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).

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)).