YES

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

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