YES

Problem:
 a(a(x1)) -> b(c(x1))
 b(b(x1)) -> c(d(x1))
 b(x1) -> a(x1)
 c(c(x1)) -> d(f(x1))
 d(d(x1)) -> f(f(f(x1)))
 d(x1) -> b(x1)
 f(f(x1)) -> g(a(x1))
 g(g(x1)) -> a(x1)

Proof:
 DP Processor:
  DPs:
   a#(a(x1)) -> c#(x1)
   a#(a(x1)) -> b#(c(x1))
   b#(b(x1)) -> d#(x1)
   b#(b(x1)) -> c#(d(x1))
   b#(x1) -> a#(x1)
   c#(c(x1)) -> f#(x1)
   c#(c(x1)) -> d#(f(x1))
   d#(d(x1)) -> f#(x1)
   d#(d(x1)) -> f#(f(x1))
   d#(d(x1)) -> f#(f(f(x1)))
   d#(x1) -> b#(x1)
   f#(f(x1)) -> a#(x1)
   f#(f(x1)) -> g#(a(x1))
   g#(g(x1)) -> a#(x1)
  TRS:
   a(a(x1)) -> b(c(x1))
   b(b(x1)) -> c(d(x1))
   b(x1) -> a(x1)
   c(c(x1)) -> d(f(x1))
   d(d(x1)) -> f(f(f(x1)))
   d(x1) -> b(x1)
   f(f(x1)) -> g(a(x1))
   g(g(x1)) -> a(x1)
  TDG Processor:
   DPs:
    a#(a(x1)) -> c#(x1)
    a#(a(x1)) -> b#(c(x1))
    b#(b(x1)) -> d#(x1)
    b#(b(x1)) -> c#(d(x1))
    b#(x1) -> a#(x1)
    c#(c(x1)) -> f#(x1)
    c#(c(x1)) -> d#(f(x1))
    d#(d(x1)) -> f#(x1)
    d#(d(x1)) -> f#(f(x1))
    d#(d(x1)) -> f#(f(f(x1)))
    d#(x1) -> b#(x1)
    f#(f(x1)) -> a#(x1)
    f#(f(x1)) -> g#(a(x1))
    g#(g(x1)) -> a#(x1)
   TRS:
    a(a(x1)) -> b(c(x1))
    b(b(x1)) -> c(d(x1))
    b(x1) -> a(x1)
    c(c(x1)) -> d(f(x1))
    d(d(x1)) -> f(f(f(x1)))
    d(x1) -> b(x1)
    f(f(x1)) -> g(a(x1))
    g(g(x1)) -> a(x1)
   graph:
    g#(g(x1)) -> a#(x1) -> a#(a(x1)) -> b#(c(x1))
    g#(g(x1)) -> a#(x1) -> a#(a(x1)) -> c#(x1)
    f#(f(x1)) -> g#(a(x1)) -> g#(g(x1)) -> a#(x1)
    f#(f(x1)) -> a#(x1) -> a#(a(x1)) -> b#(c(x1))
    f#(f(x1)) -> a#(x1) -> a#(a(x1)) -> c#(x1)
    d#(d(x1)) -> f#(f(f(x1))) -> f#(f(x1)) -> g#(a(x1))
    d#(d(x1)) -> f#(f(f(x1))) -> f#(f(x1)) -> a#(x1)
    d#(d(x1)) -> f#(f(x1)) -> f#(f(x1)) -> g#(a(x1))
    d#(d(x1)) -> f#(f(x1)) -> f#(f(x1)) -> a#(x1)
    d#(d(x1)) -> f#(x1) -> f#(f(x1)) -> g#(a(x1))
    d#(d(x1)) -> f#(x1) -> f#(f(x1)) -> a#(x1)
    d#(x1) -> b#(x1) -> b#(x1) -> a#(x1)
    d#(x1) -> b#(x1) -> b#(b(x1)) -> c#(d(x1))
    d#(x1) -> b#(x1) -> b#(b(x1)) -> d#(x1)
    b#(b(x1)) -> d#(x1) -> d#(x1) -> b#(x1)
    b#(b(x1)) -> d#(x1) -> d#(d(x1)) -> f#(f(f(x1)))
    b#(b(x1)) -> d#(x1) -> d#(d(x1)) -> f#(f(x1))
    b#(b(x1)) -> d#(x1) -> d#(d(x1)) -> f#(x1)
    b#(b(x1)) -> c#(d(x1)) -> c#(c(x1)) -> d#(f(x1))
    b#(b(x1)) -> c#(d(x1)) -> c#(c(x1)) -> f#(x1)
    b#(x1) -> a#(x1) -> a#(a(x1)) -> b#(c(x1))
    b#(x1) -> a#(x1) -> a#(a(x1)) -> c#(x1)
    c#(c(x1)) -> f#(x1) -> f#(f(x1)) -> g#(a(x1))
    c#(c(x1)) -> f#(x1) -> f#(f(x1)) -> a#(x1)
    c#(c(x1)) -> d#(f(x1)) -> d#(x1) -> b#(x1)
    c#(c(x1)) -> d#(f(x1)) -> d#(d(x1)) -> f#(f(f(x1)))
    c#(c(x1)) -> d#(f(x1)) -> d#(d(x1)) -> f#(f(x1))
    c#(c(x1)) -> d#(f(x1)) -> d#(d(x1)) -> f#(x1)
    a#(a(x1)) -> b#(c(x1)) -> b#(x1) -> a#(x1)
    a#(a(x1)) -> b#(c(x1)) -> b#(b(x1)) -> c#(d(x1))
    a#(a(x1)) -> b#(c(x1)) -> b#(b(x1)) -> d#(x1)
