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)
  Usable Rule Processor:
   DPs:
    f#(1()) -> g#(1())
    f#(1()) -> f#(g(1()))
    g#(0()) -> f#(0())
    g#(0()) -> g#(f(0()))
   TRS:
    
   TDG Processor:
    DPs:
     f#(1()) -> g#(1())
     f#(1()) -> f#(g(1()))
     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())
     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())
    Restore Modifier:
     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)
     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