Definition Evar_env' : Ensemble var_env' :=
fun ve : var_env' => forall n : nat, nat_lu n = false -> ve n = false.
Definition Evar_env'' : Ensemble var_env'' :=
fun ve : var_env'' =>
mapcanon _ ve /\ (forall x : ad, var_lu x = false -> in_dom _ x ve = false).

Lemma cardinal_Union :
forall (U : Type) (X : Ensemble U) (m : nat),
cardinal U X m ->
forall (Y : Ensemble U) (n : nat),
cardinal U Y n -> forall p : nat, cardinal U (Union U X Y) p -> p <= m + n.