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

Lemma increasing_be_seq_1 : forall (L U : nat) (f : nat -> bool_expr), (forall k : nat, be_ok (var_lu L U) (f k)) -> (forall k : nat, be_le (f k) (f (S k))) -> exists m : nat, m <= two_power (U - L) /\ be_eq (f m) (f (S m)).