Inductive larger_bound : positive -> positive -> Type :=
| larger_1  : forall b, larger_bound xH b
| larger_xI : forall x b, larger_bound x b -> larger_bound (xI x) (xO b)
| larger_xO : forall x b, larger_bound x b -> larger_bound (xO x) (xO b).

Lemma larger_bound_xO p b:
larger_bound p b -> larger_bound p b~0.