Section PositiveInductionPrinciple.
Variable P : positive -> Prop.
Proof using.
induction q;
intros;
inversion H1;
auto;
apply H0;
apply IHq.
Qed.
Proof using.
intros.
apply (H p p).
constructor.
Qed.

Theorem strongind_pos :
P xH ->
(forall q, ((forall p, le_pos p q -> P p) -> P (xI q)) /\
((forall p, le_pos p q -> P p) -> P (xO q))) ->
forall p, P p.