Ltac precondition_closed :=
 match goal with |- precondition_closed _ _ => idtac end;
 let x := fresh "x" in intro x;
 split;
  repeat match goal with
          | |- closed_wrt_vars _ (let (y,z) := ?x in _) => is_var x; destruct x
          | |- closed_wrt_lvars _ (let (y,z) := ?x in _) => is_var x; destruct x
          end;
  [simpl not_a_param; auto 50 with closed
  | simpl is_a_local; auto 50 with closed ].

Lemma Forall_map_nil: forall {A B} (F: A -> Prop) (g: B -> A), Forall F (map g nil).