NO

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

Proof:
 Containment Processor: loop length: 1
                        terms:
                         c()
                        context: h([],g(c()))
                        substitution:
                         
  Qed