YES

Problem:
 f(f(x)) -> f(x)
 f(s(x)) -> f(x)
 g(s(0())) -> g(f(s(0())))

Proof:
 DP Processor:
  DPs:
   f#(s(x)) -> f#(x)
   g#(s(0())) -> f#(s(0()))
   g#(s(0())) -> g#(f(s(0())))
  TRS:
   f(f(x)) -> f(x)
   f(s(x)) -> f(x)
   g(s(0())) -> g(f(s(0())))
  Usable Rule Processor:
   DPs:
    f#(s(x)) -> f#(x)
    g#(s(0())) -> f#(s(0()))
    g#(s(0())) -> g#(f(s(0())))
   TRS:
    f(s(x)) -> f(x)
    f(f(x)) -> f(x)
   TDG Processor:
    DPs:
     f#(s(x)) -> f#(x)
     g#(s(0())) -> f#(s(0()))
     g#(s(0())) -> g#(f(s(0())))
    TRS:
     f(s(x)) -> f(x)
     f(f(x)) -> f(x)
    graph:
     g#(s(0())) -> g#(f(s(0()))) -> g#(s(0())) -> g#(f(s(0())))
     g#(s(0())) -> g#(f(s(0()))) -> g#(s(0())) -> f#(s(0()))
     g#(s(0())) -> f#(s(0())) -> f#(s(x)) -> f#(x)
     f#(s(x)) -> f#(x) -> f#(s(x)) -> f#(x)
    Restore Modifier:
     DPs:
      f#(s(x)) -> f#(x)
      g#(s(0())) -> f#(s(0()))
      g#(s(0())) -> g#(f(s(0())))
     TRS:
      f(f(x)) -> f(x)
      f(s(x)) -> f(x)
      g(s(0())) -> g(f(s(0())))
     SCC Processor:
      #sccs: 2
      #rules: 2
      #arcs: 4/9
      DPs:
       g#(s(0())) -> g#(f(s(0())))
      TRS:
       f(f(x)) -> f(x)
       f(s(x)) -> f(x)
       g(s(0())) -> g(f(s(0())))
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [g#](x0) = x0,
        
        [g](x0) = 0,
        
        [0] = 0,
        
        [s](x0) = 1,
        
        [f](x0) = 0
       orientation:
        g#(s(0())) = 1 >= 0 = g#(f(s(0())))
        
        f(f(x)) = 0 >= 0 = f(x)
        
        f(s(x)) = 0 >= 0 = f(x)
        
        g(s(0())) = 0 >= 0 = g(f(s(0())))
       problem:
        DPs:
         
        TRS:
         f(f(x)) -> f(x)
         f(s(x)) -> f(x)
         g(s(0())) -> g(f(s(0())))
       Qed
      
      DPs:
       f#(s(x)) -> f#(x)
      TRS:
       f(f(x)) -> f(x)
       f(s(x)) -> f(x)
       g(s(0())) -> g(f(s(0())))
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [f#](x0) = x0 + 1,
        
        [g](x0) = 0,
        
        [0] = 0,
        
        [s](x0) = x0 + 1,
        
        [f](x0) = 0
       orientation:
        f#(s(x)) = x + 2 >= x + 1 = f#(x)
        
        f(f(x)) = 0 >= 0 = f(x)
        
        f(s(x)) = 0 >= 0 = f(x)
        
        g(s(0())) = 0 >= 0 = g(f(s(0())))
       problem:
        DPs:
         
        TRS:
         f(f(x)) -> f(x)
         f(s(x)) -> f(x)
         g(s(0())) -> g(f(s(0())))
       Qed