Hint Resolve in_map_fst2 : closed.
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).