YES

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

Proof:
 DP Processor:
  DPs:
   a#(b(x1)) -> d#(x1)
   d#(d(x1)) -> e#(x1)
   d#(d(x1)) -> b#(e(x1))
   b#(x1) -> d#(c(x1))
   e#(c(x1)) -> a#(x1)
   e#(c(x1)) -> d#(a(x1))
   a#(x1) -> d#(x1)
   a#(x1) -> e#(d(x1))
  TRS:
   a(b(x1)) -> c(d(x1))
   d(d(x1)) -> b(e(x1))
   b(x1) -> d(c(x1))
   d(x1) -> x1
   e(c(x1)) -> d(a(x1))
   a(x1) -> e(d(x1))
  TDG Processor:
   DPs:
    a#(b(x1)) -> d#(x1)
    d#(d(x1)) -> e#(x1)
    d#(d(x1)) -> b#(e(x1))
    b#(x1) -> d#(c(x1))
    e#(c(x1)) -> a#(x1)
    e#(c(x1)) -> d#(a(x1))
    a#(x1) -> d#(x1)
    a#(x1) -> e#(d(x1))
   TRS:
    a(b(x1)) -> c(d(x1))
    d(d(x1)) -> b(e(x1))
    b(x1) -> d(c(x1))
    d(x1) -> x1
    e(c(x1)) -> d(a(x1))
    a(x1) -> e(d(x1))
   graph:
    b#(x1) -> d#(c(x1)) -> d#(d(x1)) -> b#(e(x1))
    b#(x1) -> d#(c(x1)) -> d#(d(x1)) -> e#(x1)
    e#(c(x1)) -> d#(a(x1)) -> d#(d(x1)) -> b#(e(x1))
    e#(c(x1)) -> d#(a(x1)) -> d#(d(x1)) -> e#(x1)
    e#(c(x1)) -> a#(x1) -> a#(x1) -> e#(d(x1))
    e#(c(x1)) -> a#(x1) -> a#(x1) -> d#(x1)
    e#(c(x1)) -> a#(x1) -> a#(b(x1)) -> d#(x1)
    d#(d(x1)) -> b#(e(x1)) -> b#(x1) -> d#(c(x1))
    d#(d(x1)) -> e#(x1) -> e#(c(x1)) -> d#(a(x1))
    d#(d(x1)) -> e#(x1) -> e#(c(x1)) -> a#(x1)
    a#(b(x1)) -> d#(x1) -> d#(d(x1)) -> b#(e(x1))
    a#(b(x1)) -> d#(x1) -> d#(d(x1)) -> e#(x1)
    a#(x1) -> e#(d(x1)) -> e#(c(x1)) -> d#(a(x1))
    a#(x1) -> e#(d(x1)) -> e#(c(x1)) -> a#(x1)
    a#(x1) -> d#(x1) -> d#(d(x1)) -> b#(e(x1))
    a#(x1) -> d#(x1) -> d#(d(x1)) -> e#(x1)
   EDG Processor:
    DPs:
     a#(b(x1)) -> d#(x1)
     d#(d(x1)) -> e#(x1)
     d#(d(x1)) -> b#(e(x1))
     b#(x1) -> d#(c(x1))
     e#(c(x1)) -> a#(x1)
     e#(c(x1)) -> d#(a(x1))
     a#(x1) -> d#(x1)
     a#(x1) -> e#(d(x1))
    TRS:
     a(b(x1)) -> c(d(x1))
     d(d(x1)) -> b(e(x1))
     b(x1) -> d(c(x1))
     d(x1) -> x1
     e(c(x1)) -> d(a(x1))
     a(x1) -> e(d(x1))
    graph:
     e#(c(x1)) -> d#(a(x1)) -> d#(d(x1)) -> e#(x1)
     e#(c(x1)) -> d#(a(x1)) -> d#(d(x1)) -> b#(e(x1))
     e#(c(x1)) -> a#(x1) -> a#(b(x1)) -> d#(x1)
     e#(c(x1)) -> a#(x1) -> a#(x1) -> d#(x1)
     e#(c(x1)) -> a#(x1) -> a#(x1) -> e#(d(x1))
     d#(d(x1)) -> b#(e(x1)) -> b#(x1) -> d#(c(x1))
     d#(d(x1)) -> e#(x1) -> e#(c(x1)) -> a#(x1)
     d#(d(x1)) -> e#(x1) -> e#(c(x1)) -> d#(a(x1))
     a#(b(x1)) -> d#(x1) -> d#(d(x1)) -> e#(x1)
     a#(b(x1)) -> d#(x1) -> d#(d(x1)) -> b#(e(x1))
     a#(x1) -> e#(d(x1)) -> e#(c(x1)) -> a#(x1)
     a#(x1) -> e#(d(x1)) -> e#(c(x1)) -> d#(a(x1))
     a#(x1) -> d#(x1) -> d#(d(x1)) -> e#(x1)
     a#(x1) -> d#(x1) -> d#(d(x1)) -> b#(e(x1))
    SCC Processor:
     #sccs: 1
     #rules: 6
     #arcs: 14/64
     DPs:
      e#(c(x1)) -> d#(a(x1))
      d#(d(x1)) -> e#(x1)
      e#(c(x1)) -> a#(x1)
      a#(x1) -> e#(d(x1))
      a#(x1) -> d#(x1)
      a#(b(x1)) -> d#(x1)
     TRS:
      a(b(x1)) -> c(d(x1))
      d(d(x1)) -> b(e(x1))
      b(x1) -> d(c(x1))
      d(x1) -> x1
      e(c(x1)) -> d(a(x1))
      a(x1) -> e(d(x1))
     Arctic Interpretation Processor:
      dimension: 2
      usable rules:
       a(b(x1)) -> c(d(x1))
       d(d(x1)) -> b(e(x1))
       b(x1) -> d(c(x1))
       d(x1) -> x1
       e(c(x1)) -> d(a(x1))
       a(x1) -> e(d(x1))
      interpretation:
       [e#](x0) = [0 0]x0,
       
