Definition var_env''le (ve1 ve2 : var_env'') :=
  forall x : ad, in_dom _ x ve1 = true -> in_dom _ x ve2 = true.

Lemma decreasing_ens_seq : forall (U : Type) (n : nat) (f : nat -> Ensemble U), cardinal _ (f 0) n -> (forall k : nat, Finite _ (f k)) -> (forall k : nat, Included _ (f (S k)) (f k)) -> exists m : nat, (exists c : nat, m <= n /\ cardinal _ (f m) c /\ cardinal _ (f (S m)) c).