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

Lemma alloc_perm_eq: forall f m m' lo hi m2 m2' b b' (Hobs_eq: mem_obs_eq f m m') (Halloc: Mem.alloc m lo hi = (m2, b)) (Halloc': Mem.alloc m' lo hi = (m2', b')) b1 b2 ofs (Hf: (if proj_sumbool (valid_block_dec m b1) then f b1 else if proj_sumbool (valid_block_dec m2 b1) then Some b' else None) = Some b2), permission_at m2 b1 ofs Cur = permission_at m2' b2 ofs Cur.