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.