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())
   EDG 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:
     
    Qed