YES

Problem:
 f(f(x)) -> f(c(f(x)))
 f(f(x)) -> f(d(f(x)))
 g(c(x)) -> x
 g(d(x)) -> x
 g(c(0())) -> g(d(1()))
 g(c(1())) -> g(d(0()))

Proof:
 DP Processor:
  DPs:
   f#(f(x)) -> f#(c(f(x)))
   f#(f(x)) -> f#(d(f(x)))
   g#(c(0())) -> g#(d(1()))
   g#(c(1())) -> g#(d(0()))
  TRS:
   f(f(x)) -> f(c(f(x)))
   f(f(x)) -> f(d(f(x)))
   g(c(x)) -> x
   g(d(x)) -> x
   g(c(0())) -> g(d(1()))
   g(c(1())) -> g(d(0()))
  TDG Processor:
   DPs:
    f#(f(x)) -> f#(c(f(x)))
    f#(f(x)) -> f#(d(f(x)))
    g#(c(0())) -> g#(d(1()))
    g#(c(1())) -> g#(d(0()))
   TRS:
    f(f(x)) -> f(c(f(x)))
    f(f(x)) -> f(d(f(x)))
    g(c(x)) -> x
    g(d(x)) -> x
    g(c(0())) -> g(d(1()))
    g(c(1())) -> g(d(0()))
   graph:
    g#(c(1())) -> g#(d(0())) -> g#(c(1())) -> g#(d(0()))
    g#(c(1())) -> g#(d(0())) -> g#(c(0())) -> g#(d(1()))
    g#(c(0())) -> g#(d(1())) -> g#(c(1())) -> g#(d(0()))
    g#(c(0())) -> g#(d(1())) -> g#(c(0())) -> g#(d(1()))
    f#(f(x)) -> f#(d(f(x))) -> f#(f(x)) -> f#(d(f(x)))
    f#(f(x)) -> f#(d(f(x))) -> f#(f(x)) -> f#(c(f(x)))
    f#(f(x)) -> f#(c(f(x))) -> f#(f(x)) -> f#(d(f(x)))
    f#(f(x)) -> f#(c(f(x))) -> f#(f(x)) -> f#(c(f(x)))
   SCC Processor:
    #sccs: 2
    #rules: 4
    #arcs: 8/16
    DPs:
     f#(f(x)) -> f#(d(f(x)))
     f#(f(x)) -> f#(c(f(x)))
    TRS:
     f(f(x)) -> f(c(f(x)))
     f(f(x)) -> f(d(f(x)))
     g(c(x)) -> x
     g(d(x)) -> x
     g(c(0())) -> g(d(1()))
     g(c(1())) -> g(d(0()))
    EDG Processor:
     DPs:
      f#(f(x)) -> f#(d(f(x)))
      f#(f(x)) -> f#(c(f(x)))
     TRS:
      f(f(x)) -> f(c(f(x)))
      f(f(x)) -> f(d(f(x)))
      g(c(x)) -> x
      g(d(x)) -> x
      g(c(0())) -> g(d(1()))
      g(c(1())) -> g(d(0()))
     graph:
      
     Qed
    
    DPs:
     g#(c(1())) -> g#(d(0()))
     g#(c(0())) -> g#(d(1()))
    TRS:
     f(f(x)) -> f(c(f(x)))
     f(f(x)) -> f(d(f(x)))
     g(c(x)) -> x
     g(d(x)) -> x
     g(c(0())) -> g(d(1()))
     g(c(1())) -> g(d(0()))
    EDG Processor:
     DPs:
      g#(c(1())) -> g#(d(0()))
      g#(c(0())) -> g#(d(1()))
     TRS:
      f(f(x)) -> f(c(f(x)))
      f(f(x)) -> f(d(f(x)))
      g(c(x)) -> x
      g(d(x)) -> x
      g(c(0())) -> g(d(1()))
      g(c(1())) -> g(d(0()))
     graph:
      
     Qed