(** * [equal] *)
Definition check (cmp : elt -> elt -> bool)(k:key)(e:elt)(m': t elt) :=
match find k m' with
| None => false
| Some e' => cmp e e'
end.
Definition submap (cmp : elt -> elt -> bool)(m m' : t elt) : bool :=
fold (fun k e b => andb (check cmp k e m') b) m true.
Definition equal (cmp : elt -> elt -> bool)(m m' : t elt) : bool :=
andb (submap cmp m m') (submap (fun e' e => cmp e e') m' m).
Definition Submap cmp m m' :=
(forall k, In k m -> In k m') /\
(forall k e e', MapsTo k e m -> MapsTo k e' m' -> cmp e e' = true).
Definition Equivb cmp m m' :=
(forall k, In k m <-> In k m') /\
(forall k e e', MapsTo k e m -> MapsTo k e' m' -> cmp e e' = true).

Lemma submap_1 : forall m (Hm:NoDupA m) m' (Hm': NoDupA m') cmp,
Submap cmp m m' -> submap cmp m m' = true.