(** The following results have already been proved earlier, but we can now prove them with one hypothesis less: no need for [(transpose eqA f)]. *)
Section FoldOpt.
Variables (A:Type)(eqA:A->A->Prop)(st:Equivalence eqA).
Variables (f:elt->A->A)(Comp:compat_op E.eq eqA f).

Lemma add_fold : forall i s x, In x s ->
eqA (fold f (add x s) i) (fold f s i).