Definition from_elements A (l : list (key * A))
: t A (List.fold_left (fun acc xv => Dom.add (fst xv) acc) l Dom.empty) :=
exist (OK (List.fold_left (fun acc xv => Dom.add (fst xv) acc) l Dom.empty))
(List.fold_left (fun acc xv => S.add (fst xv) (snd xv) acc) l (@S.empty A))
(from_elements_OK l).
(** ***  Specifications  **)

Lemma map_spec : forall A B (f : A -> B) dom (m : t A dom) x v,
find x (map f m) = Some v <-> exists u, find x m = Some u ∧ f u = v.