](x0) = x0,
       
       [b#](x0) = 4x0,
       
       [d](x0) = 4x0
      orientation:
       c#(d(d(x1))) = 8x1 >= 8x1 = a#(x1)
       
       a#(a(x1)) = 22x1 >= 4x1 = b#(x1)
       
       b#(b(x1)) = 13x1 >= x1 = c#(x1)
       
       b#(b(x1)) = 13x1 >= 6x1 = c#(c(x1))
       
       b#(b(x1)) = 13x1 >= 12x1 = c#(c(c(x1)))
       
       a#(a(x1)) = 22x1 >= 13x1 = b#(b(x1))
       
       a#(a(x1)) = 22x1 >= 22x1 = b#(b(b(x1)))
       
       a#(x1) = 8x1 >= 4x1 = c#(d(x1))
       
       c(c(x1)) = 12x1 >= 12x1 = d(d(d(x1)))
       
       c(d(d(x1))) = 14x1 >= 14x1 = a(x1)
       
       a(a(x1)) = 28x1 >= 27x1 = b(b(b(x1)))
       
       a(x1) = 14x1 >= 10x1 = c(d(x1))
       
       b(b(x1)) = 18x1 >= 18x1 = c(c(c(x1)))
       
       b(x1) = 9x1 >= 8x1 = d(d(x1))
      problem:
       DPs:
        c#(d(d(x1))) -> a#(x1)
        a#(a(x1)) -> b#(b(b(x1)))
       TRS:
        c(c(x1)) -> d(d(d(x1)))
        c(d(d(x1))) -> a(x1)
        a(a(x1)) -> b(b(b(x1)))
        a(x1) -> c(d(x1))
        b(b(x1)) -> c(c(c(x1)))
        b(x1) -> d(d(x1))
      Restore Modifier:
       DPs:
        c#(d(d(x1))) -> a#(x1)
        a#(a(x1)) -> b#(b(b(x1)))
       TRS:
        a(a(x1)) -> b(b(b(x1)))
        a(x1) -> c(d(x1))
        b(b(x1)) -> c(c(c(x1)))
        c(c(x1)) -> d(d(d(x1)))
        e(d(x1)) -> a(b(c(d(e(x1)))))
        b(x1) -> d(d(x1))
        e(c(x1)) -> b(a(a(e(x1))))
        c(d(d(x1))) -> a(x1)
       EDG Processor:
        DPs:
         c#(d(d(x1))) -> a#(x1)
         a#(a(x1)) -> b#(b(b(x1)))
        TRS:
         a(a(x1)) -> b(b(b(x1)))
         a(x1) -> c(d(x1))
         b(b(x1)) -> c(c(c(x1)))
         c(c(x1)) -> d(d(d(x1)))
         e(d(x1)) -> a(b(c(d(e(x1)))))
         b(x1) -> d(d(x1))
         e(c(x1)) -> b(a(a(e(x1))))
         c(d(d(x1))) -> a(x1)
        graph:
         c#(d(d(x1))) -> a#(x1) -> a#(a(x1)) -> b#(b(b(x1)))
        SCC Processor:
         #sccs: 0
         #rules: 0
         #arcs: 1/4