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
    Bounds Processor:
     bound: 2
     enrichment: match
     automaton:
      final states: {8}
      transitions:
       g{#,2}(19) -> 8*
       d0(7) -> 7*
       f1(7) -> 14*
       f1(13) -> 14,7
       d1(14) -> 13*
       c1(7,11) -> 13*
       c1(11,7) -> 12*
       c1(11,11) -> 12*
       s1(7) -> 11*
       s1(11) -> 11*
       g1(12) -> 7*
       g{#,1}(12) -> 8*
       f2(16) -> 14*
       f2(13) -> 15*
       g{#,0}(7) -> 8*
       d2(15) -> 16*
       c0(7,7) -> 7*
       g2(19) -> 7,14
       s0(7) -> 7*
       c2(18,7) -> 19*
       c2(18,11) -> 19*
       g0(7) -> 7*
       s2(11) -> 18*
       s2(18) -> 18*
       f0(7) -> 7*
       7 -> 14*
       13 -> 15,7
       16 -> 14*
     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
    
    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