End Induction_Principles.
Section Fold_properties.
(** The following lemma has already been proved on Weak Maps, but with one additional hypothesis (some [transpose] fact). *)

Lemma fold_Add_Below : forall m1 m2 x e (A:Type)(eqA:A->A->Prop)(st:Equivalence eqA)
(f:key->elt->A->A)(i:A) (P:Proper (E.eq==>eq==>eqA==>eqA) f),
Below x m1 -> Add x e m1 m2 ->
eqA (fold f m2 i) (fold f m1 (f x e i)).