Proof with eauto with renamings id_renamings val_renamings.
intros.
constructor; constructor;
eauto with id_renamings; unfold id_ren; intros;
repeat match goal with
| [H: context[valid_block_dec ?M ?B] |- _] =>
destruct (valid_block_dec M B); simpl in *
| [H: _ = Some _ |- _] => inv H; clear H
end; auto.
now apply po_refl.
destruct (ZMap.get ofs (Mem.mem_contents m) # b2) eqn:Hget;
constructor.
destruct v0; constructor.
specialize (H _ v _ _ Hget).
simpl in H.
destruct (valid_block_dec m b); simpl; tauto.
Qed.

Lemma pigeon_positive:
forall (n m: positive) (f: positive -> option positive),
(forall i, (i < n)%positive ->
exists j, (j < m)%positive /\ f i = Some j) ->
(forall i i' j j',
f i = Some j -> f i' = Some j' ->
i<>i' -> j<>j') ->
(n <= m)%positive.