YES

Problem:
 f(1()) -> f(g(1()))
 f(f(x)) -> f(x)
 g(0()) -> g(f(0()))
 g(g(x)) -> g(x)

Proof:
 DP Processor:
  DPs:
   f#(1()) -> g#(1())
   f#(1()) -> f#(g(1()))
   g#(0()) -> f#(0())
   g#(0()) -> g#(f(0()))
  TRS:
   f(1()) -> f(g(1()))
   f(f(x)) -> f(x)
   g(0()) -> g(f(0()))
   g(g(x)) -> g(x)
  TDG Processor:
   DPs:
    f#(1()) -> g#(1())
    f#(1()) -> f#(g(1()))
    g#(0()) -> f#(0())
    g#(0()) -> g#(f(0()))
   TRS:
    f(1()) -> f(g(1()))
    f(f(x)) -> f(x)
    g(0()) -> g(f(0()))
    g(g(x)) -> g(x)
   graph:
    g#(0()) -> g#(f(0())) -> g#(0()) -> g#(f(0()))
    g#(0()) -> g#(f(0())) -> g#(0()) -> f#(0())
    g#(0()) -> f#(0()) -> f#(1()) -> f#(g(1()))
    g#(0()) -> f#(0()) -> f#(1()) -> g#(1())
    f#(1()) -> g#(1()) -> g#(0()) -> g#(f(0()))
    f#(1()) -> g#(1()) -> g#(0()) -> f#(0())
    f#(1()) -> f#(g(1())) -> f#(1()) -> f#(g(1()))
    f#(1()) -> f#(g(1())) -> f#(1()) -> g#(1())
   SCC Processor:
    #sccs: 1
    #rules: 4
    #arcs: 8/16
    DPs:
     g#(0()) -> g#(f(0()))
     g#(0()) -> f#(0())
     f#(1()) -> g#(1())
     f#(1()) -> f#(g(1()))
    TRS:
     f(1()) -> f(g(1()))
     f(f(x)) -> f(x)
     g(0()) -> g(f(0()))
     g(g(x)) -> g(x)
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [g#](x0) = x0,
      
      [f#](x0) = x0,
      
      [0] = 0,
      
      [g](x0) = 0,
      
      [f](x0) = 0,
      
      [1] = 1
     orientation:
      g#(0()) = 0 >= 0 = g#(f(0()))
      
      g#(0()) = 0 >= 0 = f#(0())
      
      f#(1()) = 1 >= 1 = g#(1())
      
      f#(1()) = 1 >= 0 = f#(g(1()))
      
      f(1()) = 0 >= 0 = f(g(1()))
      
      f(f(x)) = 0 >= 0 = f(x)
      
      g(0()) = 0 >= 0 = g(f(0()))
      
      g(g(x)) = 0 >= 0 = g(x)
     problem:
      DPs:
       g#(0()) -> g#(f(0()))
       g#(0()) -> f#(0())
       f#(1()) -> g#(1())
      TRS:
       f(1()) -> f(g(1()))
       f(f(x)) -> f(x)
       g(0()) -> g(f(0()))
       g(g(x)) -> g(x)
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [g#](x0) = 0,
       
       [f#](x0) = x0,
       
       [0] = 0,
       
       [g](x0) = 0,
       
       [f](x0) = 0,
       
       [1] = 1
      orientation:
       g#(0()) = 0 >= 0 = g#(f(0()))
       
       g#(0()) = 0 >= 0 = f#(0())
       
       f#(1()) = 1 >= 0 = g#(1())
       
       f(1()) = 0 >= 0 = f(g(1()))
       
       f(f(x)) = 0 >= 0 = f(x)
       
       g(0()) = 0 >= 0 = g(f(0()))
       
       g(g(x)) = 0 >= 0 = g(x)
      problem:
       DPs:
        g#(0()) -> g#(f(0()))
        g#(0()) -> f#(0())
       TRS:
        f(1()) -> f(g(1()))
        f(f(x)) -> f(x)
        g(0()) -> g(f(0()))
        g(g(x)) -> g(x)
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [g#](x0) = 1,
        
        [f#](x0) = 0,
        
        [0] = 0,
        
        [g](x0) = 0,
        
        [f](x0) = 0,
        
        [1] = 0
       orientation:
        g#(0()) = 1 >= 1 = g#(f(0()))
        
        g#(0()) = 1 >= 0 = f#(0())
        
        f(1()) = 0 >= 0 = f(g(1()))
        
        f(f(x)) = 0 >= 0 = f(x)
        
        g(0()) = 0 >= 0 = g(f(0()))
        
        g(g(x)) = 0 >= 0 = g(x)
       problem:
        DPs:
         g#(0()) -> g#(f(0()))
        TRS:
         f(1()) -> f(g(1()))
         f(f(x)) -> f(x)
         g(0()) -> g(f(0()))
         g(g(x)) -> g(x)
       Matrix Interpretation Processor:
        dimension: 1
        interpretation:
         [g#](x0) = x0,
         
         [0] = 1,
         
         [g](x0) = 0,
         
         [f](x0) = 0,
         
         [1] = 0
        orientation:
         g#(0()) = 1 >= 0 = g#(f(0()))
         
         f(1()) = 0 >= 0 = f(g(1()))
         
         f(f(x)) = 0 >= 0 = f(x)
         
         g(0()) = 0 >= 0 = g(f(0()))
         
         g(g(x)) = 0 >= 0 = g(x)
        problem:
         DPs:
          
         TRS:
          f(1()) -> f(g(1()))
          f(f(x)) -> f(x)
          g(0()) -> g(f(0()))
          g(g(x)) -> g(x)
        Qed