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
  EDG 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#(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: 2
    #arcs: 2/9
    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
    Usable Rule Processor:
     DPs:
      g#(c(x,s(y))) -> g#(c(s(x),y))
     TRS:
      
     Bounds Processor:
      bound: 1
      enrichment: match-dp
      automaton:
       final states: {4}
       transitions:
        s1(5) -> 7*
        s1(7) -> 7*
        g{#,0}(6) -> 4*
        c0(5,3) -> 6*
        c0(1,2) -> 2*
        c0(2,1) -> 2*
        c0(1,1) -> 2*
        c0(2,2) -> 2*
        s0(2) -> 1*
        s0(1) -> 1*
        s0(3) -> 5*
        g{#,1}(8) -> 4*
        c1(7,1) -> 8*
        c1(7,2) -> 8*
        1 -> 3*
        2 -> 3*
      problem:
       DPs:
        
       TRS:
        
      Qed
    
    DPs:
     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
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [f#](x0) = x0,
      
      [d](x0) = x0,
      
      [f](x0) = x0,
      
      [g](x0) = 0,
      
      [c](x0, x1) = x0,
      
      [s](x0) = x0 + 1
     orientation:
      f#(c(s(x),y)) = x + 1 >= x = f#(c(x,s(y)))
      
      g(c(x,s(y))) = 0 >= 0 = g(c(s(x),y))
      
      f(c(s(x),y)) = x + 1 >= x = f(c(x,s(y)))
      
      f(f(x)) = x >= x = f(d(f(x)))
      
      f(x) = x >= x = x
     problem:
      DPs:
       
      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
     Qed