Fixpoint apply_hist S h :=
match h with
| [] => S
| Add e :: h' => if in_dec Z_eq_dec e S then apply_hist S h' else apply_hist (sort (e :: S)) h'
| Remove e :: h' => apply_hist (remove Z_eq_dec e S) h'
end.

(* Specifications: *)
Lemma apply_hist_app : forall h1 S h2, apply_hist S (h1 ++ h2) = apply_hist (apply_hist S h1) h2.
(* up *)
Lemma In_remove : forall A (a : A) dec l, incl (remove dec a l) l.
Lemma remove_sorted : forall a l, sorted l -> sorted (remove Z_eq_dec a l).
Lemma apply_hist_sorted : forall h S, sorted S -> sorted (apply_hist S h).
Definition hist := hist_part set_op.