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: 1
     enrichment: match-dp
     automaton:
      final states: {7}
      transitions:
       g{#,0}(9) -> 7*
       c0(3,1) -> 2*
       c0(3,3) -> 2*
       c0(3,5) -> 2*
       c0(4,2) -> 2*
       c0(4,4) -> 2*
       c0(5,1) -> 2*
       c0(5,3) -> 2*
       c0(5,5) -> 2*
       c0(1,2) -> 2*
       c0(1,4) -> 2*
       c0(2,1) -> 2*
       c0(2,3) -> 2*
       c0(2,5) -> 2*
       c0(3,2) -> 2*
       c0(3,4) -> 2*
       c0(8,6) -> 9*
       c0(4,1) -> 2*
       c0(4,3) -> 2*
       c0(4,5) -> 2*
       c0(5,2) -> 2*
       c0(5,4) -> 2*
       c0(1,1) -> 2*
       c0(1,3) -> 2*
       c0(1,5) -> 2*
       c0(2,2) -> 2*
       c0(2,4) -> 2*
       s0(5) -> 1*
       s0(2) -> 1*
       s0(4) -> 1*
       s0(6) -> 8*
       s0(1) -> 1*
       s0(3) -> 1*
       g0(5) -> 3*
       g0(2) -> 3*
       g0(4) -> 3*
       g0(1) -> 3*
       g0(3) -> 3*
       f0(5) -> 4*
       f0(2) -> 4*
       f0(4) -> 4*
       f0(1) -> 4*
       f0(3) -> 4*
       d0(5) -> 5*
       d0(2) -> 5*
       d0(4) -> 5*
       d0(1) -> 5*
       d0(3) -> 5*
       g{#,1}(13) -> 7*
       c1(12,1) -> 13*
       c1(12,3) -> 13*
       c1(12,5) -> 13*
       c1(12,2) -> 13*
       c1(12,4) -> 13*
       s1(12) -> 12*
       s1(8) -> 12*
       1 -> 4,6
       2 -> 4,6
       3 -> 4,6
       4 -> 6*
       5 -> 4,6
     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