       [d#](x0) = [0 0]x0,
       
       [a#](x0) = [1 1]x0,
       
                 [0  -&]  
       [e](x0) = [0  -&]x0,
       
                 [1 1]  
       [c](x0) = [0 1]x0,
       
                 [0 0]  
       [d](x0) = [1 1]x0,
       
                 [0 0]  
       [a](x0) = [0 0]x0,
       
                 [1 1]  
       [b](x0) = [2 2]x0
      orientation:
       e#(c(x1)) = [1 1]x1 >= [0 0]x1 = d#(a(x1))
       
       d#(d(x1)) = [1 1]x1 >= [0 0]x1 = e#(x1)
       
       e#(c(x1)) = [1 1]x1 >= [1 1]x1 = a#(x1)
       
       a#(x1) = [1 1]x1 >= [1 1]x1 = e#(d(x1))
       
       a#(x1) = [1 1]x1 >= [0 0]x1 = d#(x1)
       
       a#(b(x1)) = [3 3]x1 >= [0 0]x1 = d#(x1)
       
                  [2 2]      [2 2]             
       a(b(x1)) = [2 2]x1 >= [2 2]x1 = c(d(x1))
       
                  [1 1]      [1  -&]             
       d(d(x1)) = [2 2]x1 >= [2  -&]x1 = b(e(x1))
       
               [1 1]      [1 1]             
       b(x1) = [2 2]x1 >= [2 2]x1 = d(c(x1))
       
               [0 0]             
       d(x1) = [1 1]x1 >= x1 = x1
       
                  [1 1]      [0 0]             
       e(c(x1)) = [1 1]x1 >= [1 1]x1 = d(a(x1))
       
