Definition all_compatible (l : list (A -> option B)) := forall m1 m2, In m1 l -> In m2 l -> compatible m1 m2.

(* Specifications: *)
Lemma all_compatible_cons : forall (m : A -> option B) l, all_compatible (m :: l) -> compatible m (maps_add l) /\ all_compatible l.
Lemma maps_add_in : forall l m (k : A) (v : B) (Hcompat : all_compatible l), In m l -> m k = Some v -> maps_add l k = Some v.
Lemma fold_right_maps_add : forall l (e : A -> option B), fold_right map_add e l = map_add (maps_add l) e.