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())))
  TDG 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())))
   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)
   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