(** 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 := (compat_op E.eq Logic.eq).Notation transposeL := (transpose Logic.eq).

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