Definition map_leq {A B} (m1: PTree.t A)(m2: PTree.t B): Prop :=
  forall p, m1 ! p -> m2 ! p.

Lemma map_leq_apply: forall {A B} (m1: PTree.t A)(m2: PTree.t B) p f1, map_leq m1 m2 -> m1 ! p = Some f1 -> exists f2, m2 ! p = Some f2.