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

Lemma be_le_ens_inc : forall (be1 be2 : bool_expr) (L U : nat), be_le be1 be2 -> Included _ (bool_expr_to_var_env'' L U be1) (bool_expr_to_var_env'' L U be2).