               [0 0]      [0 0]             
       a(x1) = [0 0]x1 >= [0 0]x1 = e(d(x1))
      problem:
       DPs:
        e#(c(x1)) -> a#(x1)
        a#(x1) -> e#(d(x1))
       TRS:
        a(b(x1)) -> c(d(x1))
        d(d(x1)) -> b(e(x1))
        b(x1) -> d(c(x1))
        d(x1) -> x1
        e(c(x1)) -> d(a(x1))
        a(x1) -> e(d(x1))
      Restore Modifier:
       DPs:
        e#(c(x1)) -> a#(x1)
        a#(x1) -> e#(d(x1))
       TRS:
        a(b(x1)) -> c(d(x1))
        d(d(x1)) -> b(e(x1))
        b(x1) -> d(c(x1))
        d(x1) -> x1
        e(c(x1)) -> d(a(x1))
        a(x1) -> e(d(x1))
       EDG Processor:
        DPs:
         e#(c(x1)) -> a#(x1)
         a#(x1) -> e#(d(x1))
        TRS:
         a(b(x1)) -> c(d(x1))
         d(d(x1)) -> b(e(x1))
         b(x1) -> d(c(x1))
         d(x1) -> x1
         e(c(x1)) -> d(a(x1))
         a(x1) -> e(d(x1))
        graph:
         e#(c(x1)) -> a#(x1) -> a#(x1) -> e#(d(x1))
         a#(x1) -> e#(d(x1)) -> e#(c(x1)) -> a#(x1)
        Arctic Interpretation Processor:
         dimension: 2
         usable rules:
          a(b(x1)) -> c(d(x1))
          d(d(x1)) -> b(e(x1))
          b(x1) -> d(c(x1))
          d(x1) -> x1
          e(c(x1)) -> d(a(x1))
          a(x1) -> e(d(x1))
         interpretation:
          [e#](x0) = [0 2]x0 + [0],
          
          [a#](x0) = [2 2]x0 + [1],
          
                    [-& 0 ]     [0 ]
          [e](x0) = [-& 0 ]x0 + [-&],
          
                    [0 0]     [0]
          [c](x0) = [1 1]x0 + [1],
          
                    [1 1]     [1 ]
          [d](x0) = [0 0]x0 + [-&],
          
                    [0 0]     [0]
          [a](x0) = [0 0]x0 + [0],
          
                    [2 2]     [2]
          [b](x0) = [1 1]x0 + [1]
         orientation:
          e#(c(x1)) = [3 3]x1 + [3] >= [2 2]x1 + [1] = a#(x1)
          
          a#(x1) = [2 2]x1 + [1] >= [2 2]x1 + [1] = e#(d(x1))
          
                     [2 2]     [2]    [1 1]     [1]           
          a(b(x1)) = [2 2]x1 + [2] >= [2 2]x1 + [2] = c(d(x1))
          
                     [2 2]     [2]    [-& 2 ]     [2]           
          d(d(x1)) = [1 1]x1 + [1] >= [-& 1 ]x1 + [1] = b(e(x1))
          
                  [2 2]     [2]    [2 2]     [2]           
          b(x1) = [1 1]x1 + [1] >= [1 1]x1 + [1] = d(c(x1))
          
                  [1 1]     [1 ]           
          d(x1) = [0 0]x1 + [-&] >= x1 = x1
          
                     [1 1]     [1]    [1 1]     [1]           
          e(c(x1)) = [1 1]x1 + [1] >= [0 0]x1 + [0] = d(a(x1))
          
                  [0 0]     [0]    [0 0]     [0 ]           
          a(x1) = [0 0]x1 + [0] >= [0 0]x1 + [-&] = e(d(x1))
         problem:
          DPs:
           a#(x1) -> e#(d(x1))
          TRS:
           a(b(x1)) -> c(d(x1))
           d(d(x1)) -> b(e(x1))
           b(x1) -> d(c(x1))
           d(x1) -> x1
           e(c(x1)) -> d(a(x1))
           a(x1) -> e(d(x1))
         Restore Modifier:
          DPs:
           a#(x1) -> e#(d(x1))
          TRS:
           a(b(x1)) -> c(d(x1))
           d(d(x1)) -> b(e(x1))
           b(x1) -> d(c(x1))
           d(x1) -> x1
           e(c(x1)) -> d(a(x1))
           a(x1) -> e(d(x1))
          EDG Processor:
           DPs:
            a#(x1) -> e#(d(x1))
           TRS:
            a(b(x1)) -> c(d(x1))
            d(d(x1)) -> b(e(x1))
            b(x1) -> d(c(x1))
            d(x1) -> x1
            e(c(x1)) -> d(a(x1))
            a(x1) -> e(d(x1))
           graph:
            
           Qed