Definition meminj_no_overlap (f: meminj) (m: mem) : Prop :=
  forall b1 b1' delta1 b2 b2' delta2 ofs1 ofs2,
  b1 <> b2 ->
  f b1 = Some (b1', delta1) ->
  f b2 = Some (b2', delta2) ->
  perm m b1 ofs1 Max Nonempty ->
  perm m b2 ofs2 Max Nonempty ->
  b1' <> b2' \/ ofs1 + delta1 <> ofs2 + delta2.

Lemma alloc_left_unmapped_inj: forall f m1 m2 lo hi m1' b1, mem_inj f m1 m2 -> alloc m1 lo hi = (m1', b1) -> f b1 = None -> mem_inj f m1' m2.