NO

Problem:
 g(a()) -> f(g(a()))
 g(b()) -> c(a())
 a() -> b()
 f(x) -> h(x)
 h(x) -> c(b())

Proof:
 Containment Processor: loop length: 1
                        terms:
                         g(a())
                        context: f([])
                        substitution:
                         
  Qed