YES

Problem:
 g(c(x,s(y))) -> g(c(s(x),y))
 f(c(s(x),y)) -> f(c(x,s(y)))
 f(f(x)) -> f(d(f(x)))
 f(x) -> x

Proof:
 DP Processor:
  DPs:
   g#(c(x,s(y))) -> g#(c(s(x),y))
   f#(c(s(x),y)) -> f#(c(x,s(y)))
   f#(f(x)) -> f#(d(f(x)))
  TRS:
   g(c(x,s(y))) -> g(c(s(x),y))
   f(c(s(x),y)) -> f(c(x,s(y)))
   f(f(x)) -> f(d(f(x)))
   f(x) -> x
  TDG Processor:
   DPs:
    g#(c(x,s(y))) -> g#(c(s(x),y))
    f#(c(s(x),y)) -> f#(c(x,s(y)))
    f#(f(x)) -> f#(d(f(x)))
   TRS:
    g(c(x,s(y))) -> g(c(s(x),y))
    f(c(s(x),y)) -> f(c(x,s(y)))
    f(f(x)) -> f(d(f(x)))
    f(x) -> x
   graph:
    f#(f(x)) -> f#(d(f(x))) -> f#(f(x)) -> f#(d(f(x)))
    f#(f(x)) -> f#(d(f(x))) -> f#(c(s(x),y)) -> f#(c(x,s(y)))
    f#(c(s(x),y)) -> f#(c(x,s(y))) -> f#(f(x)) -> f#(d(f(x)))
    f#(c(s(x),y)) -> f#(c(x,s(y))) -> f#(c(s(x),y)) -> f#(c(x,s(y)))
    g#(c(x,s(y))) -> g#(c(s(x),y)) -> g#(c(x,s(y))) -> g#(c(s(x),y))
   SCC Processor:
    #sccs: 2
    #rules: 3
    #arcs: 5/9
    DPs:
     g#(c(x,s(y))) -> g#(c(s(x),y))
    TRS:
     g(c(x,s(y))) -> g(c(s(x),y))
     f(c(s(x),y)) -> f(c(x,s(y)))
     f(f(x)) -> f(d(f(x)))
     f(x) -> x
    CDG Processor:
     DPs:
      g#(c(x,s(y))) -> g#(c(s(x),y))
     TRS:
      g(c(x,s(y))) -> g(c(s(x),y))
      f(c(s(x),y)) -> f(c(x,s(y)))
      f(f(x)) -> f(d(f(x)))
      f(x) -> x
     graph:
      
     Qed
    
    DPs:
     f#(f(x)) -> f#(d(f(x)))
     f#(c(s(x),y)) -> f#(c(x,s(y)))
    TRS:
     g(c(x,s(y))) -> g(c(s(x),y))
     f(c(s(x),y)) -> f(c(x,s(y)))
     f(f(x)) -> f(d(f(x)))
     f(x) -> x
    EDG Processor:
     DPs:
      f#(f(x)) -> f#(d(f(x)))
      f#(c(s(x),y)) -> f#(c(x,s(y)))
     TRS:
      g(c(x,s(y))) -> g(c(s(x),y))
      f(c(s(x),y)) -> f(c(x,s(y)))
      f(f(x)) -> f(d(f(x)))
      f(x) -> x
     graph:
      f#(c(s(x),y)) -> f#(c(x,s(y))) -> f#(c(s(x),y)) -> f#(c(x,s(y)))
     CDG Processor:
      DPs:
       f#(f(x)) -> f#(d(f(x)))
       f#(c(s(x),y)) -> f#(c(x,s(y)))
      TRS:
       g(c(x,s(y))) -> g(c(s(x),y))
       f(c(s(x),y)) -> f(c(x,s(y)))
       f(f(x)) -> f(d(f(x)))
       f(x) -> x
      graph:
       
      Qed