Definition activation_function (function_default : function) (p: cbv) :=
  match p with
    | cbv_update _ _ _ _ (fapply f _) _ _ => f
    | _                                   => function_default
  end.

Lemma FA_path_app_destruct p p' lp1 lp2 : FA_path p p' (lp1 ++ lp2) -> exists p'', FA_path p p'' lp1 /\ FA_path p'' p' lp2.