Definition var_env'_to_env'' (L U : nat) (ve : var_env') :=
  match var_env'_to_var_env''_lemma2 (U - L) L U ve (refl_equal _) with
  | exist ve'' _ => ve''
  end.

Lemma be_le1_le : forall (L U : nat) (be1 be2 : bool_expr), be_ok (var_lu L U) be1 -> be_ok (var_lu L U) be2 -> be_le1 L U be1 be2 -> be_le be1 be2.