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 mapped_dec :
forall (f : positive -> option positive) m j
(Hdomain_invalid : forall b, ~ (b < m)%positive -> f b = None)
(Hdomain_valid : forall b, (b < m)%positive -> exists b', f b = Some b'),
(exists i, f i = Some j) \/ ~ exists i, f i = Some j.