(** Adding a valuation function on all elements of a set. *)

Definition sum (f:elt -> nat)(s:t) := fold (fun x => plus (f x)) s 0.Notation compat_opL := (Proper (E.eq==>Logic.eq==>Logic.eq)).Notation transposeL := (transpose Logic.eq).

Lemma sum_filter : forall f : elt -> bool, Proper (E.eq==>Logic.eq) f -> forall s, (sum (fun x => if f x then 1 else 0) s) = (cardinal (filter f s)).