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

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.