End Bool'.
Section Sum.
(** 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).

(* Specifications: *)
Lemma sum_plus :
forall f g, Proper (E.eq==>Logic.eq) f -> Proper (E.eq==>Logic.eq) g ->
forall s, sum (fun x =>f x+g x) s = sum f s + sum g s.
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)).
Lemma fold_compat :
forall (A:Type)(eqA:A->A->Prop)(st:Equivalence eqA)
(f g:elt->A->A),
(compat_op E.eq eqA f) -> (transpose eqA f) ->
(compat_op E.eq eqA g) -> (transpose eqA g) ->
forall (i:A)(s:t),(forall x:elt, (In x s) -> forall y, (eqA (f x y) (g x y))) ->
(eqA (fold f s i) (fold g s i)).
Lemma sum_compat :
forall f g, Proper (E.eq==>Logic.eq) f -> Proper (E.eq==>Logic.eq) g ->
forall s, (forall x, In x s -> f x=g x) -> sum f s=sum g s.