Definition for_all (f : elt -> bool) (s : t) : bool :=
if for_all (P:=fun x => f x = true) (f_dec f) s
then true
else false.

(* Specifications: *)
Lemma for_all_1 :
forall (s : t) (f : elt -> bool),
compat_bool E.eq f ->
For_all (fun x => f x = true) s -> for_all f s = true.
Lemma for_all_2 :
forall (s : t) (f : elt -> bool),
compat_bool E.eq f ->
for_all f s = true -> For_all (fun x => f x = true) s.