Notation nil := (nil K V).Definition fnd k (s : fmap) :=
  if (filter (predk k) (seq_of s)) is (_, v):: _
  then Some v else None.

(* Specifications: *)
Lemma path_ins' s k1 k2 v : ord k1 k2 -> path ord k1 (map key s) -> path ord k1 (map key (ins' k2 v s)).
Lemma sorted_ins' s k v : sorted ord (map key s) -> sorted ord (map key (ins' k v s)).
Lemma sorted_filter k s : sorted ord (map key s) -> sorted ord (map key (filter (predCk k) s)).