Definition be_le2 (be1 be2 : bool_expr) :=
  forall ve : var_env,
  bool_fun_of_bool_expr be1 ve = true -> bool_fun_of_bool_expr be2 ve = true.

Lemma be_le2_le : forall be1 be2 : bool_expr, be_le2 be1 be2 -> be_le be1 be2.