NO

Problem:
 f(x,x) -> h(x,x)
 c() -> h(c(),g(c()))
 h(g3(x),g1(g2(x))) -> f(x,h(x,g(x)))

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