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

Lemma decreasing_be_seq : forall (n L U : nat) (f : nat -> bool_expr), cardinal _ (bool_expr_to_var_env'' L U (f 0)) n -> (forall k : nat, be_le (f (S k)) (f k)) -> (forall k : nat, be_ok (var_lu L U) (f k)) -> exists m : nat, m <= n /\ be_le (f m) (f (S m)).