YES

Problem:
 strict:
  f(g(f(x))) -> f(g(g(g(f(x)))))
 weak:
  g(x) -> g(g(x))

Proof:
 Bounds Processor:
  bound: 1
  enrichment: match-rt
  automaton:
   final states: {2,1}
   transitions:
    f0(2) -> 1*
    f0(1) -> 1*
    g0(2) -> 2*
    g0(1) -> 2*
    g1(22) -> 23*
    g1(28) -> 29*
    g1(36) -> 37*
    g1(6) -> 7*
    g1(27) -> 28*
    g1(7) -> 8*
    g1(5) -> 6*
    g1(20) -> 21*
    f1(18) -> 19*
    f1(4) -> 5*
    f1(8) -> 9*
    19 -> 5*
    7 -> 21*
    2 -> 36,4
    21 -> 6*
    8 -> 22*
    1 -> 27,18
    37 -> 29,28
    9 -> 5,20,1
    29 -> 28,2
    23 -> 8*
  problem:
   strict:
    
   weak:
    g(x) -> g(g(x))
  Qed