    a#(a(x1)) -> c#(x1) -> c#(c(x1)) -> d#(f(x1))
    a#(a(x1)) -> c#(x1) -> c#(c(x1)) -> f#(x1)
   CDG Processor:
    DPs:
     a#(a(x1)) -> c#(x1)
     a#(a(x1)) -> b#(c(x1))
     b#(b(x1)) -> d#(x1)
     b#(b(x1)) -> c#(d(x1))
     b#(x1) -> a#(x1)
     c#(c(x1)) -> f#(x1)
     c#(c(x1)) -> d#(f(x1))
     d#(d(x1)) -> f#(x1)
     d#(d(x1)) -> f#(f(x1))
     d#(d(x1)) -> f#(f(f(x1)))
     d#(x1) -> b#(x1)
     f#(f(x1)) -> a#(x1)
     f#(f(x1)) -> g#(a(x1))
     g#(g(x1)) -> a#(x1)
    TRS:
     a(a(x1)) -> b(c(x1))
     b(b(x1)) -> c(d(x1))
     b(x1) -> a(x1)
     c(c(x1)) -> d(f(x1))
     d(d(x1)) -> f(f(f(x1)))
     d(x1) -> b(x1)
     f(f(x1)) -> g(a(x1))
     g(g(x1)) -> a(x1)
    graph:
     g#(g(x1)) -> a#(x1) -> a#(a(x1)) -> c#(x1)
     g#(g(x1)) -> a#(x1) -> a#(a(x1)) -> b#(c(x1))
     f#(f(x1)) -> a#(x1) -> a#(a(x1)) -> c#(x1)
     f#(f(x1)) -> a#(x1) -> a#(a(x1)) -> b#(c(x1))
     d#(d(x1)) -> f#(f(f(x1))) -> f#(f(x1)) -> a#(x1)
     d#(d(x1)) -> f#(f(f(x1))) -> f#(f(x1)) -> g#(a(x1))
     d#(d(x1)) -> f#(f(x1)) -> f#(f(x1)) -> a#(x1)
     d#(d(x1)) -> f#(f(x1)) -> f#(f(x1)) -> g#(a(x1))
     d#(d(x1)) -> f#(x1) -> f#(f(x1)) -> a#(x1)
     d#(d(x1)) -> f#(x1) -> f#(f(x1)) -> g#(a(x1))
     d#(x1) -> b#(x1) -> b#(b(x1)) -> d#(x1)
     d#(x1) -> b#(x1) -> b#(b(x1)) -> c#(d(x1))
     d#(x1) -> b#(x1) -> b#(x1) -> a#(x1)
     b#(b(x1)) -> d#(x1) -> d#(d(x1)) -> f#(x1)
     b#(b(x1)) -> d#(x1) -> d#(d(x1)) -> f#(f(x1))
     b#(b(x1)) -> d#(x1) -> d#(d(x1)) -> f#(f(f(x1)))
     b#(b(x1)) -> d#(x1) -> d#(x1) -> b#(x1)
     b#(b(x1)) -> c#(d(x1)) -> c#(c(x1)) -> f#(x1)
     b#(b(x1)) -> c#(d(x1)) -> c#(c(x1)) -> d#(f(x1))
     b#(x1) -> a#(x1) -> a#(a(x1)) -> c#(x1)
     b#(x1) -> a#(x1) -> a#(a(x1)) -> b#(c(x1))
     c#(c(x1)) -> f#(x1) -> f#(f(x1)) -> a#(x1)
     c#(c(x1)) -> f#(x1) -> f#(f(x1)) -> g#(a(x1))
     c#(c(x1)) -> d#(f(x1)) -> d#(x1) -> b#(x1)
     a#(a(x1)) -> b#(c(x1)) -> b#(b(x1)) -> d#(x1)
     a#(a(x1)) -> b#(c(x1)) -> b#(b(x1)) -> c#(d(x1))
     a#(a(x1)) -> b#(c(x1)) -> b#(x1) -> a#(x1)
     a#(a(x1)) -> c#(x1) -> c#(c(x1)) -> f#(x1)
     a#(a(x1)) -> c#(x1) -> c#(c(x1)) -> d#(f(x1))
    SCC Processor:
     #sccs: 1
     #rules: 12
     #arcs: 29/196
     DPs:
      a#(a(x1)) -> b#(c(x1))
      b#(x1) -> a#(x1)
      a#(a(x1)) -> c#(x1)
      c#(c(x1)) -> d#(f(x1))
      d#(x1) -> b#(x1)
      b#(b(x1)) -> c#(d(x1))
      c#(c(x1)) -> f#(x1)
      f#(f(x1)) -> a#(x1)
      b#(b(x1)) -> d#(x1)
      d#(d(x1)) -> f#(f(f(x1)))
      d#(d(x1)) -> f#(f(x1))
      d#(d(x1)) -> f#(x1)
     TRS:
      a(a(x1)) -> b(c(x1))
      b(b(x1)) -> c(d(x1))
      b(x1) -> a(x1)
      c(c(x1)) -> d(f(x1))
      d(d(x1)) -> f(f(f(x1)))
      d(x1) -> b(x1)
      f(f(x1)) -> g(a(x1))
      g(g(x1)) -> a(x1)
     Matrix Interpretation Processor: dim=1
      
