Corollary mem_obs_eq_domain_ren:
    forall f m m',
      mem_obs_eq f m m' ->
      domain_memren f m.

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.