Qed.
Notation compatb := (Proper (E.eq==>Logic.eq)) (only parsing).

Lemma filter_aux_lemma:
forall f s n l lo hi,
searchtree lo hi s -> filter_aux f s 1%positive nil = (n,l) ->
HdRel' K.lt lo l /\ Sorted K.lt l /\ S (length l) = nat_of_P n /\
(compatb f -> forall x, interp s x /\ f x = true <-> InA K.eq x l).