      interpretation:
       [f#](x0) = 2x0,
       
       [d#](x0) = 2x0 + 14,
       
       [b#](x0) = 2x0 + 12,
       
       [c#](x0) = 2x0 + 9,
       
       [a#](x0) = 2x0 + 11,
       
       [g](x0) = x0 + 8,
       
       [f](x0) = x0 + 12,
       
       [d](x0) = x0 + 18,
       
       [b](x0) = x0 + 17,
       
       [c](x0) = x0 + 15,
       
       [a](x0) = x0 + 16
      orientation:
       a#(a(x1)) = 2x1 + 43 >= 2x1 + 42 = b#(c(x1))
       
       b#(x1) = 2x1 + 12 >= 2x1 + 11 = a#(x1)
       
       a#(a(x1)) = 2x1 + 43 >= 2x1 + 9 = c#(x1)
       
       c#(c(x1)) = 2x1 + 39 >= 2x1 + 38 = d#(f(x1))
       
       d#(x1) = 2x1 + 14 >= 2x1 + 12 = b#(x1)
       
       b#(b(x1)) = 2x1 + 46 >= 2x1 + 45 = c#(d(x1))
       
       c#(c(x1)) = 2x1 + 39 >= 2x1 = f#(x1)
       
       f#(f(x1)) = 2x1 + 24 >= 2x1 + 11 = a#(x1)
       
       b#(b(x1)) = 2x1 + 46 >= 2x1 + 14 = d#(x1)
       
       d#(d(x1)) = 2x1 + 50 >= 2x1 + 48 = f#(f(f(x1)))
       
       d#(d(x1)) = 2x1 + 50 >= 2x1 + 24 = f#(f(x1))
       
       d#(d(x1)) = 2x1 + 50 >= 2x1 = f#(x1)
       
       a(a(x1)) = x1 + 32 >= x1 + 32 = b(c(x1))
       
       b(b(x1)) = x1 + 34 >= x1 + 33 = c(d(x1))
       
       b(x1) = x1 + 17 >= x1 + 16 = a(x1)
       
       c(c(x1)) = x1 + 30 >= x1 + 30 = d(f(x1))
       
       d(d(x1)) = x1 + 36 >= x1 + 36 = f(f(f(x1)))
       
       d(x1) = x1 + 18 >= x1 + 17 = b(x1)
       
       f(f(x1)) = x1 + 24 >= x1 + 24 = g(a(x1))
       
       g(g(x1)) = x1 + 16 >= x1 + 16 = a(x1)
      problem:
       DPs:
        
       TRS:
        a(a(x1)) -> b(c(x1))
        b(b(x1)) -> c(d(x1))
        b(x1) -> a(x1)
        c(c(x1)) -> d(f(x1))
        d(d(x1)) -> f(f(f(x1)))
        d(x1) -> b(x1)
        f(f(x1)) -> g(a(x1))
        g(g(x1)) -> a(x1)
      Qed