(** * [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).

(* Specifications: *)
Lemma submap_1 : forall m (Hm:NoDupA m) m' (Hm': NoDupA m') cmp, Submap cmp m m' -> submap cmp m m' = true.
Lemma submap_2 : forall m (Hm:NoDupA m) m' (Hm': NoDupA m') cmp, submap cmp m m' = true -> Submap cmp m m'.
Lemma equal_1 : forall m (Hm:NoDupA m) m' (Hm': NoDupA m') cmp, Equivb cmp m m' -> equal cmp m m' = true.
Lemma equal_2 : forall m (Hm:NoDupA m) m' (Hm':NoDupA m') cmp, equal cmp m m' = true -> Equivb cmp m m'.