Section MapSubsetDef.
Variables A B : Type.
Definition MapSubset (m:Map A) (m':Map B) :=
forall a:ad, in_dom A a m = true -> in_dom B a m' = true.
Definition MapSubset_1 (m:Map A) (m':Map B) :=
match MapSweep A (fun (a:ad) (_:A) => negb (in_dom B a m')) m with
| None => true
| _ => false
end.
Definition MapSubset_2 (m:Map A) (m':Map B) :=
eqmap A (MapDomRestrBy A B m m') (M0 A).

Lemma map_dom_empty_1 :
forall m:Map A, eqmap A m (M0 A) -> forall a:ad, in_dom _ a m = false.